en Artistic License 2.0 This is a core version which contains all but deprecated classes. Kinetic Simulation Algorithm Ontology (KiSAO) http://co.mbine.org/standards/kisao http://identifiers.org/pubmed/22027554 2.5 The Kinetic Simulation Algorithm Ontology (KiSAO) classifies algorithms available for the simulation and analysis of models in biology, and their characteristics and the parameters required for their use. has characteristic 'has characteristic' links algorithms to the characteristics, they possess. is hybrid of http://identifiers.org/doi/10.1093/bib/bbn050 The basic idea of hybrid simulation methods is to combine the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time. 'is hybrid of' is a relation between the complex simulation method and algorithms it applies to the different parts of the system. http://identifiers.org/doi/10.1093/bib/bbn050 Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Pahle J. Brief Bioinform. 2009 Jan;10(1):53-64. Epub 2009 Jan 16. is parameter of 'is parameter of' is an inverse of 'has parameter' and links parameters to the algorithms which use them. has parameter 'has parameter' links algorithms to the parameters they use. is characteristic of 'is characteristic of' is an inverse of 'has characteristic' and links characteristics to the algorithms which possess them. 2011-06-10 AZ is similar to A relationship indicating that two entities are similar to each other. 2011-06-10 AZ uses A relation between algorithms indicating that one algorithm uses another one (for example, if the algorithm switches between several algorithms). 2011-06-10 AZ is generalization of A relation between kinetic simulation algorithms, indicating that one is a generalization of another (e.g.if one extends another to suit systems with more general characteristics ). 2011-07-19 AZ is used by A relation between algorithms indicating that one algorithm is used by another one (for example, if the algorithm switches between several algorithms). 'is used by' is inverse of 'uses'. has type Indicates the type of algorithm parameter value, such as, for example, xsd:double for 'absolute tolerance'. has Runge-Kutta method order Indicates the order of a 'Runge-Kutta based method' [urn.miriam.biomodels.kisao:KISAO_0000064]. A 'Runge-Kutta based method' has order p if for sufficiently smooth initial value problems (y'=f(x,y), y(x0)=y0), ||y(x0 + h)-y1||<=Kh^(p+h). 2008-05-26 dk true modelling and simulation algorithm modeling and simulation algorithm Algorithm used to instantiate a simulation from a mathematical model. 24JAN2009 NLN weighted stochastic simulation algorithm http://identifiers.org/pubmed/19045316 weighted SSA The weighted Stochastic Simulation Algorithm manipulates the probabilities measure of biochemical systems by sampling, in order to increase the fraction of simulation runs exhibiting rare events. weighted SSA EXACT http://identifiers.org/pubmed/19045316 Kuwahara H, Mura I. (2008) An efficient and exact stochastic simulation method to analyse rare events in biochemical systems. J Chem Phys. 129(16):165101. 2007-11-09 NLN Cain Gillespie first reaction algorithm http://identifiers.org/doi/10.1016/0021-9991(76)90041-3 http://identifiers.org/doi/10.1021/j100540a008 Gillespie's first reaction method Stochastic simulation algorithm using the reaction probability density function (next-reaction density function), giving the probability that the next reaction will happen in a given time interval. To choose the next reaction to fire, the algorithm calculates a tentative reaction time for each reaction and then select the smallest. http://identifiers.org/doi/10.1016/0021-9991(76)90041-3 Gillespie DT. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics, Volume 2 , pages 403-434 (1976). http://identifiers.org/doi/10.1021/j100540a008 Gillespie DT. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, Vol. 81, No. 25. (1977), pp. 2340-2361. Gillespie's first reaction method EXACT StochSim multi-state agent-based simulation method http://identifiers.org/pubmed/9628844 Morton-Firth The agent-based simulation method instantiates each molecule as an individual software object. The interactions between those objects are determined by interaction probabilities according to experimental data. The probability is depended on the state the molecule is in at that specific time (molecules have multiple-state). Additionally, ''pseudo-molecules'' are introduced to the system in order to simulate unimolecular reactions. For simulation, continuous time is broken down into discrete, independent ''slices''. During each time slice one molecule is selected randomly, a second molecule or pseudo-molecule is selected afterwards (leading to either a unimolecular or a bimolecular reaction). The reaction will only take place if a produced random number exceeds the reaction probability calculated beforehand. In that case, the system is updated after that reaction. http://identifiers.org/pubmed/9628844 Morton-Firth, C.J, Bray, D. Predicting temporal fluctuations in an intracellular signalling pathway. Journal of Theoretical Biology, volume 192, pages 117-128 (1998). Morton-Firth EXACT 1 1 1 1 1 1 1 1 1 1 2007-11-30 dk BioNetGen JSim OpenCOR SBML-SAT SUNDIALS VCell CVODE citeulike:1832863 http://identifiers.org/doi/10.1145/1089014.1089020 VODE VODEPK code value ordinary differential equation solver code value ordinary differential equation solver The CVODE is a package written in C that solves ODE initial value problems, in real N-space, written as y'=f(t,y), y(t0)=y0. It is capable for stiff and non-stiff systems and uses two different linear multi-step methods, namely the Adam-Moulton [http://identifiers.org/biomodels.kisao/KISAO_0000280] method and the backward differentiation formula [http://identifiers.org/biomodels.kisao/KISAO_0000288]. citeulike:1832863 Cohen S, Hindmarsh C. Cvode, A Stiff/nonstiff Ode Solver In C. Computers in Physics, Vol. 10 (2), pages 138-143 (1996). http://identifiers.org/doi/10.1145/1089014.1089020 Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005. VODE RELATED VODEPK RELATED code value ordinary differential equation solver EXACT SUNDIALS PVODE http://identifiers.org/doi/10.1145/1089014.1089020 http://identifiers.org/doi/10.1177/109434209901300405 parallel code value ordinary differential equation solver PVODE is a general-purpose solver for ordinary differential equation (ODE) systems that implements methods for both stiff and nonstiff systems. [...] In the stiff case, PVODE uses a backward differentiation formula method [http://identifiers.org/biomodels.kisao/KISAO_0000288] combined with preconditioned GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000253] iteration. Parallelism is achieved by distributing the ODE solution vector into user-specified segments and parallelizing a set of vector kernels accordingly. For PDE-based ODE systems, we provide a module that generates a band block-diagonal preconditioner for use with the GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000253] iteration. PVODE is based on CVODE [http://identifiers.org/biomodels.kisao/KISAO_0000019]. http://identifiers.org/doi/10.1145/1089014.1089020 Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005. http://identifiers.org/doi/10.1177/109434209901300405 Byrne GD, Hindmarsh AC. PVODE, an ODE Solver for Parallel Computers. International Journal of High Performance Computing Applications, Vol. 13 (4), pages 354-365 (1999). parallel code value ordinary differential equation solver EXACT Stochsim 1.2 and more recent versions StochSim nearest-neighbour algorithm http://identifiers.org/pubmed/11395441 The nearest-neighbour algorithm allows for the representation of spatial information, by adding a two-dimensional lattice in the form of a probabilistic cellular automata. That way, nearest neighbour interactions do additionally influence reactions taking place in the systems. Reactions between entities are calculated using the agent-based simulation algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000017]. http://identifiers.org/pubmed/11395441 Le Novere N, Shimizu TS. STOCHSIM: modelling of stochastic biomolecular processes. Bioinformatics, volume 17 (6), pages 575-576 (2001). MesoRD SmartCell Elf and Ehrenberg method http://identifiers.org/doi/10.1049/sb:20045021 Elf algorithm NSM next-subvolume method Sub-volume stochastic reaction-diffusion method that is a combination of the Direct Method [http://identifiers.org/biomodels.kisao/KISAO_0000029] for sampling the time for a next reaction or diffusion event in each subvolume, with Gibson and Bruck's Next Reaction Method [http://identifiers.org/biomodels.kisao/KISAO_0000027], which is used to keep track of in which subvolume an event occurs next. The subvolumes are kept sorted in a queue, implemented as a binary tree, according to increasing time of the next event. When an event has occurred in the subvolume at the top of the queue, new event times need to be sampled only for one (the event is a chemical reaction) or two (the event is a diffusion jump) subvolume(s). http://identifiers.org/doi/10.1049/sb:20045021 Elf J, Ehrenberg M. Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. Systems Biology, IEE Proceedings, volume 1 (2), pages 230-236 (2004). Elf algorithm EXACT NSM EXACT next-subvolume method EXACT 2007-11-10 dk Cain E-Cell SmartCell VCell Gibson-Bruck next reaction algorithm http://identifiers.org/doi/10.1021/jp993732q Gibson and Bruck algorithm Gibson-Bruck's next reaction algorithm Gillespie-Gibson stochastic simulation algorithm SSA-GB next reaction method As with the first reaction method [http://identifiers.org/biomodels.kisao/KISAO_0000015], a putative reaction time is calculated for each reaction, and the reaction with the shortest reaction time will be realized. However, the unused calculated reaction times are not wasted. The set of reactions is organized in a priority queue to allow for the efficient search for the fastest reaction. In addition, by using a so-called dependency graph only those reaction times are recalculated in each step, that are dependent on the reaction, which has been realized. http://identifiers.org/doi/10.1021/jp993732q Gibson MA, Bruck J. Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. Journal of Physical Chemistry A, Vol. 104, pages 1876-1889 (2000). Gibson and Bruck algorithm EXACT Gibson-Bruck's next reaction algorithm EXACT Gillespie-Gibson stochastic simulation algorithm EXACT SSA-GB EXACT next reaction method EXACT slow-scale stochastic simulation algorithm http://identifiers.org/pubmed/15638651 slow-scale stochastic SSA ssSSA Attempt to overcome the problem of stiff systems by developing an ''approximate theory that allows one to stochastically advance the system in time by simulating the firings of only the slow reaction events''. http://identifiers.org/pubmed/15638651 Cao Y, Gillespie DT, Petzold LR. The slow-scale stochastic simulation algorithm. Journal of Chemical Physics, Vol. 122, No. 1. (1 January 2005). slow-scale stochastic SSA EXACT ssSSA EXACT 2007-11-10 dk BetaWB BioNetGen ByoDyn Cain iBioSim Gillespie direct algorithm http://identifiers.org/doi/10.1016/0021-9991(76)90041-3 http://identifiers.org/doi/10.1021/j100540a008 DM Doob-Gillespie method Gillespie's direct method SSA stochastic simulation algorithm Stochastic simulation algorithm using the reaction probability density function (next-reaction density function), giving the probability that the next reaction will happen in a given time interval. To choose the next reaction to fire, the algorithm directly and separately calculates the identity of the reaction and the time it will fire. http://identifiers.org/doi/10.1016/0021-9991(76)90041-3 Gillespie DT. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics, Volume 2 , pages 403-434 (1976). http://identifiers.org/doi/10.1021/j100540a008 Gillespie DT. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, Vol. 81, No. 25. (1977), pp. 2340-2361. DM EXACT Doob-Gillespie method EXACT Gillespie's direct method EXACT SSA EXACT stochastic simulation algorithm EXACT 2007-11-10 dk VCell iBioSim Euler forward method http://identifiers.org/isbn/052143064X explicit Euler method explicit Gaussian first order Runge-Kutta The Euler method is an explicit one-step method for the numerical integration of ODES with a given initial value. The calculation of the next integration step at time t+1 is based on the state of the system at time point t. http://identifiers.org/isbn/052143064X Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes. Cambridge University Press, New York, 2nd edition (1992). explicit Euler method EXACT explicit Gaussian first order Runge-Kutta EXACT 2007-11-02 dk GSL Euler backward method http://identifiers.org/isbn/052143064X implicit Euler method implicit Gaussian first order Runge-Kutta The Euler backward method is an implicit one-step method for the numerical integration of ODES with a given initial value. The next state of a system is calculated by solving an equation that considers both, the current state of the system and the later one. http://identifiers.org/isbn/052143064X Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001). implicit Euler method EXACT implicit Gaussian first order Runge-Kutta EXACT 4 2007-11-12 dk GSL JSim VCell explicit fourth-order Runge-Kutta method http://identifiers.org/isbn/0-471-91046-5 ERK4 RK4 Runge-Kutta method The Runge-Kutta method is a method for the numerical integration of ODES with a given initial value. The calculation of the next integration step at time t+1 is based on the state of the system at time point t, plus the product of the size of the interval and an estimated slope. The slope is a weighted average of 4 single slope points (beginning of interval-midpoint-midpoint-end of interval). http://identifiers.org/isbn/0-471-91046-5 Butcher JC. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987). ERK4 EXACT RK4 EXACT Runge-Kutta method EXACT 2007-11-12 dk E-Cell Rosenbrock method http://identifiers.org/doi/10.1093/comjnl/5.4.329 http://identifiers.org/isbn/052143064X Kaps-Rentrop method generalized fourth order Runge-Kutta method Some general implicit processes are given for the solution of simultaneous first-order differential equations. These processes, which use successive substitution, are implicit analogues of the (explicit) Runge-Kutta processes. They require the solution in each time step of one or more set of simultaneous linear equations, usually of a special and simple form. Processes of any required order can be devised, and they can be made to have a wide margin of stability when applied to a linear problem. http://identifiers.org/doi/10.1093/comjnl/5.4.329 Rosenbrock HH. Some general implicit processes for the numerical solution of differential equations. The Computer Journal, pages 329-330 (1963). http://identifiers.org/isbn/052143064X Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes. Cambridge University Press, New York, 2nd edition, pages 742-746 (1992). Kaps-Rentrop method EXACT generalized fourth order Runge-Kutta method EXACT sorting stochastic simulation algorithm http://identifiers.org/doi/10.1016/j.compbiolchem.2005.10.007 sorting SSA sorting direct method In order to overcome the problem of high complexity of the Stochastic Simulation Algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000029] when simulating large systems, the sorting direct method maintains a loosely sorted order of the reactions as the simulation executes. http://identifiers.org/doi/10.1016/j.compbiolchem.2005.10.007 McCollum J M, Peterson G D, Cox C D, Simpson M L, Samatova N F. The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behaviour. Computational Biology and Chemistry, Volume 30 (1), pages 39-49 (2006). sorting SSA EXACT sorting direct method EXACT 1 2008-07-08 NLN ByoDyn Cain SmartCell tau-leaping method http://identifiers.org/doi/10.1063/1.1378322 tauL Approximate acceleration procedure of the Stochastic Simulation Algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000029] that divides the time into subintervals and ''leaps'' from one to another, firing all the reaction events in each subinterval. http://identifiers.org/doi/10.1063/1.1378322 Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section V. tauL EXACT ByoDyn Poisson tau-leaping method http://identifiers.org/doi/10.1063/1.1378322 explicit tau-leaping explicit tau-leaping method with basic pre-leap check explicit tau-leaping method with basic preleap check poisson tau-leaping Explicit tau-leaping method with basic pre leap check. http://identifiers.org/doi/10.1063/1.1378322 Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section V. explicit tau-leaping EXACT explicit tau-leaping method with basic preleap check EXACT poisson tau-leaping EXACT 1 2007-10-12 dk implicit tau-leaping method http://identifiers.org/doi/10.1063/1.1627296 Contrary to the explicit tau-leaping [http://identifiers.org/biomodels.kisao/KISAO_0000039 and http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000239] , the implicit tau-leaping allows for much larger time-steps when simulating stiff systems. http://identifiers.org/doi/10.1063/1.1627296 Rathinam M, Petzold L R, Cao Y, Gillespie D T. Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. The Journal of Chemical Physics, Volume 119 (24), pages 12784-12794 (2003). 2007-10-16 dk trapezoidal tau-leaping method citeulike:1755561 trapezoidal implicit tau-leaping method trapezoidal implicit tau-leaping method Formula for accelerated discrete efficient stochastic simulation of chemically reacting system [which] has better accuracy and stiff stability properties than the explicit and implicit [http://identifiers.org/biomodels.kisao/KISAO_0000045] tau-leaping formulas for discrete stochastic systems, and it limits to the trapezoidal rule in the deterministic regime. citeulike:1755561 Cao Y, Petzold L. Trapezoidal tau-leaping formula for the stochastic simulation of biochemical systems. Foundations of Systems Biology in Engineering (FOSBE), pages 149-152 (2005). trapezoidal implicit tau-leaping method EXACT adaptive explicit-implicit tau-leaping method http://identifiers.org/doi/10.1063/1.2745299 Modification of the original tau-selection strategy [http://identifiers.org/biomodels.kisao/KISAO_0000040], designed for explicit tau-leaping, is modified to apply to implicit tau-leaping, allowing for longer steps when the system is stiff. Further, an adaptive strategy is proposed that identifies stiffness and automatically chooses between the explicit and the (new) implicit tau-selection methods to achieve better efficiency. http://identifiers.org/doi/10.1063/1.2745299 Cao Y, Gillespie DT, Petzold LR. Adaptive explicit-implicit tau-leaping method with automatic tau selection. The Journal of Chemical Physics, Vol. 126 (22) (2007). Bortz-Kalos-Lebowitz algorithm http://identifiers.org/doi/10.1016/0021-9991(75)90060-1 BKL DMC KMC dynamic Monte Carlo dynamic Monte Carlo method kinetic Monte Carlo kinetic Monte Carlo method n-fold way The Bortz-Kalos-Lebowitz (or: kinetic Monte-Carlo-) method is a stochastic method for the simulation of time evolution of processes using (pseudo-)random numbers. http://identifiers.org/doi/10.1016/0021-9991(75)90060-1 Bortz AB, Kalos MH, Lebowitz JL. A new algorithm for Monte Carlo simulation of Ising spin systems. Journal of Computational Physics, Vol. 17 (1), pages 10-18 (1975). BKL EXACT DMC EXACT KMC EXACT dynamic Monte Carlo EXACT dynamic Monte Carlo method EXACT kinetic Monte Carlo EXACT kinetic Monte Carlo method EXACT n-fold way EXACT 2007-10-29 dk true Smoluchowski equation based method urn:issn:0942-9352 Method based on the Smoluchowski equation. urn:issn:0942-9352 Smoluchowski M. Mathematical theory of the kinetics of the coagulation of colloidal solutions. Z. Phys. Chem, Vol. 92, No. 129. (1917). 1 1 1 1 Smoldyn Brownian diffusion Smoluchowski method http://identifiers.org/pubmed/16204833 In the Brownian diffusion Smoluchowski method, ''each molecule is treated as a point-like particle that diffuses freely in three-dimensional space. When a pair of reactive molecules collide, such as an enzyme and its substrate, a reaction occurs and the simulated reactants are replaced by products. [..] Analytic solutions are presented for some simulation parameters while others are calculated using look-up tables''. Supported chemical processes include molecular diffusion, treatment of surfaces, zeroth-order-, unimolecular-, and bimolecular reactions. http://identifiers.org/pubmed/16204833 Andrews SS, Bray D. Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Phys Biol, volume 1 (3-4), pages 137-151 (December 2004). Greens function reaction dynamics http://identifiers.org/doi/10.1063/1.2137716 GFRD Green's function reaction dynamics Method that simulates biochemical networks on particle level. It considers both changes in time and space by ''exploiting both the exact solution of the Smoluchowski Equation to set up an event-driven algorithm'' which allows for large jumps in time when the considered particles are far away from each other [in space] and thus cannot react. GFRD combines the propagation of particles in space with the reactions taking place between them in one simulation step. http://identifiers.org/doi/10.1063/1.2137716 Van Zon JS, Ten Wolde PR. Green's-function reaction dynamics: A particle-based approach for simulating biochemical networks in time and space. Journal of Chemical Physics, Volume 123 (23) (2005). GFRD EXACT Green's function reaction dynamics EXACT 2007-11-12 dk ByoDyn true Runge-Kutta based method http://identifiers.org/isbn/0-471-91046-5 modified Euler method A method of numerically integrating ordinary differential equations, which uses a sampling of slopes through an interval and takes a weighted average to determine the right end point. This averaging gives a very accurate approximation. http://identifiers.org/isbn/0-471-91046-5 Butcher JC. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987). modified Euler method RELATED 2007-11-30 dk deterministic cellular automata update algorithm http://identifiers.org/isbn/978-0252000232 A cellular automaton is a discrete model of a regular grid of cells with a finite number of dimensions. Each cell has a finite number of defined states. The automaton changes its state in a discrete manner, meaning that the state of a cell at time t is determined by a function of the states of its neighbours at time t - 1. These neighbours are a selection of cells relative to the specified cell. Famous examples for deterministic cellular automata are Conway's game of life or Wolfram's elementary cellular automata. http://identifiers.org/isbn/978-0252000232 Burks, Arthur W (Editor). Essays on Cellular Automata. University of Illinois Press (1970). 2007-11-16 dk LSODE citeulike:1774586 http://identifiers.org/doi/10.1145/1218052.1218054 Livermore solver for ordinary differential equations LSODE solves stiff and nonstiff systems of the form dy/dt = f. In the stiff case, it treats the Jacobian matrix sf/dy as either a dense (full) or a banded matrix, and as either user-supplied or internally approximated by difference quotients. It uses Adams methods (predictor-corrector) [http://identifiers.org/biomodels.kisao/KISAO_0000364] in the nonstiff case, and Backward Differentiation Formula (BDF) methods (the Gear methods) [http://identifiers.org/biomodels.kisao/KISAO_0000288] in the stiff case. citeulike:1774586 Radhakrishnan K, Hindmarsh AC. Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations. Lawrence Livermore National Laboratory Report, Vol. UCRL-ID-113855 (1993). http://identifiers.org/doi/10.1145/1218052.1218054 Hindmarsh AC. LSODE and LSODI, two new initial value ordinary differential equation solvers. SIGNUM Newsletter, Volume 15 (4), pages 10-11 (1980). Livermore solver for ordinary differential equations EXACT 1 2007-10-16 dk binomial tau-leaping method http://identifiers.org/doi/10.1063/1.2771548 BtauL binomial tau-leap spatial stochastic simulation algorithm binomial tau-leap spatial stochastic simulation algorithm Coarse grained modified version of the next subvolume method [http://identifiers.org/biomodels.kisao/KISAO_0000022] that allows the user to consider both diffusion and reaction events in relatively long simulation time spans as compared with the original method and other commonly used fully stochastic computational methods. http://identifiers.org/doi/10.1063/1.2771548 Marquez-Lago T., Burrage K. Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics. The Journal of Chemical Physics, Vol. 127 (10) (2007). BtauL EXACT binomial tau-leap spatial stochastic simulation algorithm EXACT Gillespie multi-particle method http://identifiers.org/pubmed/16731694 GMP Gillespie's multi-particle method particle-based spatial stochastic method Combination of the multiparticle method for diffusion [http://identifiers.org/biomodels.kisao/KISAO_0000334] and the SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029]. http://identifiers.org/pubmed/16731694 Rodriguez VJ, Kaandorp JA, Dobrzynski M, Blom JG. Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli. Bioinformatics, Vol. 22 (15), pages 1895-1901 (2006). GMP EXACT Gillespie's multi-particle method EXACT particle-based spatial stochastic method EXACT Stundzia and Lumsden method http://identifiers.org/doi/10.1006/jcph.1996.0168 RD SSA reaction-diffusion stochastic simulation algorithm Sub-volume stochastic reaction-diffusion method that using Green's function to link the bulk diffusion coefficient D in Fick's differential law to the corresponding transition rate probability for diffusion of a particle between finite volume elements. This generalized stochastic algorithm enables to numerically calculate the time evolution of a spatially inhomogeneous mixture of reaction-diffusion species in a finite volume. The time step is stochastic and is generated by a probability distribution determined by the intrinsic reaction kinetics and diffusion dynamics. http://identifiers.org/doi/10.1006/jcph.1996.0168 Stundzia AB, Lumsden CJ. Stochastic simulation of coupled reaction-diffusion processes. J Comput Phys, Vol. 127 (1), pages 196-207 (1996). RD SSA EXACT reaction-diffusion stochastic simulation algorithm EXACT estimated midpoint tau-leaping method http://identifiers.org/doi/10.1063/1.1378322 explicit tau-leaping method with estimated-mid point technique Estimated-Midpoint tau-Leap Method: For the selected leaping time tau which satisfies the Leap Condition, compute the expected state change lambda' = tau sumj( aj(x)vj ) during [t, t + tau). Then, with x' =x + [lambda'/2], generate for each j = 1,...,M a sample value kj of the Poisson random variable P(aj(x'), tau). Compute the actual state change, lambda = sumj( kjvj ), and effect the leap by replacing t by t + tau and x by x + lambda. http://identifiers.org/doi/10.1063/1.1378322 Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section VI. explicit tau-leaping method with estimated-mid point technique EXACT k-alpha leaping method http://identifiers.org/doi/10.1063/1.1378322 Alternative to the tau-leaping [http://identifiers.org/biomodels.kisao/KISAO_0000039], where one leaps a fixed number of reaction-events. http://identifiers.org/doi/10.1063/1.1378322 Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section VIII. 1 nonnegative Poisson tau-leaping method http://identifiers.org/doi/10.1063/1.1992473 modified poisson tau-leaping The explicit tau-leaping procedure attempts to speed up the stochastic simulation of a chemically reacting system by approximating the number of firings of each reaction channel during a chosen time increment Tau as a Poisson random variable. Since the Poisson random variable can have arbitrarily large sample values, there is always the possibility that this procedure will cause one or more reaction channels to fire so many times during Tau that the population of some reactant species will be driven negative. Two recent papers have shown how that unacceptable occurrence can be avoided by replacing the Poisson random variables with binomial random variables, whose values are naturally bounded. This paper describes a modified Poisson tau-leaping procedure that also avoids negative populations, but is easier to implement than the binomial procedure. The new Poisson procedure also introduces a second control parameter, whose value essentially dials the procedure from the original Poisson tau-leaping at one extreme to the exact stochastic simulation algorithm at the other; therefore, the modified Poisson procedure will generally be more accurate than the original Poisson procedure [http://identifiers.org/biomodels.kisao/KISAO_0000040]. http://identifiers.org/doi/10.1063/1.1992473 Cao Y, Gillespie DT, Petzold LR. Avoiding negative populations in explicit Poisson tau-leaping. Journal of Chemical Physics, Vol. 123, 4104 (2005). E-Cell GSL JSim VCell iBioSim Fehlberg method http://identifiers.org/doi/10.1007/BF02241732 http://identifiers.org/pubmed/14990450 RKF45 Runge-Kutta-Fehlberg method The method was developed by the German mathematician Erwin Fehlberg and is based on the class of Runge-Kutta methods. The Runge-Kutta-Fehlberg method uses an O(h4) method together with an O(h5) method that uses all of the points of the O(h4) method, and hence is often referred to as an RKF45 method. Similar schemes with different orders have since been developed. By performing one extra calculation that would be required for an RK5 method, the error in the solution can be estimated and controlled and an appropriate step size can be determined automatically, making this method efficient for ordinary problems of automated numerical integration of ordinary differential equations. http://identifiers.org/doi/10.1007/BF02241732 Erwin Fehlberg (1970). Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme, Computing (Arch. Elektron. Rechnen), vol. 6, pp. 61-71. http://identifiers.org/pubmed/14990450 Takahashi K, Kaizu K, Hu B, Tomita M. A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics volume 20(4), pages 538-46 (2004). RKF45 EXACT Runge-Kutta-Fehlberg method EXACT 2007-11-12 dk ECell3 GSL JSim Matlab iBioSim Dormand-Prince method http://identifiers.org/doi/10.1016/0771-050X(80)90013-3 http://identifiers.org/pubmed/14990450 DOPRI Prince-Dormand method Dormand-Prince is an explicit method for the numerical integration of ODES with a given initial value. It is an 'embedded Runge-Kutta method' [http://identifiers.org/biomodels.kisao/KISAO_0000302] RK5 (4) which (a) has a 'small' principal truncation term in the Fifth order and (b) has an extended region of absolute stability. http://identifiers.org/doi/10.1016/0771-050X(80)90013-3 Dormand J, Prince P. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, Volume 6, pages 19-26 (1980). http://identifiers.org/pubmed/14990450 Takahashi K, Kaizu K, Hu B, Tomita M. A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics volume 20(4), pages 538-46 (2004). Prince-Dormand method EXACT 1 1 1 1 2007-11-30 dk LSODA http://identifiers.org/doi/10.1137/0904010 http://identifiers.org/isbn/978-0444866073 http://www.nea.fr/abs/html/uscd1227.html Livermore solver for ordinary differential equations with automatic method switching LSODA solves systems dy/dt = f with a dense or banded Jacobian when the problem is stiff, but it automatically selects between non-stiff (Adams [http://identifiers.org/biomodels.kisao/KISAO_0000289]) and stiff (BDF [http://identifiers.org/biomodels.kisao/KISAO_0000288]) methods. It uses the non-stiff method initially, and dynamically monitors data in order to decide which method to use. http://identifiers.org/doi/10.1137/0904010 Petzold LR. Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. SIAM Journal on Scientific and Statistical Computing, Vol. 4 (1), pages 136-148 (1983). http://identifiers.org/isbn/978-0444866073 A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64. http://www.nea.fr/abs/html/uscd1227.html Petzold LR, Hindmarsh, AC. LSODAR: Livermore solver of ordinary differential equations with automatic method switching and root finding, Computing and Mathematics Research Division, 1-316 Lawerence Livermore National Laboratory (1987). Livermore solver for ordinary differential equations with automatic method switching EXACT 2007-10-27 dk LSODAR http://identifiers.org/isbn/978-0444866073 http://www.nea.fr/abs/html/uscd1228.html Livermore solver for ordinary differential equations with automatic method switching and root finding ordinary differential equation solver for stiff or non-stiff systems with root finding LSODAR is a variant of LSODA [http://identifiers.org/biomodels.kisao/KISAO_0000088] with a root finding capability added. Thus it solves problems dy/dt = f with dense or banded Jacobian and automatic method selection, and at the same time, it finds the roots of any of a set of given functions of the form g(t,y). This is often useful for finding stop conditions, or for finding points at which a switch is to be made in the function f. http://identifiers.org/isbn/978-0444866073 A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64. http://www.nea.fr/abs/html/uscd1228.html Petzold LR, Hindmarsh, AC. LSODAR: Livermore solver of ordinary differential equations with automatic method switching and root finding, Computing and Mathematics Research Division, 1-316 Lawerence Livermore National Laboratory (1987). Livermore solver for ordinary differential equations with automatic method switching and root finding EXACT ordinary differential equation solver for stiff or non-stiff systems with root finding EXACT LSODI http://identifiers.org/doi/10.1145/1218052.1218054 http://identifiers.org/isbn/978-0444866073 Livermore solver for ordinary differential equations, implicit version LSODI solves systems given in linearly implicit form, including differential-algebraic systems. http://identifiers.org/doi/10.1145/1218052.1218054 Hindmarsh AC. LSODE and LSODI, two new initial value ordinary differential equation solvers. SIGNUM Newsletter, Volume 15 (4), pages 10-11 (1980). http://identifiers.org/isbn/978-0444866073 A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64. Livermore solver for ordinary differential equations, implicit version EXACT 2007-11-30 dk LSODIS http://identifiers.org/isbn/978-0444866073 http://www.nea.fr/abs/html/uscd1225.html Livermore solver for ordinary differential equations, implicit sparse version LSODIS is a set of general-purpose FORTRAN routines solver for the initial value problem for ordinary differential equation systems. It is suitable for both stiff and nonstiff systems. LSODIS treat systems in the linearly implicit form A(t,y) dy/dt = g(t,y), A = a square matrix, i.e. with the derivative dy/dt implicit, but linearly so. http://identifiers.org/isbn/978-0444866073 A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64. http://www.nea.fr/abs/html/uscd1225.html Seager M, Balsdon S. LSODIS - A sparse implicit ODE solver. 10th World Congress on System Simulation and Scientific Computation, pages 437-439 (1983). Livermore solver for ordinary differential equations, implicit sparse version EXACT 2008-07-08 NLN LSODPK http://identifiers.org/isbn/978-0444866073 Livermore solver for ordinary differential equations for stiff and nonstiff systems with krylov corrector iteration LSODPK is a set of FORTRAN subroutines for solving the initial value problem for stiff and nonstiff systems of ordinary differential equations. In solving stiff systems, LSODPK uses a corrector iteration composed of Newton iteration and one of four preconditioned Krylov subspace iteration methods [http://identifiers.org/biomodels.kisao/KISAO_0000354]. The user must select the desired Krylov method and supply a pair of routine to evaluate, preprocess, and solve the (left and/or right) preconditioner matrices. Aside from preconditioning, the implementation is matrix-free, meaning that explicit storage of the Jacobian (or related) matrix is not required. The method is experimental because the scope of problems for which it is effective is not well-known, and users are forewarned that LSODPK may or may not be competitive with traditional methods on a given problem. LSODPK also includes an option for a user-supplied linear system solver to be used without Krylov iteration. http://identifiers.org/isbn/978-0444866073 A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64. Livermore solver for ordinary differential equations for stiff and nonstiff systems with krylov corrector iteration EXACT 2008-07-08 NLN true Livermore solver Method to solve ordinary differential equations developed at the Lawrence Livermore National Laboratory. 2008-07-08 NLN true sub-volume stochastic reaction-diffusion algorithm Stochastic method using a combination of discretisation of compartment volumes into voxels and Gillespie-like algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000241] to simulate the evolution of the system. AZ true modelling and simulation algorithm characteristic modeling and simulation algorithm characteristic Simulation algorithm property, which can, for example, describe the model, such as the type of variables (discrete or continuous), and information on the treatment of spatial descriptions, or can be a numerical characteristic, such as the system's behaviour (deterministic or stochastic) as well as the progression mechanism (fixed or adaptive time steps). AZ true type of variable Type of variables used for the simulation. AZ true type of system behaviour A characteristic describing the rules the algorithm uses to simulate the temporal evolution of a system, specifically whether or not the final state is uniquely determined from a precise initial state. AZ true type of progression time step Type of time steps used by the algorithm. 2008-07-08 NLN spatial description Algorithm, possessing this characteristic, takes into account the location of the reacting components. 2008-07-08 NLN deterministic system behaviour Algorithm, possessing this characteristic, simulates the temporal evolution of a system deterministically, so that from a precise initial state the algorithm will always end up in the same final state. 2008-07-08 NLN stochastic system behaviour Algorithm, possessing this characteristic, simulates the temporal evolution of a system using probabilistic rules, so that between two simulations, the same precise initial state may result in a different final state. 2008-07-08 NLN discrete variable Algorithm, possessing this characteristic, allows values of system's variables to change by discrete (integral) amounts. 2008-07-08 NLN continuous variable Algorithm, possessing this characteristic, allows the values of a system's variables to change by continuous (non-integral) amounts. 2008-07-08 NLN progression with adaptive time step Algorithm, possessing this characteristic, does not use fixed time steps to update the state of a system during the whole simulation, but on the contrary adapts the length of the time steps to the local situation. 2008-07-08 NLN progression with fixed time step Algorithm, possessing this characteristic, uses time steps of constant length to update the state of a system during the whole simulation. AZ true modelling and simulation algorithm parameter modeling and simulation algorithm parameter Parameter that can be used in the simulation experiment settings. particle number lower limit This parameter of 'Pahle hybrid method' [http://identifiers.org/biomodels.kisao/KISAO_0000231] is a double value specifying the lower limit for particle numbers. Species with a particle number below this value are considered as having a low particle number. The 'particle number lower limit' cannot be higher than the 'particle number upper limit' [http://identifiers.org/biomodels.kisao/KISAO_0000204]. particle number upper limit This parameter of 'Pahle hybrid method' [http://identifiers.org/biomodels.kisao/KISAO_0000231] is a double value specifying the upper limit for particle numbers. Species with a particle number above this value are considered as having a high particle number. The 'particle number upper limit' cannot be lower than the 'particle number lower limit' [http://identifiers.org/biomodels.kisao/KISAO_0000203]. partitioning interval This positive integer value specifies after how many steps the internal partitioning of the system should be recalculated. relative tolerance RTOL This parameter is a numeric value specifying the desired relative tolerance the user wants to achieve. A smaller value means that the trajectory is calculated more accurately. RTOL EXACT absolute tolerance ATOL This parameter is a positive numeric value specifying the desired absolute tolerance the user wants to achieve. ATOL EXACT integrate reduced model http://identifiers.org/pubmed/14990466 This parameter is a boolean value to determine whether the integration shall be performed using the mass conservation laws (true), i.e., reducing the number of system variables or to use the complete model (false). maximum Adams order Adams max order maximum non-stiff order This parameter is a positive integer value specifying the maximal order the non-stiff Adams integration method [http://identifiers.org/biomodels.kisao/KISAO_0000289] shall attempt before switching to the stiff BDF method [http://identifiers.org/biomodels.kisao/KISAO_0000288]. maximum BDF order BDF max order maximum stiff order This parameter is a positive integer value specifying the maximal order the stiff BDF integration method [http://identifiers.org/biomodels.kisao/KISAO_0000288] shall attempt before switching to smaller internal step sizes. number of history bins The 'number of history bins' is only enabled for models that contain delayed or multistep reactions for specifying the granularity with which the delayed reaction solver should retain the history of species values, for species that participate in delayed reactions. tau-leaping epsilon http://identifiers.org/doi/10.1063/1.1378322 epsilon tolerance The leap condition is chosen such that the expected change in the propensity function aj(x) is bounded by Epsilon * a0 where Epsilon is an error control parameter between 0 and 1. This parameter is the basic error control mechanism for the Tau-Leaping algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000039]. As Epsilon decreases the leaps become shorter and the simulation is more accurate. http://identifiers.org/doi/10.1063/1.1378322 Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section V. epsilon EXACT minimum reactions per leap threshold 'minimum reactions per leap' parameter is used in hybrid methods, which adaptively switch between the tau-leaping algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000039] to the SSA Direct Method [http://identifiers.org/biomodels.kisao/KISAO_0000029] when the number of reactions in a single tau-leaping leap step is less than the threshold. 1 1 AZ COPASI Pahle hybrid method http://identifiers.org/doi/10.1093/bioinformatics/btl485 The hybrid method combines the stochastic 'Gibson-Bruck's next reaction method' [http://identifiers.org/biomodels.kisao/KISAO_0000027] with different algorithms for the numerical integration of ODEs [http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000374]. The biochemical network is dynamically partitioned into a deterministic and a stochastic subnet depending on the current particle numbers in the system. The user can define limits for when a particle number should be considered low or high. The stochastic subnet contains reactions involving low numbered species as substrate or product. All the other reactions form the deterministic subnet. The two subnets are then simulated in parallel using the stochastic and deterministic solver, respectively. The reaction probabilities in the stochastic subnet are approximated as constant between two stochastic reaction events. http://identifiers.org/doi/10.1093/bioinformatics/btl485 COPASI--a COmplex PAthway SImulator. Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N, Singhal M, Xu L, Mendes P, Kummer U. Bioinformatics. 2006 Dec 15;22(24):3067-74. Epub 2006 Oct 10. LSOIBT http://identifiers.org/isbn/978-0444866073 Livermore solver for ordinary differential equations given in implicit form, with block-tridiagonal Jacobian treatment LSOIBT solves linearly implicit systems in which the matrices involved are all assumed to be block-tridiagonal. Linear systems are solved by the LU method. http://identifiers.org/isbn/978-0444866073 A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64. Livermore solver for ordinary differential equations given in implicit form, with block-tridiagonal Jacobian treatment EXACT LSODES http://identifiers.org/isbn/978-0444866073 Livermore solver for ordinary differential equations with general sparse Jacobian matrix LSODES solves systems dy/dt = f and in the stiff case treats the Jacobian matrix in general sparse form. It determines the sparsity structure on its own, or optionally accepts this information from the user. It then uses parts of the Yale Sparse Matrix Package (YSMP) to solve the linear systems that arise, by a sparse (direct) LU factorization/backsolve method. http://identifiers.org/isbn/978-0444866073 A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64. Livermore solver for ordinary differential equations with general sparse Jacobian matrix EXACT LSODKR http://identifiers.org/isbn/978-0444866073 Livermore solver for ordinary differential equations, with preconditioned Krylov iteration methods for the Newton correction linear systems, and with root finding. LSODKR is an initial value ODE solver for stiff and nonstiff systems. It is a variant of the LSODPK [http://identifiers.org/biomodels.kisao/KISAO_0000093] and LSODE [http://identifiers.org/biomodels.kisao/KISAO_0000071] solvers, intended mainly for large stiff systems. The main differences between LSODKR and LSODE [http://identifiers.org/biomodels.kisao/KISAO_0000071] are the following: a) for stiff systems, LSODKR uses a corrector iteration composed of Newton iteration and one of four preconditioned Krylov subspace iteration methods. The user must supply routines for the preconditioning operations, b) within the corrector iteration, LSODKR does automatic switching between functional (fixpoint) iteration and modified Newton iteration, c) LSODKR includes the ability to find roots of given functions of the solution during the integration. http://identifiers.org/isbn/978-0444866073 A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64. Livermore solver for ordinary differential equations, with preconditioned Krylov iteration methods for the Newton correction linear systems, and with root finding. EXACT AZ true type of solution A characteristic describing the type of the solution produced by the method, specifically whether it is exact or approximate. exact solution Algorithm, possessing this characteristic, provides an exact solution to the initial problem. approximate solution Approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial time exact algorithms solving NP-hard problems, one settles for polynomial time sub-optimal solutions. Unlike heuristics, which usually only find reasonably good solutions reasonably fast, one wants provable solution quality and provable run time bounds. Ideally, the approximation is optimal up to a small constant factor (for instance within 5% of the optimal solution). Approximation algorithms are increasingly being used for problems where exact polynomial-time algorithms are known but are too expensive due to the input size. AZ true type of method A characteristic describing the way the method finds a solution, specifically whether it solves an equation involving only the current state of the system (explicit) or both the current and the later one (implicit). AZ explicit method type Explicit methods calculate the state of a system at a later time from the state of the system at the current time. Mathematically, if Y(t) is the current system state and Y((t+delta t) is the state at the later time (delta t is a small time step), then, for an explicit method Y(t+delta t) = F(Y(t)), to find Y(t+delta t). AZ implicit method type Implicit methods find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if Y(t) is the current system state and Y((t+delta t) is the state at the later time (delta t is a small time step), then, for an implicit method one solves an equation G(Y(t), Y(t+delta t))=0, to find Y(t+delta t). true Gillespie-like method Stochastic simulation algorithm using an approach alike the one described in Gillespie's papers of 1976 and 1977. AZ true error control parameter Parameter controlling method accuracy. AZ true method switching control parameter Parameters describing threshold conditions for algorithms that switch between different methods. AZ true granularity control parameter Parameter controlling granularity. tau-leaping delta Tau-leaping delta specifies how close two symmetric transition rates must be before we classify them as in partial-equilibrium. Only applies to the implicit tau routine [http://identifiers.org/biomodels.kisao/KISAO_0000045]. critical firing threshold nonnegative tau-leaping second control parameter The 'nonnegative Poisson tau-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000084] is based on the fact that negative populations typically arise from multiple firings of reactions that are only a few firings away from consuming all the molecules of one of their reactants. To focus on those reaction channels, the modified tau-leaping algorithm introduces a second control parameter nc, a positive integer that is usually set somewhere between 5 and 20. Any reaction channel with a positive propensity function that is currently within nc firings of exhausting one of its reactants is then classified as a critical reaction. The modified algorithm chooses tau in such a way that no more than one firing of all the critical reactions can occur during the leap. nonnegative tau-leaping second control parameter EXACT Parameter describing partitioning of the system. true partitioning control parameter coarse-graining factor http://identifiers.org/pubmed/15638577 The time in each Monte-Carlo iteration of 'binomial tau-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000074] is updated with the time increments tau=f/(a1+a2+...+aM). Here 1/(a1+a2+...+aM) is the averaged microscopic increment of the SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] and f is a coarse-graining factor, controlling the speed-up. http://identifiers.org/pubmed/15638577 Chatterjee A, Vlachos DG, Katsoulakis MA. Binomial distribution based tau-leap accelerated stochastic simulation. J Chem Phys. 2005;122(2):024112. Smoldyn Brownian diffusion accuracy Accuracy code of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057], which sets which neighbouring boxes are checked for potential bi-molecular reactions. Consider the reaction A + B -> C and suppose that A and B are within a binding radius of each other. This reaction will always be performed if A and B are in the same virtual box. If accuracy is set to at least 3, then it will also occur if A and B are in nearest-neighbour virtual boxes. If it is at least 7, then the reaction will happen if they are in nearest-neighbour boxes that are separated by periodic boundary conditions. And if it is 9 or 10, then all edge and corner boxes are checked for reactions, which means that no potential reactions are overlooked. Smoldyn molecules per virtual box Target molecules per virtual box is a parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057], which sets the box sizes so that the average number of molecules per box, at simulation initiation, is close to the requested number. Smoldyn virtual box side length The 'virtual box side length' is a parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. It requests the length of one side of a box. surface-bound epsilon A parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. Molecules that are bound to a surface are given locations that are extremely close to that surface. However, this position does not need to be exactly at the surface, and in fact it usually cannot be exactly at the surface due to round-off error. The tolerance for how far a surface-bound molecule is allowed to be away from the surface can be set with the epsilon statement. neighbour distance A parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. When a surface-bound molecule diffuses off of one surface panel, it can sometimes diffuse onto the neighbouring surface tile. It does so only if the neighbouring panel is declared to be a neighbour and also the neighbour is within a distance that is set with the neighbour distance statement. virtual box size Target size of virtual boxes for 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. 1 AZ ByoDyn JSim Euler method http://identifiers.org/isbn/052143064X The Euler method, named after Leonhard Euler, is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. http://identifiers.org/isbn/052143064X Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001). 2011-04-07 AZ NFSim NFSim agent-based simulation method http://identifiers.org/doi/10.1038/nmeth.1546 A generalization a rule-based version of 'Gillespie's direct method' (SSA) [http://identifiers.org/biomodels.kisao/KISAO_0000029]. The method is guaranteed to produce the same results as the exact SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] by cycling over three primary steps. First, NFsim calculates the probability or propensity for each rule to take effect given the current molecular states. Second, it samples the time to the next reaction event and selects the corresponding reaction rule. Finally, NFsim executes the selected reaction by applying the rule and updating the molecular agents accordingly. http://identifiers.org/doi/10.1038/nmeth.1546 Sneddon MW, Faeder JR and Emonet T. Efficient modelling, simulation and coarse-graining of biological complexity with NFsim. Nature Methods (2011) 8(2):177-83. 2011-04-07 AZ cellular automata update method http://identifiers.org/doi/10.1103/RevModPhys.55.601 CA cellular automata cellular spaces cellular structures homogeneous structures iterative arrays tessellation automata tessellation structures Cellular automata are mathematical idealizations of physical systems in which space and time are discrete, and physical quantities take on a finite set of discrete values. A cellular automaton consists of a regular uniform lattice (or ''array''), usually infinite in extent, with a discrete variable at each site (''cell''). A cellular automaton evolves in discrete time steps, with the value of the variable at one site being affected by the values of variables at sites in its ''neighbourhood'' on the previous time step. The neighbourhood of a site is typically taken to be the site itself and all immediately adjacent sites. The variables at each site are updated simultaneously (''synchronously''), based on the values of the variables in their neighbourhood at the preceding time step, and according to a definite set of ''local rules''. http://identifiers.org/doi/10.1103/RevModPhys.55.601 Wolfram, Stephen (1983) Statistical mechanics of cellular automata. Reviews of Modern Physics 55 (3): 601-644. CA EXACT cellular automata EXACT cellular spaces EXACT cellular structures EXACT homogeneous structures EXACT iterative arrays EXACT tessellation automata EXACT tessellation structures EXACT 2011-05-05 AZ hard-particle molecular dynamics http://identifiers.org/doi/10.1016/j.jcp.2004.08.014 A collision-driven molecular dynamics algorithm for a system of non-spherical particles. http://identifiers.org/doi/10.1016/j.jcp.2004.08.014 Aleksandar Donev, Salvatore Torquato, Frank H. Stillinger, Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details, Journal of Computational Physics, v.202 n.2, p.737-764, 20 January 2005 2011-05-05 AZ first-passage Monte Carlo algorithm http://identifiers.org/doi/10.1103/PhysRevLett.97.230602 AED DKMC AED diffusion kinetic Monte Carlo method asynchronous event-driven diffusion Monte Carlo We present a novel Monte Carlo algorithm for N diffusing finite particles that react on collisions. Using the theory of first-passage processes and time dependent Green's functions, we break the difficult N-body problem into independent single- and two-body propagations circumventing numerous diffusion hops used in standard Monte Carlo simulations. The new algorithm is exact, extremely efficient, and applicable to many important physical situations in arbitrary integer dimensions. http://identifiers.org/doi/10.1103/PhysRevLett.97.230602 T. Oppelstrup, V. V. Bulatov, G. H. Gilmer, M. H. Kalos, and B. Sadigh. First-passage Monte- Carlo algorithm: Diffusion without all the hops. Physical Review Letters, 97(23):230602, 2006. AED DKMC EXACT AED diffusion kinetic Monte Carlo method EXACT asynchronous event-driven diffusion Monte Carlo EXACT 4 2011-05-09 AZ Gill method http://identifiers.org/doi/10.1017/S0305004100026414 Gill's method Runge-Kutta-Gill method Gill's fourth order method is a Runge-Kutta method for approximating the solution of the initial value problem y'(x) = f(x,y); y(x0) = y0 which evaluates the integrand,f(x,y), four times per step. This method is a fourth order procedure for which Richardson extrapolation can be used. http://identifiers.org/doi/10.1017/S0305004100026414 Gill, S. 1951. A process for the step-by-step integration of differential equations in an automatic digital computing machine. Proc. Cambridge Philos. Soc., 47, pp 96-108. Gill's method EXACT Runge-Kutta-Gill method EXACT 2011-05-09 AZ CompuCell3D Metropolis Monte Carlo algorithm http://identifiers.org/doi/10.1063/1.1699114 Metropolis algorithm Metropolis-Hastings algorithm A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration [http://identifiers.org/biomodels.kisao/KISAO_0000051] over configuration space. http://identifiers.org/doi/10.1063/1.1699114 Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. (1953). Equations of State Calculations by Fast Computing Machines. Journal of Chemical Physics 21 (6): 1087-1092. Metropolis algorithm EXACT Metropolis-Hastings algorithm EXACT 2011-05-09 AZ Adams-Bashforth method http://identifiers.org/isbn/978-3-540-56670-0 explicit Adams method Given an initial value problem: y' = f(x,y), y(x0) = y0 together with additional starting values y1 = y(x0 + h), . . . , yk-1 = y(x0 + (k-1) h) the k-step Adams-Bashforth method is an explicit linear multistep method that approximates the solution, y(x) at x = x0+kh, of the initial value problem by yk = yk - 1 + h * ( a0 f(xk - 1,yk - 1) + a1 f(xk - 2,yk - 2) + . . . + ak - 1 f(x0,y0) ), where a0, a1, . . . , ak - 1 are constants. http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag explicit Adams method EXACT 2011-05-09 AZ VCell Adams-Moulton method http://identifiers.org/isbn/978-3-540-56670-0 implicit Adams method The (k-1)-step Adams-Moulton method is an implicit linear multistep method that iteratively approximates the solution, y(x) at x = x0+kh, of the initial value problem by yk = yk - 1 + h * ( b0 f(xk,yk) + b1 f(xk - 1,yk - 1) + . . . + bk - 1 f(x1,y1) ), where b1, . . . , bk - 1 are constants. http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag implicit Adams method EXACT 2011-05-09 AZ true multistep method http://identifiers.org/isbn/978-3-540-56670-0 multi-value method A numerical method for differential equations which is based on several values of the solution. http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag multi-value method EXACT 2011-05-09 AZ SUNDIALS KINSOL http://identifiers.org/doi/10.1137/0911026 http://identifiers.org/doi/10.1145/1089014.1089020 http://identifiers.org/doi/10.4249/scholarpedia.2860 FKINSOL NKSOL Newton-Krylov solver for nonlinear algebraic systems KINSOL solves algebraic systems in real N-space, written as F(u)=0, F:RN->RN, given an initial guess u0. The basic method is either a modified or an inexact Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408]. The linear systems that arise are solved with either a direct (dense or banded) solver (serial version only), or one of the Krylov iterative solvers [http://identifiers.org/biomodels.kisao/KISAO_0000354]. In the Krylov case, the user can (optionally) supply a right preconditioner. http://identifiers.org/doi/10.1137/0911026 BROWN, P. N. AND SAAD, Y. 1990. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11, 450-481. http://identifiers.org/doi/10.1145/1089014.1089020 Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005. http://identifiers.org/doi/10.4249/scholarpedia.2860 Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860. FKINSOL RELATED NKSOL RELATED Newton-Krylov solver for nonlinear algebraic systems RELATED 1 1 1 1 2011-05-09 AZ OpenCOR SUNDIALS VCell IDA http://identifiers.org/doi/10.1137/0915088 http://identifiers.org/doi/10.1145/1089014.1089020 http://identifiers.org/doi/10.4249/scholarpedia.2860 implicit differential-algebraic solver solver for differential-algebraic equation systems IDA solves real differential-algebraic systems in N-space, in the general form F(t,y,y')=0, y(t0)=y0, y'(t0)=y'0. At each step, a Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408] leads to linear systems Jx=b, which are solved by one of five methods - two direct (dense or band; serial version only) and three Krylov [http://identifiers.org/biomodels.kisao/KISAO_0000354] (GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000353], BiCGStab [http://identifiers.org/biomodels.kisao/KISAO_0000392], or TFQMR [http://identifiers.org/biomodels.kisao/KISAO_0000396]). IDA is written in C, but derived from the package DASPK [http://identifiers.org/biomodels.kisao/KISAO_0000355] which is written in Fortran. IDA Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860. http://identifiers.org/doi/10.1137/0915088 BROWN, P. N., HINDMARSH, A. C., AND PETZOLD, L. R. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 1467-1488. http://identifiers.org/doi/10.1145/1089014.1089020 Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005. http://identifiers.org/doi/10.4249/scholarpedia.2860 Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860. implicit differential-algebraic solver EXACT solver for differential-algebraic equation systems RELATED 2011-05-09 AZ VCell finite volume method http://identifiers.org/isbn/0898715342 FVM The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations, which attempts to emulate continuous conservation laws of physics. http://identifiers.org/isbn/0898715342 Y. Saad. 2003. Iterative Methods for Sparse Linear Systems (2nd ed.). Soc. for Industrial and Applied Math., Philadelphia, PA, USA. FVM EXACT 2011-05-09 AZ Euler-Maruyama method http://identifiers.org/doi/10.1098/rspa.2003.1247 stochastic Euler scheme The Euler-Maruyama method is a method for the approximate numerical solution of a stochastic differential equation, which truncates the Ito and Stratonovich Taylor series of the exact solution after the first order stochastic terms. This converges to the Ito solution with strong global order accuracy 1/2 or weak global order accuracy 1. It is a simple generalization of the Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261] for ordinary differential equations to stochastic differential equations. http://identifiers.org/doi/10.1098/rspa.2003.1247 Burrage, K., Burrage, P.M. and Tian, T. (2004) Numerical methods for strong solutions of stochastic differential equations: an overview. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 460 (2041). pp. 373-402. stochastic Euler scheme EXACT 2011-05-10 AZ Milstein method http://identifiers.org/isbn/079233213X The Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. http://identifiers.org/isbn/079233213X Milstein M. G. Numerical Integration of Stochastic Differential Equations (Mathematics and Its Applications) Springer, 2004. 2011-05-10 AZ ByoDyn GSL iBioSim backward differentiation formula http://identifiers.org/isbn/0136266061 BDF Gear method Gear's method The backward differentiation formulas (BDF) are implicit multistep methods based on the numerical differentiation of a given function and are wildly used for integration of stiff differential equations. http://identifiers.org/isbn/0136266061 C. William Gear. 1971. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall PTR, Upper Saddle River, NJ, USA. BDF EXACT Gear method EXACT Gear's method EXACT 2011-05-10 AZ ByoDyn Adams method http://identifiers.org/isbn/978-3-540-56670-0 Adams' methods are multi-step methods used for the numerical integration of initial value problems in Ordinary Differential Equations (ODE's). Adams' algorithm consists of two parts: firstly, a starting procedure which provides y1, ... , yk-1 ( approximations to the exact solution at the points x0 + h, ... , x0 + (k - 1)h ) and, secondly, a multistep formula to obtain an approximation to the exact solution y(x0 + kh). This is then applied recursively, based on the numerical approximation of k successive steps, to compute y(x0 + (k + 1)h). http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag 4 2011-05-10 AZ JSim Merson method http://nla.gov.au/nla.cat-vn870866 KM Kutta-Merson method Merson's method Runge-Kutta-Merson method A five-stage Runge-Kutta method with fourth-order accuracy. http://nla.gov.au/nla.cat-vn870866 R.H. Merson, An operational method for the study of integration processes, in: Proceedings of the Symposium on Data Processing, Weapons Research Establishment, Salisbury, Australia, 1957, pp. 110-1-110-25. KM EXACT Kutta-Merson method EXACT Merson's method EXACT Runge-Kutta-Merson method EXACT 2011-05-10 AZ Hammer-Hollingsworth method urn:issn:0891-6837 The numerical integration of ordinary differential equations by the use of Gaussian quadrature methods. urn:issn:0891-6837 P. C. Hammer & J. W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations, MTAC, v. 9, 1955, pp. 92-96. MR 17, 302. 2011-05-10 AZ Lobatto method urn:issn:1088-6842(e) implicit Runge-Kutta method based on Lobatto quadrature There are three families of Lobatto methods, called IIIA, IIIB and IIIC. These are named after Rehuel Lobatto. All are implicit Runge-Kutta methods, have order 2s − 2 and they all have c1 = 0 and cs = 1. urn:issn:1088-6842(e) J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comp., v. 18, 1964, pp. 233-244. implicit Runge-Kutta method based on Lobatto quadrature EXACT 2011-05-10 AZ Butcher-Kuntzmann method http://identifiers.org/doi/10.1002/zamm.19660460519 urn:issn:1088-6842(e) Gauss method From a theoretical point of view, the Butcher-Kuntzmann Runge-Kutta methods belong to the best step-by-step methods for nonstiff problems. These methods integrate first-order initial-value problems by means of formulas based on Gauss-Legendre quadrature, and combine excellent stability features with the property of superconvergence at the step points. http://identifiers.org/doi/10.1002/zamm.19660460519 F. Ceschin, J. Kuntzmann, Problèmes différentiels de conditions initiales. Paris 1963. Dunod. urn:issn:1088-6842(e) Butcher, J. C. (1964) Implicit Runge-Kutta processes. Math. Comput. 18, 50-64. Gauss method EXACT 2 2011-05-10 AZ Heun method http://identifiers.org/isbn/978-3-540-56670-0 Heun's method The method is named after Karl L. W. M. Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It can be seen as extension of the Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261] into two-stage second-order Runge-Kutta method. http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag Heun's method EXACT 2011-05-10 AZ true embedded Runge-Kutta method http://identifiers.org/isbn/978-3-540-56670-0 embedded RK An embedded Runge-Kutta method is a method in which two Runge-Kutta estimates are obtained using the same auxiliary functions ki but with a different linear combination of these functions so that one estimate has an order one greater than the other. http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag embedded RK EXACT 3 4 2011-05-10 AZ Zonneveld method http://trove.nla.gov.au/work/21424455 An embedded Runge-Kutta method [http://identifiers.org/biomodels.kisao/KISAO_0000302] of order 4(3), proposed by J.A. Zonneveld in 1964. http://trove.nla.gov.au/work/21424455 Zonneveld, J. A (1970). Automatic numerical integration Mathematisch Centrum, Amsterdam 2011-05-11 AZ JSim Radau method urn:issn:1088-6842(e) implicit Runge-Kutta method based on Radau quadrature Implicit Runge-Kutta methods based on Radau quadrature. urn:issn:1088-6842(e) J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comput., 18(1964), 233-244. implicit Runge-Kutta method based on Radau quadrature EXACT 5 6 2011-05-10 AZ Verner method http://identifiers.org/doi/10.1137/0728027 Verner's method The first high order (6(5)) embedded Runge-Kutta formulas that avoid the drawback of giving identically zero error estimates for quadrature problems y' = f(x) were constructed by Verner in 1978. http://identifiers.org/doi/10.1137/0728027 J.H. Verner, Some Runge-Kutta formula pairs, SIAM J. Numer. Anal 28 (1991) 496-511 Verner's method EXACT 2011-05-11 AZ JSim Lagrangian sliding fluid element algorithm http://identifiers.org/pubmed/1449234 BTEX LSFEA blood-tissue exchange method Because the analytic solutions to the partial differential equations require convolution integration, solutions are obtained relatively efficiently by a fast numerical method. Our approach centers on the use of a sliding fluid element algorithm for capillary convection, with the time step set equal to the length step divided by the fluid velocity. Radial fluxes by permeation between plasma, interstitial fluid, and cells and axial diffusion exchanges within each time step are calculated analytically. The method enforces mass conservation unless there is regional consumption. http://identifiers.org/pubmed/1449234 Bassingthwaighte JB, Chan IS, and Wang CY. Computationally efficient algorithms for capillary convection-permeation-diffusion models for blood-tissue exchange. Ann Biomed Eng 20: 687-725, 1992. BTEX EXACT LSFEA EXACT blood-tissue exchange method EXACT 2011-05-11 AZ finite difference method http://identifiers.org/isbn/0898715342 FDM The finite difference method is based on local approximations of the partial derivatives in a Partial Differential Equation, which are derived by low order Taylor series expansions. http://identifiers.org/isbn/0898715342 Y. Saad. 2003. Iterative Methods for Sparse Linear Systems (2nd ed.). Soc. for Industrial and Applied Math., Philadelphia, PA, USA. FDM EXACT 2011-05-11 AZ JSim MacCormack method http://identifiers.org/doi/10.2514/2.6901 In computational fluid dynamics, the MacCormack method is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations. This second-order finite difference method [http://identifiers.org/biomodels.kisao/KISAO_0000307] is introduced by R. W. MacCormack in 1969. http://identifiers.org/doi/10.2514/2.6901 MacCormack, R. W., The Effect of viscosity in hypervelocity impact cratering, AIAA Paper, 69-354 (1969). 2011-05-11 AZ Crank-Nicolson method http://identifiers.org/doi/10.1007/BF02127704 In numerical analysis, the Crank-Nicolson method is a finite difference method [http://identifiers.org/biomodels.kisao/KISAO_0000307] used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time, implicit in time, and is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century. http://identifiers.org/doi/10.1007/BF02127704 Crank, J.; Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc. Camb. Phil. Soc. 43 (1): 50-67. 2011-05-11 AZ method of lines http://identifiers.org/isbn/0126241309 MOL NMOL NUMOL The method of lines is a general technique for solving partial differential equations (PDEs) by typically using finite difference relationships for the spatial derivatives and ordinary differential equations for the time derivative. method of lines EXACT http://identifiers.org/isbn/0126241309 Schiesser, W. E. (1991). The Numerical Method of Lines. Academic Press. MOL EXACT NMOL EXACT NUMOL EXACT AZ true type of domain geometry handling 2011-05-20 AZ E-Cell S-System power-law canonical differential equations solver http://identifiers.org/doi/10.1137/0727042 ESSYNS GMA Ordinary differential equations can be recast into a nonlinear canonical form called an S-system. Evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions, recognized as among the most general, are special cases of S-systems. Identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations. By capitalizing on the regular structure of S-systems, efficient formulas for a variable-order, variable-step Taylor-series method are developed. http://identifiers.org/doi/10.1137/0727042 Irvine, D.H. and Savageau, M.A., Efficient solution of nonlinear ordinary differential equations expressed in S-System canonical form, Siam. J. Numer. Anal., 27:704-735, 1990. ESSYNS GMA EXACT 2011-05-23 AZ lattice gas automata http://identifiers.org/isbn/3-540-66973-6 LGA LGCA lattice gas cellular automata Lattice gas automata methods are a series of cellular automata methods used to simulate fluid flows. From the LGCA, it is possible to derive the macroscopic Navier-Stokes equations. http://identifiers.org/isbn/3-540-66973-6 Dieter A. Wolf-Gladrow (2000). Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer. LGA EXACT LGCA EXACT lattice gas cellular automata EXACT 2011-05-23 AZ enhanced Greens function reaction dynamics http://identifiers.org/doi/10.1007/978-3-540-88562-7_3 eGFRD enhanced Greens function reaction dynamics GFRD [http://identifiers.org/biomodels.kisao/KISAO_0000058] decomposes the multi­body reaction diffusion problem to a set of single and two body problems. Analytical solutions for two body reaction diffusion are available via Smoluchowski equation. eGFRD allows to solve each sub­problem asynchronously by introducing the concept of first passage processes. http://identifiers.org/doi/10.1007/978-3-540-88562-7_3 K. Takahashi, An exact Brownian dynamics method for cell simulation, in Computational Methods in Systems Biology, vol. 5307/2008. eGFRD EXACT enhanced Greens function reaction dynamics EXACT 2011-05-23 AZ E-Cell E-Cell multi-algorithm simulation method http://identifiers.org/pubmed/14990450 A modular meta-algorithm with a discrete event scheduler that can incorporate any type of time-driven simulation algorithm. It was shown that this meta-algorithm can efficiently drive simulation models with different simulation algorithms with little intrusive modification to the algorithms themselves. Only a few additional methods to handle communications between computational modules are required. http://identifiers.org/pubmed/14990450 Takahashi, K., Kaizu, K., Hu, B., and Tomita, M., A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics, in press. 2011-05-26 AZ iBioSim Gauss-Legendre Runge-Kutta method http://identifiers.org/isbn/960-8457-54-8 Open Formula So called 'Open Formula', two points formula, three points formula, four points formula, five points formula and six points formula of the Runge-Kutta method to solve the initial value problem of the ordinary differential equation. These formulas use the points and weights from the Gauss-Legendre Quadrature formulas for finding the value of the definite integral. http://identifiers.org/isbn/960-8457-54-8 Maitree Podisuk, Sirirat Khuntidilokwongsa, and Witchaya Rattanametawee. 2006. Gauss-Legendre Quadrature formula in Runge-Kutta method with modified model of Newton cooling law. In Proceedings of the 8th WSEAS international conference on Mathematical methods and computational techniques in electrical engineering (MMACTEE'06). World Scientific and Engineering Academy and Society (WSEAS), Stevens Point, Wisconsin, USA, 312-317. Open Formula EXACT 2011-05-26 AZ true Monte Carlo method http://identifiers.org/doi/10.2307/2280232 MC Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results. http://identifiers.org/doi/10.2307/2280232 Metropolis, N.; Ulam, S. (1949). "The Monte Carlo Method". Journal of the American Statistical Association (American Statistical Association) 44 (247): 335-341. MC EXACT 2011-05-26 AZ BioRica BioRica hybrid method http://identifiers.org/pubmed/19425152 The simulation schema for a given BioRica node is given by a hybrid algorithm that deals with continuous time and allows for discrete events that roll back the time according to these discrete interruptions. http://identifiers.org/pubmed/19425152 Cvijovic M, Soueidan H, Sherman DJ, Klipp E, Nikolski M. Exploratory simulation of cell ageing using hierarchical models. Genome Inform. 2008;21:114-25. 2011-05-26 AZ Cain GSL Cash-Karp method http://identifiers.org/doi/10.1145/79505.79507 An family of explicit Runge-Kutta formulas, which are very efficient for problems with smooth solution as well as problems having rapidly varying solutions. Each member of this family consists of a fifty-order formula that contains embedded formulas of all orders 1 through 4. By computing solutions at several different orders, it is possible to detect sharp fronts or discontinuities before all the function evaluations defining the full Runge-Kutta step have been computed. http://identifiers.org/doi/10.1145/79505.79507 J. R. Cash, A. H. Karp. "A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides", ACM Transactions on Mathematical Software 16: 201-222, 1990. 2011-05-27 AZ hybridity The basic idea of hybrid simulation methods is to combine the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time. 2011-06-02 AZ equation-free probabilistic steady-state approximation http://identifiers.org/doi/10.1063/1.2131050 We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. <...> The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques. http://identifiers.org/doi/10.1063/1.2131050 Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106. 2011-06-02 AZ nested stochastic simulation algorithm http://identifiers.org/pubmed/16321076 nested SSA This multiscale method is a small modification of the Gillespie's direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029], in the form of a nested SSA, with inner loops for the fast reactions, and outer loop for the slow reactions. The number of groups can be more than two, and the grouping into fast and slow variables can be done dynamically in an adaptive version of the scheme. http://identifiers.org/pubmed/16321076 E W., Liu D. and Vanden-Eijnden E. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J Chem Phys. 2005 Nov 15;123(19):194107. nested SSA EXACT 2011-06-02 AZ minimum fast/discrete reaction occurrences number http://identifiers.org/doi/10.1063/1.2131050 Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which describes the minimum number of fast/discrete reaction occurrences before their effects cause convergence to a quasi-steady-state distribution. http://identifiers.org/doi/10.1063/1.2131050 Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106. 2011-06-02 AZ number of samples http://identifiers.org/doi/10.1063/1.2131050 Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which determines the number of samples taken from the distribution. http://identifiers.org/doi/10.1063/1.2131050 Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106. 2011-06-02 AZ maximum discrete number http://identifiers.org/doi/10.1063/1.2131050 Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which controls the maximum number of molecules of some reactant species in order for the reaction to be considered discrete. http://identifiers.org/doi/10.1063/1.2131050 Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106. 2011-06-02 AZ minimum fast rate http://identifiers.org/doi/10.1063/1.2131050 Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which controls the minimum rate of the reaction in order for it to be considered fast. http://identifiers.org/doi/10.1063/1.2131050 Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106. 2011-06-03 AZ constant-time kinetic Monte Carlo algorithm http://identifiers.org/pubmed/18513044 SSA-CR The computational cost of the original SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] scaled linearly with the number of reactions in the network. Gibson and Bruck developed a logarithmic scaling version of the SSA which uses a priority queue or binary tree for more efficient reaction selection [http://identifiers.org/biomodels.kisao/KISAO_0000027]. More generally, this problem is one of dynamic discrete random variate generation which finds many uses in kinetic Monte Carlo and discrete event simulation. We present here a constant-time algorithm, whose cost is independent of the number of reactions, enabled by a slightly more complex underlying data structure. http://identifiers.org/pubmed/18513044 Slepoy A, Thompson AP, Plimpton SJ. A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks. J Chem Phys. 2008 May 28;128(20):205101. SSA-CR EXACT 2011-06-03 AZ R-leaping algorithm http://identifiers.org/pubmed/16964997 R-leap method A novel algorithm is proposed for the acceleration of the exact stochastic simulation algorithm by a predefined number of reaction firings (R-leaping) that may occur across several reaction channels. In the present approach, the numbers of reaction firings are correlated binomial distributions and the sampling procedure is independent of any permutation of the reaction channels. This enables the algorithm to efficiently handle large systems with disparate rates, providing substantial computational savings in certain cases. http://identifiers.org/pubmed/16964997 Auger A, Chatelain P, Koumoutsakos P. R-leaping: accelerating the stochastic simulation algorithm by reaction leaps. J Chem Phys. 2006 Aug 28;125(8):084103. R-leap method EXACT 2011-06-03 AZ exact R-leaping algorithm http://identifiers.org/pubmed/2852436 ER-leap method exact R-leap method exact accelerated stochastic simulation algorithm We present a SSA which, similar to R-leap [http://identifiers.org/biomodels.kisao/KISAO_0000330], accelerates SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] by executing multiple reactions per algorithmic step, but which samples the reactant trajectories from the same probability distribution as the SSA. This 'exact R-leap' or 'ER-leap' algorithm is a modification of the R-leap algorithm which is both exact and capable of substantial speed-up over SSA. http://identifiers.org/pubmed/2852436 Eric Mjolsness, David Orendorff, Philippe Chatelain, and Petros Koumoutsakos. An exact accelerated stochastic simulation algorithm. J Chem Phys. 2009 April 14; 130(14): 144110. ER-leap method EXACT exact R-leap method EXACT exact accelerated stochastic simulation algorithm EXACT 2011-06-03 AZ ER-leap initial leap http://identifiers.org/pubmed/2852436 L L (initial step) is a parameter of 'exact R-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000331]. ''We will assume that the reaction event to be bounded occurs within a run of L events in the SSA algorithm[http://identifiers.org/biomodels.kisao/KISAO_0000029], in order to execute L reactions at once in the manner of the R-leap algorithm[http://identifiers.org/biomodels.kisao/KISAO_0000230]''. http://identifiers.org/pubmed/2852436 Eric Mjolsness, David Orendorff, Philippe Chatelain, and Petros Koumoutsakos. An exact accelerated stochastic simulation algorithm. J Chem Phys. 2009 April 14; 130(14): 144110. L EXACT 2011-06-03 AZ true accelerated stochastic simulation algorithm accelerated SSA An algorithm, which accelerates SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] either at the expense of its accuracy or exact. accelerated SSA EXACT 2011-06-03 AZ multiparticle lattice gas automata http://identifiers.org/doi/10.1142/S0129183194000052 multiparticle lattice gas cellular automata An algorithm which allows for an arbitrary number of particles, while keeping the benefits of the cellular automata approach [http://identifiers.org/biomodels.kisao/KISAO_0000315]. http://identifiers.org/doi/10.1142/S0129183194000052 Chopard B., et al.Multiparticle lattice gas automata for reaction diffusion systems. Int. J. Mod. Phys. C 1994;5:47-63. multiparticle lattice gas cellular automata EXACT 2011-06-03 AZ true generalized stochastic simulation algorithm Gillespie direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029] follows unit-by-unit changes in the total numbers of each reactant species, it is especially well suited to the study of systems in which reactant densities are low and the application of methods based on continuum approximations, such as the traditional ordinary differential equations of chemical kinetics, is questionable. The 'generalized stochastic simulation algorithm' branch presents methods, which extend Gillespie direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029] to suit to systems with other characteristics. 2011-06-03 AZ D-leaping method http://identifiers.org/doi/10.1016/j.jcp.2009.05.004 We propose a novel, accelerated algorithm for the approximate stochastic simulation of biochemical systems with delays. The present work extends existing accelerated algorithms by distributing, in a time adaptive fashion, the delayed reactions so as to minimize the computational effort while preserving their accuracy. http://identifiers.org/doi/10.1016/j.jcp.2009.05.004 Bayati B, Chatelain P, Koumoutsakos. D-leaping: Accelerating stochastic simulation algorithms for reactions with delays. Journal of Computational Physics. 2009 Sept;228(16):5909-5918. 2011-06-07 AZ finite element method http://identifiers.org/doi/10.1109/MAP.2007.376627 FEA FEM finite element analysis A numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261], Runge-Kutta [http://identifiers.org/biomodels.kisao/KISAO_0000064], etc. http://identifiers.org/doi/10.1109/MAP.2007.376627 Pelosi G. The finite-element method, Part I: R. L. Courant: Historical Corner. IEEE Antennas and Propagation Magazine, Vol. 49, No. 2. (April 2007), pp. 180-182. FEA EXACT FEM EXACT finite element analysis EXACT 2011-06-07 AZ h-version of the finite element method http://identifiers.org/doi/10.1016/0168-874X(94)90003-5 h-FEM h-method Classical form of the 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337], in which polynomials of fixed degree p are used and the mesh is refined to increase accuracy. Can be considered as a special case of the h-p version [http://identifiers.org/biomodels.kisao/KISAO_0000340]. http://identifiers.org/doi/10.1016/0168-874X(94)90003-5 Babuska, I., T. Strouboulis, A. Mathur, C. S. Upadhyay, Pollution error in the h-version of the finite element method and the local quality of a-posteriori error estimators, Finite Elements in Analysis and Design, 17: 273-321, 1994. h-FEM EXACT h-method EXACT 2011-06-07 AZ p-version of the finite element method http://identifiers.org/doi/10.1137/0718033 p-FEM p-method The p version of 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] uses a fixed mesh but increases the polynomial degree p to increase accuracy. Can be considered as a special case of the h-p version [http://identifiers.org/biomodels.kisao/KISAO_0000340]. http://identifiers.org/doi/10.1137/0718033 Babuska, I., B.A. Szabo and I.N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal., 18: 515-545, 1981. p-FEM EXACT p-method EXACT 2011-06-07 AZ h-p version of the finite element method http://identifiers.org/doi/10.1007/BF00272624 hp-FEM hp-method In h-p version of 'finite difference method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] the two approaches of mesh refinement and degree enchacement are combined. http://identifiers.org/doi/10.1007/BF00272624 Babuska, I., and B. Guo, The h-p version of the finite element method, Comput. Mech., 1: 203-220, 1986. hp-FEM EXACT hp-method EXACT 2011-06-07 AZ mixed finite element method http://identifiers.org/doi/10.1016/0045-7825(90)90168-L A 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] in which both stress and displacement fields are approximated as primary variables. http://identifiers.org/doi/10.1016/0045-7825(90)90168-L D. N. Arnold. 1990. Mixed finite element methods for elliptic problems. Comput. Methods Appl. Mech. Eng. 82, 1-3 (September 1990), 281-300. 2011-06-07 AZ level set method http://identifiers.org/doi/10.1016/0021-9991(88)90002-2 LSM level-set method An algorithm for moving surfaces under their curvature. This algorithm rely on numerically solving Hamilton-Jacobi equations with viscous terms, using approximation techniques from hyperbolic conservation laws. http://identifiers.org/doi/10.1016/0021-9991(88)90002-2 Osher, S.; Sethian, J. A. (1988), "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations", J. Comput. Phys. 79: 12-49. LSM EXACT level-set method EXACT 2011-06-07 AZ generalized finite element method http://identifiers.org/doi/10.1142/S0219876204000083 GFEM PUM partition of unity method The GFEM is a generalization of the classical 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] — in its h [http://identifiers.org/biomodels.kisao/KISAO_0000338], p [http://identifiers.org/biomodels.kisao/KISAO_0000339], and h-p versions [http://identifiers.org/biomodels.kisao/KISAO_0000340]— as well as of the various forms of meshless methods used in engineering. http://identifiers.org/doi/10.1142/S0219876204000083 Babuska, Ivo; Uday Banerjee, John E. Osborn (June 2004). Generalized Finite Element Methods: Main Ideas, Results, and Perspective. International Journal of Computational Methods 1 (1): 67-103. GFEM EXACT PUM EXACT partition of unity method EXACT 2011-06-09 AZ h-p cloud method http://identifiers.org/doi/10.1002/(SICI)1098-2426(199611)12:6<673::AID-NUM3>3.0.CO;2-P h-p clouds method of clouds A meshless method, which uses a partition of unity to construct the family of h-p cloud functions. http://identifiers.org/doi/10.1002/(SICI)1098-2426(199611)12:6<673::AID-NUM3>3.0.CO;2-P Duarte, C. A. and Oden, J. T. (1996), H-p clouds—an h-p meshless method. Numerical Methods for Partial Differential Equations, 12: 673-705. h-p clouds EXACT method of clouds EXACT AZ mesh-based geometry handling In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational effort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion and separation to model fracture, fragmentation, free surfaces, etc. AZ meshless geometry handling Most meshless methods require a scattered set of nodal points in the domain of interest. In these methods, there may be no fixed connectivities between the nodes, unlike the finite element or finite difference methods. This feature has significant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis. 2011-06-09 AZ extended finite element method http://identifiers.org/doi/10.1007/s00466-002-0391-2 X-FEM XFEM A numerical method to model arbitrary discontinuities in continuous bodies that does not require the mesh to conform to the discontinuities nor significant mesh refinement near singularities. In X-FEM the standard finite element approximation [http://identifiers.org/biomodels.kisao/KISAO_0000337] is enriched and the approximation space is extended by an additional family of functions. http://identifiers.org/doi/10.1007/s00466-002-0391-2 Stazi, F. L.; Budyn, E.; Chessa, J.; Belytschko, T. An extended finite element method with higher-order elements for curved cracks. Computational Mechanics, Volume 31, Issue 1-2, pp. 38-48 (2003). X-FEM EXACT XFEM EXACT 2011-06-09 AZ method of finite spheres http://identifiers.org/doi/10.1007/s004660050481 MFS Method of finite spheres is truly meshless in the sense that the nodes are placed and the numerical integration is performed without a mesh. Some of the novel features of the method of finite spheres are the numerical integration scheme and the way in which the Dirichlet boundary conditions are incorporated. http://identifiers.org/doi/10.1007/s004660050481 S. De and K. J. Bathe. The method of finite spheres. Computational Mechanics, Volume 25, Number 4, 329-345. MFS EXACT 2011-06-09 AZ probability-weighted dynamic Monte Carlo method http://identifiers.org/doi/10.1021/jp011404w PW-DMC probability-weighted DMC We have developed a probability-weighted DMC method by incorporating the weighted sampling algorithm of equilibrium molecular simulations. This new algorithm samples the slow reactions very efficiently and makes it possible to simulate in a computationally efficient manner the reaction kinetics of physical systems in which the rates of reactions vary by several orders of magnitude. http://identifiers.org/doi/10.1021/jp011404w Resat H, Wiley HS, Dixon DA. Probability-weighted dynamic Monte Carlo method for reaction kinetics simulations. J Phys Chem B. 2001;105:11026-11034. PW-DMC EXACT probability-weighted DMC EXACT 2011-06-09 AZ multinomial tau-leaping method http://identifiers.org/pubmed/17343434 MtauL The multinomial tau-leaping method is an extension of the binomial tau-leaping method [http://identifiers.org/biomodels.kisao/KISAO_0000074] to networks with arbitrary multiple-channel reactant dependencies. Improvements were achieved by a combination of three factors: First, tau-leaping steps are determined simply and efficiently using a-priori information and Poisson distribution based estimates of expectation values for reaction numbers. Second, networks are partitioned into closed groups of reactions and corresponding reactants in which no group reactant set is found in any other group. Third, product formation is factored into upper bound estimation of the number of times a particular reaction occurs. http://identifiers.org/pubmed/17343434 Pettigrew MF, Resat H. Multinomial tau-leaping method for stochastic kinetic simulations. J Chem Phys. 2007 Feb 28;126(8):084101. MtauL EXACT 2011-06-09 AZ true hybrid method A simulation methods which combines the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time. 2011-06-10 AZ generalized minimal residual algorithm http://identifiers.org/doi/10.1137/0907058 GMRES An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The generalized minimal residual method extends the minimal residual method (MINRES) [http://identifiers.org/biomodels.kisao/KISAO_0000388], which is only applicable to symmetric systems, to non-symmetric systems. http://identifiers.org/doi/10.1137/0907058 Youcef Saad and Martin H. Schultz, GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J. Sci. and Stat. Comput. 7, 856 (1986). GMRES EXACT 2011-06-10 AZ Krylov subspace projection method http://identifiers.org/isbn/0898715342 Krylov subspace method Krylov subspace method is an iterative linear equation method, which builds up Krylov subspaces and look for good approximations to eigenvectors and invariant subspaces within the Krylov spaces. http://identifiers.org/isbn/0898715342 Y. Saad. 2003. Iterative Methods for Sparse Linear Systems (2nd ed.). Soc. for Industrial and Applied Math., Philadelphia, PA, USA. Krylov subspace method EXACT 2011-06-10 AZ DASPK http://identifiers.org/doi/10.1137/0915088 DDASPK SDASPK differential algebraic system solver with Krylov preconditioning In DASPK, we have combined the time-stepping methods of DASSL [http://identifiers.org/biomodels.kisao/KISAO_0000255] with preconditioned iterative method GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000386], for solving large-scale systems of DAEs of the form F(t, y, y') = 0, where F, y, y' are N-dimensional vectors, and a consistent set of initial conditions y(t0) = y0, y'(t0) = y'0 is given. DASPK is written in Fortran. http://identifiers.org/doi/10.1137/0915088 Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 6 (November 1994), 1467-1488. DDASPK NARROW SDASPK NARROW differential algebraic system solver with Krylov preconditioning EXACT 2011-06-10 AZ DASSL http://www.nea.fr/abs/html/nesc9918.html DDASSL SDASSL differential algebraic system solver DASSL is designed for the numerical solution of implicit systems of differential/algebraic equations written in the form F(t,y,y')=0, where F, y, and y' are vectors, and initial values for y and y' are given. http://www.nea.fr/abs/html/nesc9918.html Linda R. Petzold: A Description of DASSL: A Differential/Algebraic System Solver, SAND82-8637 (September 1982). DDASSL NARROW SDASSL NARROW differential algebraic system solver EXACT 2011-06-10 AZ conjugate gradient method urn:issn:0091-0635 CG Conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too large to be handled by direct methods. Such systems often arise when numerically solving partial differential equations. urn:issn:0091-0635 M. R. Hestenes and E. L. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49 (1952), p. 409. CG EXACT 2011-06-10 AZ biconjugate gradient method http://identifiers.org/doi/10.1007/BFb0080109 BCG Bi-CG BiCG The biconjugate gradient method provides a generalization of conjugate gradient method [http://identifiers.org/biomodels.kisao/KISAO_0000357] to non-symmetric matrices. http://identifiers.org/doi/10.1007/BFb0080109 Fletcher, R. (1976). "Conjugate gradient methods for indefinite systems". Numerical Analysis. Lecture Notes in Mathematics (Springer Berlin / Heidelberg) 506: 73-89. BCG EXACT Bi-CG EXACT BiCG EXACT 2011-06-13 AZ implicit-state Doob-Gillespie algorithm http://identifiers.org/isbn/3-540-76636-7 978-3-540-76636-0 The algorithm uses a representation of the system together with a super-approximation of its ‘event horizon’ (all events that may happen next), and a specific correction scheme to obtain exact timings. Being completely local and not based on any kind of enumeration, this algorithm has a per event time cost which is independent of (i) the size of the set of generable species (which can even be infinite), and (ii) independent of the size of the system (ie, the number of agent instances). The algorithm can be refined, using concepts derived from the classical notion of causality, so that in addition to the above one also has that the even cost is depending (iii) only logarithmically on the size of the model (ie, the number of rules). http://identifiers.org/isbn/3-540-76636-7 978-3-540-76636-0 V. Danos, J. Feret, W. Fontana, and J. Krivine. 2007. Scalable simulation of cellular signalling networks. In Proceedings of the 5th Asian conference on Programming languages and systems (APLAS'07), Zhong Shao (Ed.). Springer-Verlag, Berlin, Heidelberg, 139-157. 2011-06-13 AZ true rule-based simulation method Rule-based models provide a powerful alternative to approaches that require explicit enumeration of all possible molecular species of a system. Such models consist of formal rules governing interactive behaviour. Rule-based simulation methods simulate such models. 2011-06-16 AZ GSL Adams predictor-corrector method urn:asin:B0000EFQ8B The combination of evaluating a single explicit integration method ('Adams-Bashforth method' [http://identifiers.org/biomodels.kisao/KISAO_0000279]) (the predictor step) in order to provide a good initial guess for the successive evaluation of an implicit method ('Adams-Moulton method' [http://identifiers.org/biomodels.kisao/KISAO_0000280]) (the corrector step) using iteration. urn:asin:B0000EFQ8B Moulton, Forest R. (1926), New methods in exterior ballistics, University of Chicago Press. 2011-06-16 AZ Mathematica NDSolve method http://identifiers.org/isbn/978-1-57955-058-5 The Mathematica computation system function NDSolve is a general numerical differential equation solver. It can handle a wide range of ordinary differential equations as well as some partial differential equations. NDSolve can also solve some differential-algebraic equations, which are typically a mix of differential and algebraic equations. http://identifiers.org/isbn/978-1-57955-058-5 Advanced Numerical Differential Equation Solving in Mathematica, Wolfram Mathematica Tutorial Collection, 2008. 2011-06-16 AZ symplecticness http://identifiers.org/doi/10.1017/S0962492900002282 Roughly speaking, ‘symplecticness’ is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness (phase volume preservation). http://identifiers.org/doi/10.1017/S0962492900002282 J. M. Sanz-Serna (1992). Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica, 1, pp 243-286 2 2011-06-16 AZ partitioned Runge-Kutta method http://identifiers.org/doi/10.1007/BF01389456 PRK SPRK symplectic partitioned Runge-Kutta method If a Hamiltonian system possesses a natural partitioning, it is possible to integrate its certain components using one Runge-Kutta method and other components using a different Runge-Kutta method. The overall s-stage scheme is called a partitioned Runge-Kutta method. http://identifiers.org/doi/10.1007/BF01389456 P. Rentrop (1985): Partitioned Runge-Kutta methods with stiffness detection and stepsize control. Numer. Math., Vol. 47, p.545-564. PRK EXACT SPRK EXACT symplectic partitioned Runge-Kutta method EXACT 2011-06-27 AZ true partial differential equation discretization method A method which solves partial differential equations by discretizing them, i.e. approximating them by equations that involve a finite number of unknowns. 2011-06-29 AZ true type of problem A characteristic describing the type of the problems which can be solved by the algorithm. AZ stochastic differential equation problem SDE problem SDE problem EXACT AZ partial differential equation problem PDE problem PDE problem EXACT AZ differential-algebraic equation problem DAE DAE EXACT AZ ordinary differential equation problem ODE problem ODE problem EXACT AZ delay differential equation problem DDE problem 2011-07-19 AZ linearity of equation Linear differential equations are of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function of the same nature as y. 2011-06-30 AZ true one-step method http://identifiers.org/isbn/978-3-540-56670-0 A numerical method for differential equations which uses one starting value at each step. http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag 2 2011-06-30 AZ GSL implicit midpoint rule http://identifiers.org/isbn/978-3-540-56670-0 implicit Gaussian second order Runge-Kutta method The implicit midpoint rule is a second-order case of the more general implicit s-stage Runge-Kutta methods [http://identifiers.org/biomodels.kisao/KISAO_0000064 and (http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000240)]. http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag. implicit Gaussian second order Runge-Kutta method EXACT 2011-07-01 AZ Bulirsch-Stoer algorithm http://identifiers.org/doi/10.1007/BF02165234 GBS Gragg-Bulirsch-Stoer algorithm The Bulirsch-Stoer method is an adaptive method which uses Gragg's modified midpoint method [http://identifiers.org/biomodels.kisao/KISAO_0000382] to estimate the solution of an initial value problem for various step sizes. The estimates are fit to a "diagonal" rational function or a polynomial as a function of the step size and the limit as the step size tends to zero is taken as the final estimate. http://identifiers.org/doi/10.1007/BF02165234 Bulirsch R. and Stoer J. (1966) Numerical treatment of ordinary differential equations by extrapolation methods, Numerische Mathematik, Volume 8, Number 1, 1-13. GBS EXACT Gragg-Bulirsch-Stoer algorithm EXACT 2011-07-01 AZ true Richardson extrapolation based method http://identifiers.org/doi/10.1098/rsta.1911.0009 A method based on ideas of Richardson extrapolation, which is a process for obtaining increased accuracy in a discretized approximation by extrapolating results from coarse discretizations to an arbitrarily fine one. http://identifiers.org/doi/10.1098/rsta.1911.0009 Richardson, L. F. (1911). The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London, Series A 210 (459-470): 307-357. 2 2011-07-01 AZ midpoint method http://identifiers.org/isbn/978-3-540-56670-0 The midpoint method is an explicit method for approximating the solution of the initial value problem y' = f(x,y); y(x0) = y0 at x for a given step size h. For the midpoint method the derivative of y(x) is approximated by the symmetric difference y'(x) = ( y(x+h) - y(x-h) ) / 2h + O(h2). http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag. 2011-07-01 AZ modified midpoint method http://identifiers.org/isbn/052143064X Gragg's method Gragg's modified midpoint method The modified midpoint method is globally a second order method for approximating the solution of the initial value problem y' = f(x, y), y(x0) = y0, which advances a vector of dependent variables y(x) from a point x to a point x + H by a sequence of n substeps each of size h, h = H/n. http://identifiers.org/isbn/052143064X Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001). Gragg's method EXACT 2011-07-01 AZ GSL Bader-Deuflhard method http://identifiers.org/doi/10.1007/BF01418331 The Bader-Deuflhard method is an extrapolation method based on a semi-implicit discretization [http://identifiers.org/biomodels.kisao/KISAO_0000387]. It is a generalization of the Bulirsch-Stoer algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000379] for solving ordinary differential equations. http://identifiers.org/doi/10.1007/BF01418331 Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations." Numer. Math. 41, 373-398, 1983. 2011-07-01 AZ semi-implicit midpoint rule http://identifiers.org/doi/10.1007/BF01418331 A semi-implicit version of the midpoint method that has an even error series [http://identifiers.org/biomodels.kisao/KISAO_0000381]. http://identifiers.org/doi/10.1007/BF01418331 Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations." Numer. Math. 41, 373-398, 1983. 2011-07-18 AZ scaled preconditioned generalized minimal residual method http://identifiers.org/doi/10.1137/0915088 SPGMR A scaled preconditioned version of 'generalized minimal residual algorithm' [http://identifiers.org/biomodels.kisao/KISAO_0000353]. For linear system Ax = b a preconditioner matrix P that approximates A is sought, for which linear system Px = b can be solved easily. Preconditioning is applied on the left only. Scaling is done using diagonal matrix D whose diagonal elements are weights w^i = rtol|y^i| +atol^i, where rtol is 'relative tolerance' [http://identifiers.org/biomodels.kisao/KISAO_0000209] and atol is 'absolute tolerance' [http://identifiers.org/biomodels.kisao/KISAO_0000211]. http://identifiers.org/doi/10.1137/0915088 Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 6 (November 1994), 1467-1488. SPGMR EXACT 2011-07-18 AZ minimal residual method http://identifiers.org/doi/10.1137/0712047 MINRES The 'minimal residual method' is an algorithm for the numerical solution of indefinite symmertic systems of linear equations. http://identifiers.org/doi/10.1137/0712047 Paige C.C., Saunders M.A. Solution of sparse indefinite systems of linear equations, SIAM Journal on Numerical Analysis (1975) Volume: 12, Issue: 4, Publisher: SIAM, Pages: 617-629. MINRES EXACT 2011-07-18 AZ quasi-minimal residual method http://identifiers.org/doi/10.1007/BF01385726 QMR The QMR algorithm is a robust iterative solver for general nonsingular non-Hermitian linear systems. The method uses a robust implementation of the look-ahead Lanczos algorithm to generate basis vectors for the Krylov subspaces Kn(r0, A). The QMR iterates are characterized by a quasi-minimal residual property over Kn(r0, A). http://identifiers.org/doi/10.1007/BF01385726 Roland W. Freund and Noël M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. NUMERISCHE MATHEMATIK Volume 60, Number 1, 315-339. QMR EXACT biconjugate gradient stabilized method http://identifiers.org/doi/10.1137/0913035 Bi-CGSTAB BiCGSTAB An iterative method for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) [http://identifiers.org/biomodels.kisao/KISAO_0000358] and has faster and smoother convergence than the original BiCG. http://identifiers.org/doi/10.1137/0913035 Van der Vorst, H. A. (1992). Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM Journal on Scientific and Statistical Computing 13: 631-644. Bi-CGSTAB EXACT BiCGSTAB EXACT 2011-07-18 AZ ingenious conjugate gradients-squared method http://identifiers.org/doi/10.1137/0910004 CGS A Lanczos-type method for nonsymmetric sparse linear systems. The method is based on a polynomial variant of the conjugate gradients algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000357]. Although related to the so-called bi-conjugate gradients (Bi-CG) algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000358], it does not involve adjoint matrix-vector multiplications, and the expected convergence rate is about twice that of the Bi-CG algorithm. http://identifiers.org/doi/10.1137/0910004 Peter Sonneveld. 1989. CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10, 1 (January 1989), 36-52. CGS EXACT 2011-07-19 AZ quasi-minimal residual variant of biconjugate gradient stabilized method http://identifiers.org/doi/10.1137/0915023 QMRCGSTAB QMRCGSTAB is a quasi-minimal residual (QMR) variant of the Bi-CGSTAB algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000394] of van der Vorst for solving nonsymmetric linear systems. The motivation for the QMR variant is to obtain smoother convergence behavior of the underlying method. http://identifiers.org/doi/10.1137/0915023 T. F. Chan, E. Gallopoulos, V. Simoncini, T. Szeto, and C. H. Tong. 1994. A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems. SIAM J. Sci. Comput. 15, 2 (March 1994), 338-347. QMRCGSTAB EXACT 2011-07-19 AZ true improved biconjugate gradient method An 'improved biconjugate gradient method' branch contains algorithms which can be viewed as improvements over some of drawbacks of BCG [http://identifiers.org/biomodels.kisao/KISAO_0000358], such as (1) the need for matrix-vector multiplications with A^T (which can be inconvenient as well as doubling the number of matrix-vector multiplications compared to CG [http://identifiers.org/biomodels.kisao/KISAO_0000357] for each increase in the degree of the underlying Krylov subspace), (2) the possibility of breakdowns and (3) erratic convergence behavior. 2011-07-19 AZ transpose-free quasi-minimal residual algorithm http://identifiers.org/doi/10.1137/0914029 TFQMR A version of CGS [http://identifiers.org/biomodels.kisao/KISAO_0000393] which 'quasi-minimizes' the residual in the space spanned by the vectors generated by the CGS iteration. http://identifiers.org/doi/10.1137/0914029 Roland W. Freund. 1993. A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Comput. 14, 2 (March 1993), 470-482. TFQMR EXACT 2011-07-19 AZ preconditioning technique http://identifiers.org/isbn/0898715342 Preconditioning is simply a means of transforming the original linear system into one which has the same solution, but which is likely to be easier to solve with an iterative solver. http://identifiers.org/isbn/0898715342 Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM. 2011-07-19 AZ true iterative method for linear system http://identifiers.org/isbn/0898715342 http://identifiers.org/isbn/0898715342 Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM. 2011-07-19 AZ homogeneousness of equation Homogeneous equations are of the form Ly = 0, where the differential operator L is a linear operator and y is the unknown function. 2011-07-19 AZ symmetricity of matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. 2011-07-19 AZ true type of differential equation 2012-01-17 AZ true Requested by Frank T. Bergmann on Sunday, November 27, 2011 4:45:30 PM. steady state method A method looking for a steady state of a dynamic system. 2012-01-18 AZ Requested by Frank T. Bergmann on Sunday, November 27, 2011 4:45:30 PM. Newton-type method http://identifiers.org/isbn/9783540210993 A method which attacks the solution of a nonlinear problem F(x) = 0 by solving a sequence of liner problems of the same kind. The solution of the system F(x)=0 can be interpreted as a steady state of a dynamic system x'(t)=F(x(t)). The Newton approach will only work if the fixed point [http://identifiers.org/biomodels.teddy/TEDDY_0000086] of the dinamic system is attractive [http://identifiers.org/biomodels.teddy/TEDDY_0000094]. http://identifiers.org/isbn/9783540210993 Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004. 2012-01-18 AZ ordinary Newton method http://identifiers.org/isbn/9783540210993 A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which solves the general nonlinear problem F(x)=0 by applying successive linearization F'(x[k])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,... http://identifiers.org/isbn/9783540210993 Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004. 2012-01-18 AZ simlified Newton method http://identifiers.org/isbn/9783540210993 A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which is characterized by keeping the initial derivative throughout the whole iteration: F'(x)deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,... http://identifiers.org/isbn/9783540210993 Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004. 2012-01-18 AZ Newton-like method http://identifiers.org/isbn/9783540210993 A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which is characterized by the fact that, in finite dimension, the Jacodian matrices are either replaced by some fixed 'close by' Jacobian F'(z) with z not equal to the initial guess x, or by some approximation so that: M'(x)deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,... http://identifiers.org/isbn/9783540210993 Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004. 2012-01-18 AZ inexact Newton method http://identifiers.org/isbn/9783540210993 iterative Newton method truncated Newton method For extremely large scale nonlinear problems the arising linear systems for the Newton corrections can no longer be solved directly ('exactly'), but must be solved iterativly ('inexactly) - which gives the name inexact Newton methods. The whole scheme then consists of an inner iteration (at Newton step k): F'(x[k])deltaxi[k]=-F(x[k])+ri[k], k=0,1,... xi[k+1]=x[k]+deltaxi[k], i=0,1,..,imax[k] in terms of residuals ri[k] and an outer iteration where, given x, the iterates are defined as x[k+1]=xi[k+1] for i=imax[k], k=0,1,... http://identifiers.org/isbn/9783540210993 Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004. iterative Newton method EXACT truncated Newton method EXACT 2012-01-18 AZ exact Newton method http://identifiers.org/isbn/9783540210993 direct Newton method Any of the finite dimensional Newton-type methods [http://identifiers.org/biomodels.kisao/KISAO_0000408] requires the numerical solution of the linear equations F'(x[k])deltax[k]=-F(x[k]). Whenever direct elimination methods are applicable, we speak of exact Newton methods. http://identifiers.org/isbn/9783540210993 Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004. 2012-01-18 AZ maximum number of steps maximum steps The limit on number of internal steps before an output point. 1 1 2012-01-18 AZ Requested by Kristin Tøndel on Thursday, October 13, 2011 10:46:04 AM. partial least squares regression method http://identifiers.org/biomodels.kisao/10.1007/BFb0062108 PLSR PLSR method Multivariate regression method based on estimated latent variables. Related to Principal Component Analysis (PCA) and Principal Component Regression (PCR). http://identifiers.org/biomodels.kisao/10.1007/BFb0062108 Wold S, Martens H, Wold H (1983) The multivariate calibration method in chemistry solved by the PLS method. In: Lecture notes in Mathematics, Matrix Pencils. Heidelberg: Springer-Verlag. pp. 286-293. PLSR EXACT PLSR method EXACT 1 2012-01-18 AZ Requested by Kristin Tøndel on Thursday, October 13, 2011 11:13:17 AM. hierarchical cluster-based partial least squares regression method http://identifiers.org/doi/10.1186/1752-0509-5-90 HC-PLSR Multivariate regression method based on separating the observations into clusters and generating Partial Least Squares Regression (PLSR) [http://identifiers.org/biomodels.kisao/KISAO_0000416] models within each cluster. This local regression analysis is suitable for very non-linear systems. PLSR is a regression method based on estimated latent variables, related to Principal Component Analysis (PCA) and Principal Component Regression (PCR). Hierarchical cluster-based partial least squares regression method uses fuzzy C-means clustering, PLSR and Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA) or Naive Bayes for classification of new observations to be predicted. http://identifiers.org/doi/10.1186/1752-0509-5-90 Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90. 1 2012-01-18 AZ Requested by Kristin Tøndel on Thursday, October 13, 2011 11:33:06 AM. N-way partial least squares regression method http://identifiers.org/doi/10.1002/(SICI)1099-128X(199601)10:1<47::AID-CEM400>3.0.CO;2-C http://www.models.life.ku.dk/nwaytoolbox N-PLS N-way PLSR N-way partial least squares method Multivariate regression method that can be used on N-way data. Based on Partial Least Squares Regression (PLSR) [http://identifiers.org/biomodels.kisao/KISAO_0000416], which is a regression method based on estimated latent variables. PLSR is related to Principal Component Analysis (PCA) and Principal Component Regression (PCR). http://identifiers.org/doi/10.1002/(SICI)1099-128X(199601)10:1<47::AID-CEM400>3.0.CO;2-C Bro R (1996) Multiway calibration. Multilinear PLS. J. Chemometrics 10: 47-6. http://www.models.life.ku.dk/nwaytoolbox N-way toolbox website N-PLS EXACT N-way PLSR EXACT N-way partial least squares method EXACT 2012-01-18 AZ true metamodelling method http://identifiers.org/doi/10.1186/1752-0509-5-90 Deterministic dynamic models of complex biological systems contain a large number of parameters and state variables, related through nonlinear differential equations with various types of feedback. A metamodel of such a dynamic model is a statistical approximation model that maps variation in parameters and initial conditions (inputs) to variation in features of the trajectories of the state variables (outputs) throughout the entire biologically relevant input space. http://identifiers.org/doi/10.1186/1752-0509-5-90 Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90. 2012-01-18 AZ number of partial least squares components http://identifiers.org/biomodels.kisao/10.1007/BFb0062108 Parameter used by 'partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000416] describing number of PLS components to include in the regression analysis. http://identifiers.org/biomodels.kisao/10.1007/BFb0062108 Wold S, Martens H, Wold H (1983) The multivariate calibration method in chemistry solved by the PLS method. In: Lecture notes in Mathematics, Matrix Pencils. Heidelberg: Springer-Verlag. pp. 286-293. 2012-01-18 AZ type of validation Parameter of 'partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000416] describing how validation is performed. Possible values include cross-validation and test set validation. 2012-01-18 AZ number of N-way partial least squares regression factors number of factors Parameter of 'N-way partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000369] describing the number of factors to compute. number of factors EXACT 1 2012-01-18 AZ true partial least squares regression-like method Method for building regression models between independent and dependent variables. 2012-01-18 AZ mean-centring of variables http://identifiers.org/doi/10.1186/1752-0509-5-90 A boolean parameter of the 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417] specifying whether the variables were mean-centred prior to the regression analysis. http://identifiers.org/doi/10.1186/1752-0509-5-90 Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90. 2012-01-18 AZ standardising of variables http://identifiers.org/doi/10.1186/1752-0509-5-90 A boolean parameter of the 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417] specifying whether the variables were standardised (divided by their standard deviations) prior to the regression analysis. http://identifiers.org/doi/10.1186/1752-0509-5-90 Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90. 2012-01-18 AZ number of clusters http://identifiers.org/doi/10.1186/1752-0509-5-90 Parameter specifying the number of clusters used by C-means algorithm. http://identifiers.org/doi/10.1186/1752-0509-5-90 Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90. 2012-01-18 AZ matrix for clusterization http://identifiers.org/doi/10.1186/1752-0509-5-90 A matrix to do the clustering in 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417]. http://identifiers.org/doi/10.1186/1752-0509-5-90 Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90. AZ true clusterization parameter Parameter used by algorithms performing clusterization. AZ true variables preprocessing parameter 1 2012-05-24 AZ true IDA-like method http://identifiers.org/doi/10.1145/1089014.1089020 Solves real differential-algebraic systems in N-space, in the general form F(t,y,y')=0, y(t0)=y0, y'(t0)=y'0. At each step, a Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408] leads to linear systems Jx=b, which are solved by one of five methods - two direct (dense or band; serial version only) and three Krylov [http://identifiers.org/biomodels.kisao/KISAO_0000354] (GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000353], BiCGStab [http://identifiers.org/biomodels.kisao/KISAO_0000392], or TFQMR [http://identifiers.org/biomodels.kisao/KISAO_0000396]). http://identifiers.org/doi/10.1145/1089014.1089020 Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005. 1 2012-05-24 AZ true CVODE-like method http://identifiers.org/doi/10.1145/1089014.1089020 Solves ODE initial value problems, in real N-space, written as y'=f(t,y), y(t0)=y0. It is capable for stiff and non-stiff systems and uses two different linear multi-step methods, namely the Adam-Moulton [http://identifiers.org/biomodels.kisao/KISAO_0000280] method and the backward differentiation formula [http://identifiers.org/biomodels.kisao/KISAO_0000288]. http://identifiers.org/doi/10.1145/1089014.1089020 Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005. 2012-09-25 AZ Higham-Hall method http://identifiers.org/doi/10.1016/0377-0427(90)90192-3 RK5(4)7FEql The equilibrium theory of Hall and Higham (1988) can be used to determine whether a Runge-Kutta algorithm will perform smoothly when stability restricts the stepsize. Higham-Hall method is a fifth order embedded Runge-Kutta method [http://identifiers.org/biomodels.kisao/KISAO_0000302], which behaves smoothly with respect to the standard type of stepsize controllers. http://identifiers.org/doi/10.1016/0377-0427(90)90192-3 Higham, D.J. and Hall, G. (1990) Embedded Runge-Kutta formulae with stable equilibrium states. Journal of Computational and Applied Mathematics, 29 (1). pp. 25-33. 4 5 2012-09-26 AZ true embedded Runge-Kutta 5(4) method RK5(4) An embedded Runge-Kutta integrator of order 5(4). 5 8 2012-09-26 AZ Dormand-Prince 8(5,3) method http://identifiers.org/isbn/978-3-540-56670-0 This method is based on an 8(6) method by Dormand and Prince (i.e. order 8 for the integration and order 6 for error estimation) modified by Hairer and Wanner to use a 5th order error estimator with 3rd order correction. http://identifiers.org/isbn/978-3-540-56670-0 Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag 2012-11-29 AZ Requested by Frank T. Bergmann on Thursday, November 29, 2012 9:51:58 AM. flux balance analysis http://identifiers.org/doi/10.1038/nbt.1614 FBA Flux balance analysis is a mathematical approach for analyzing the flow of metabolites through a metabolic network. http://identifiers.org/doi/10.1038/nbt.1614 Jeffrey D Orth, Ines Thiele and Bernhard O Palsson (2010) What is flux balance analysis? Nature Biotechnology 28, 245–248 1 2013-01-28 AZ Requested by Mark Moeller on Friday, January 25, 2013 11:11:30 AM. COAST http://identifiers.org/pubmed/17100426 controllable approximative stochastic reaction algorithm An approximative algorithm for stochastic simulations of chemical reaction systems based on three different modeling levels: for small numbers of particles an exact [http://identifiers.org/biomodels.kisao/KISAO_0000236] stochastic [http://identifiers.org/biomodels.kisao/KISAO_0000104] model; for intermediate numbers an approximative [http://identifiers.org/biomodels.kisao/KISAO_0000237], but computationally more efficient stochastic [http://identifiers.org/biomodels.kisao/KISAO_0000104] model based on discrete Gaussian distributions; and for large numbers the deterministic [http://identifiers.org/biomodels.kisao/KISAO_0000103] reaction kinetics. In every simulation time step, the subdivision of the reaction channels into the three different modeling levels is done automatically, where all approximations applied can be controlled by a single error parameter for which an appropriate value can easily be found. http://identifiers.org/pubmed/17100426 Holger Wagner, Mark Moeller, and Klaus Prank (2006) COAST: Controllable approximative stochastic reaction algorithm. J. Chem. Phys. 125, 174104 2013-01-28 AZ logical model simulation method http://www.identifiers.org/pubmed/22141422 Qualitative (logical) models specify the evolution rules of their components. In each state a number of transitions are enabled. A 'logical model simulation method' guides the choice of the transitions processed at each step. http://www.identifiers.org/pubmed/22141422 Daniel Machado et al. Modeling formalisms in Systems Biology. AMB Express 2011, 1:45 2013-01-28 AZ synchronous logical model simulation method Qualitative (logical) models specify the evolution rules of their components. In the case of a synchronous updating all enabled transitions are processed simultaneously. 2013-01-28 AZ asynchronous logical model simulation method Qualitative (logical) models specify the evolution rules of their components. It the case of an asynchronous updating all enabled transitions are performed independently: a state has as many successors as the number of transitions enabled in this state. An 'asynchronous logical model simulation method' specifies a rule to guide the choice of a unique transition at each step (for example random). 2013-01-28 AZ true type of updating policy A rule to guide the choice of a unique transition at each step used by an 'asynchronous logical model simulation method' [http://identifiers.org/biomodels.kisao/KISAO_0000450]. 2013-01-28 AZ random updating policy An updating policy that chooses a transition randomly. 2013-01-28 AZ ordered updating policy An updating policy that chooses a transition in a definite way. 2013-01-28 AZ constant updating policy An updating policy that chooses a transition in a constant way. 2013-01-28 AZ prioritized updating policy An updating policy that chooses a transition in a prioritized way. 2013-07-05 AZ Requested by Andrew Miller on Tuesday, June 4, 2013 2:33:51 AM. maximum step size An upper limit, in the units of the bound variable over which a numerical integration is being performed, that an adaptive step size numerical integration algorithm should take. 2014-04-25 AZ Requested by Andrzej Kierzek on Thursday, April 24, 2014 12:40:35 PM Used to simulate systems involving reactions with propensities varying by many orders of magnitude. maximal timestep method http://identifiers.org/pubmed/14990466 Hybrid simulation algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000352] combining Gibson and Bruck algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000027] with the Gillespie tau-leap method [http://www.biomodels.net/kisao/KISAO#KISAO_0000039]. http://identifiers.org/pubmed/14990466 Puchalka J, Kierzek AM. Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks. Biophys J. 2004 Mar;86(3):1357-72. 2014-04-25 AZ maximal timestep http://identifiers.org/pubmed/14990466 Key parameter of the 'maximal timestep method' [http://www.biomodels.net/kisao/KISAO#KISAO_0000468]. If Gillespie [http://www.biomodels.net/kisao/KISAO#KISAO_0000027] waiting time is longer than maximal timestep, slow reaction is not fired and tau-leap [http://www.biomodels.net/kisao/KISAO#KISAO_0000039] step is executed for fast reactions. Otherwise, slow reaction is fired and tau-leap [http://www.biomodels.net/kisao/KISAO#KISAO_0000039] is executed with shorter time step. http://identifiers.org/pubmed/14990466 Puchalka J, Kierzek AM. Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks. Biophys J. 2004 Mar;86(3):1357-72. 2015-04-23 AZ true optimization algorithm http://en.wikipedia.org/wiki/Mathematical_optimization#Optimization_algorithms http://identifiers.org/isbn/0691102872 optimization method An optimization algorithm tries to find the minumum or maximum of an arbitrary function. It takes a function of one or several variables and determines the values for the variables so that the function value is optimal. http://identifiers.org/isbn/0691102872 Brinkhuis J, Tikhomirov V. Optimization: Insights and Applications Princeton Series in Applied Mathematics (illustrated), Princeton University Press, 2005. 2015-04-23 AZ local optimization algorithm local optimiation method A local optimization algorithm is an optimisation algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000470] that only finds a local optimum of a function. If several optima exist for the function, it usually depends on the starting values for the variables which optimum is found. 2015-04-23 AZ global optimization algorithm global optimization method A global optimization algorithm is an optimization algorithm [http://www.biomodels.net/kisao/KISAO#KISAO_0000470] that tries to find the global optimum of a function. If a function has several minima/maxima with in the allowed range of variable values, the global minimum/maximum is the one with the smallest/largest function value. 2015-04-23 AZ Bayesian inference algorithm http://en.wikipedia.org/wiki/Bayesian_inference http://identifiers.org/isbn/111803144X A bayesian inference algorithm calculates a posterior probability distribution from a prior probability distribution and some additional evidence in the form of a likelyhood function. http://identifiers.org/isbn/111803144X George E. P. Box, George C. Tiao. Bayesian Inference in Statistical Analysis, Volume 40 of Wiley Classics Library. John Wiley & Sons (reprint, revised), 2011. 2015-09-10 AZ OpenCOR integration method http://identifiers.org/doi/10.3389/fphys.2015.00026 the integration method used by the solver http://identifiers.org/doi/10.3389/fphys.2015.00026 Garny A and Hunter PJ (2015) OpenCOR: a modular and interoperable approach to computational biology. Front. Physiol. 6:26 2015-09-10 AZ OpenCOR Requested in https://sourceforge.net/p/kisao/feature-requests/12/ iteration type http://identifiers.org/doi/10.3389/fphys.2015.00026 the type of iteration used by the solver http://identifiers.org/doi/10.3389/fphys.2015.00026 Garny A and Hunter PJ (2015) OpenCOR: a modular and interoperable approach to computational biology. Front. Physiol. 6:26 2015-09-10 AZ OpenCOR Requested in https://sourceforge.net/p/kisao/feature-requests/12/ linear solver http://identifiers.org/doi/10.3389/fphys.2015.00026 the linear solver used by the solver during a Newton iteration http://identifiers.org/doi/10.3389/fphys.2015.00026 Garny A and Hunter PJ (2015) OpenCOR: a modular and interoperable approach to computational biology. Front. Physiol. 6:26 2015-09-10 AZ OpenCOR Requested in https://sourceforge.net/p/kisao/feature-requests/12/ preconditioner http://identifiers.org/doi/10.3389/fphys.2015.00026 the preconditioner, if any, used by the solver when using a GMRES, BiCGStab or TFQMR linear solver during a Newton iteration. http://identifiers.org/doi/10.3389/fphys.2015.00026 Garny A and Hunter PJ (2015) OpenCOR: a modular and interoperable approach to computational biology. Front. Physiol. 6:26 2015-09-10 AZ OpenCOR Requested in https://sourceforge.net/p/kisao/feature-requests/12/ upper half-bandwidth http://identifiers.org/doi/10.3389/fphys.2015.00026 the upper half-bandwidth value used by the Banded linear solver or preconditioner (a value between 0 and n-1 with n the number of ODEs/DAEs in the model). http://identifiers.org/doi/10.3389/fphys.2015.00026 Garny A and Hunter PJ (2015) OpenCOR: a modular and interoperable approach to computational biology. Front. Physiol. 6:26 2015-09-10 AZ OpenCOR Requested in https://sourceforge.net/p/kisao/feature-requests/12/ lower half-bandwidth http://identifiers.org/doi/10.3389/fphys.2015.00026 the lower half-bandwidth value used by the Banded linear solver or preconditioner (a value between 0 and n-1 with n the number of ODEs/DAEs in the model). http://identifiers.org/doi/10.3389/fphys.2015.00026 Garny A and Hunter PJ (2015) OpenCOR: a modular and interoperable approach to computational biology. Front. Physiol. 6:26 2015-09-10 AZ OpenCOR Requested in https://sourceforge.net/p/kisao/feature-requests/12/ interpolate solution http://identifiers.org/doi/10.3389/fphys.2015.00026 whether the solver returns an interpolated solution. http://identifiers.org/doi/10.3389/fphys.2015.00026 Garny A and Hunter PJ (2015) OpenCOR: a modular and interoperable approach to computational biology. Front. Physiol. 6:26 half-bandwith parameter the parameter related to the half-bandwidth value used by the Banded linear solver or preconditioner. 2015-09-10 AZ Requested in https://sourceforge.net/p/kisao/feature-requests/13/ step size the size of every step of the algorithm AZ Roadrunner maximum order Maximum order of method. For example, in Roadrunner it can be used for two parameters that one can set for deterministic runs: 'maximum_bdf_order' and 'maximum_adams_order'. AZ minimum step size A lower limit, in the units of the bound variable over which a numerical integration is being performed, that a numerical integration algorithm with variable step size should take. AZ maximum iterations iteration limit For algorithms that iterate to a solution (steady state finders in particular), a limit on the number of iterations that should be performed. Roadrunner The damping factor is a variable for at least some steady state algorithms: roadrunner allows you to set the minimum value for this. minimum damping seed random seed Random seed of a stochastic algorithm. Setting it allows one to reproduce their results while running the same algorithm on the same computer. discrete event simulation algorithm http://identifiers.org/isbn/0387333320 DES Discrete Event Simulation algorithm refers to the simulation of systems whose (countable) discrete states change over time and are event-driven. http://identifiers.org/isbn/0387333320 C.G. Cassandras & S. Lafortune "Introduction to Discrete Event Systems", Chapt 1 2nd Ed 2008. Springer asynchronous updating policy An updating policy where all enabled transitions (events) occur independently. Thus a state has as many successors as the number of transitions enabled in this state. synchronous updating policy An updating policy where all enabled transitions occur simultaneously. Thus a state will have at most one successor. fully asynchronous updating policy An updating policy where all enabled transitions occur either independently or (partially) simultaneously. (i.e. considering all possible combinations of enabled transitions). Thus a state has as many successors as the number of combinations of transitions enabled in this state. random asynchronous updating policy An updating policy where a single transition is picked randomly from the set of transitions enabled in this state. Thus a state will have at most one successor. SUNDIALS CVODES is a superset of CVODE [http://identifiers.org/biomodels.kisao/KISAO_0000019] and hence all options available to CVODE (with the exception of the FCVODE interface module) are also available for CVODES. Both integration methods (Adams-Moulton [http://identifiers.org/biomodels.kisao/KISAO_0000280] and BDF [http://identifiers.org/biomodels.kisao/KISAO_0000288]) and the corresponding nonlinear iteration methods, as well as all linear solver and preconditioner modules, are available for the integration of the original ODEs, the sensitivity systems, or the adjoint system. CVODES http://computation.llnl.gov/projects/sundials/cvodes CVODES is a solver for stiff and nonstiff ODE systems (initial value problem) given in explicit form y’ = f(t,y,p) with sensitivity analysis capabilities (both forward and adjoint modes). 2017-01-09 AZ KLU http://identifiers.org/doi/10.1145/1824801.1824814 "Clark Kent" LU factorization algorithm KLU is a software package and an algorithm for solving sparse unsymmetric linear systems of equations that arise in circuit simulation applications. It relies on a permutation to Block Triangular Form (BTF), several methods for finding a fill-reducing ordering (variants of approximate minimum degree and nested dissection), and Gilbert/Peierls’ sparse left-looking LU factorization algorithm to factorize each block. The package is written in C and includes a MATLAB interface. http://identifiers.org/doi/10.1145/1824801.1824814 Davis, Timothy A., and Ekanathan Palamadai Natarajan. "Algorithm 907: KLU, a direct sparse solver for circuit simulation problems." ACM Transactions on Mathematical Software (TOMS) 37.3 (2010): 36. 2017-07-18 AZ Requested in https://sourceforge.net/p/kisao/feature-requests/31/ number of runs The number of runs that a simulation should perform. Typically used to specify the number of runs for a stochastic simulation. 2017-09-12 AZ DFBAlab Ticket 32 dynamic flux balance analysis http://identifiers.org/doi/10.1186/s12859-014-0409-8 DFBA Dynamic flux balance analysis (DFBA) enables the simulation of dynamic biological systems by assuming organisms reach steady state rapidly in response to changes in the extracellular environment. DFBA couples flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model approaches with dynamic model approaches. http://identifiers.org/doi/10.1186/s12859-014-0409-8 DFBAlab: a fast and reliable MATLAB code for dynamic flux balance analysis. Gomez JA, Höffner K, Barton PI. BMC Bioinformatics. 2014 Dec 18;15:409. 2017-09-12 AZ DFBAlab Ticket 32 SOA-DFBA http://identifiers.org/doi/10.1186/s12859-014-0409-8 SOA static optimization approach dynamic flux balance analysis Dynamic Flux Balance Analysis (DFBA) [http://identifiers.org/biomodels.kisao/KISAO_0000499] couples flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model approaches with dynamic model approaches. The static optimization approach (SOA) uses the Euler forward method [http://identifiers.org/biomodels.kisao/KISAO_0000030], solving the embedded LPs at each time step. The FBA fluxes are assumed to be constant during the time step. http://identifiers.org/doi/10.1186/s12859-014-0409-8 DFBAlab: a fast and reliable MATLAB code for dynamic flux balance analysis. Gomez JA, Höffner K, Barton PI. BMC Bioinformatics. 2014 Dec 18;15:409. 2017-09-12 AZ DFBAlab Ticket 32 DOA-DFBA http://identifiers.org/doi/10.1186/s12859-014-0409-8 DOA dynamic optimization approach dynamic flux balance analysis Dynamic Flux Balance Analysis (DFBA) [http://identifiers.org/biomodels.kisao/KISAO_0000499] couples flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model approaches with dynamic model approaches. The dynamic optimization approach (DOA) discretizes the time horizon and optimizes simultaneously over the entire time period of interest by solving a nonlinear programming problem (NLP). http://identifiers.org/doi/10.1186/s12859-014-0409-8 DFBAlab: a fast and reliable MATLAB code for dynamic flux balance analysis. Gomez JA, Höffner K, Barton PI. BMC Bioinformatics. 2014 Dec 18;15:409. 2017-09-12 AZ DFBAlab Ticket 32 DA-DFBA http://identifiers.org/doi/10.1186/s12859-014-0409-8 DA direct approach dynamics flux balance analysis Dynamic Flux Balance Analysis (DFBA) [http://identifiers.org/biomodels.kisao/KISAO_0000499] couples flux balance analysis (FBA) [http://identifiers.org/biomodels.kisao/KISAO_0000437] model approaches with dynamic model approaches. The direct approach (DA) includes the LP solver in the right-hand side evaluator for the ordinary differential equations (ODEs) and takes advantage of reliable implicit ODE integrators with adaptive step size for error control. http://identifiers.org/doi/10.1186/s12859-014-0409-8 DFBAlab: a fast and reliable MATLAB code for dynamic flux balance analysis. Gomez JA, Höffner K, Barton PI. BMC Bioinformatics. 2014 Dec 18;15:409. 2019-01-18 AZ COPASI simulated annealing http://identifiers.org/doi/10.1126/science.220.4598.671 Simulated annealing is an optimization algorithm first proposed by Kirkpatrick et al. and was inspired by statistical mechanics and the way in which perfect crystals are formed. Perfect crystals are formed by first melting the substance of interest, and then cooling it very slowly. At large temperatures the particles vibrate with wide amplitude and this allows a search for global optimum. As the temperature decreases so do the vibrations until the system settles to the global optimum (the perfect crystal). The simulated annealing optimization algorithm uses a similar concept: the objective function is considered a measure of the energy of the system and this is maintained constant for a certain number of iterations (a temperature cycle). In each iteration, the parameters are changed to a nearby location in parameter space and the new objective function value calculated; if it decreased, then the new state is accepted, if it increased then the new state is accepted with a probability that follows a Boltzmann distribution (higher temperature means higher probability of accepting the new state). After a fixed number of iterations, the stopping criterion is checked; if it is not time to stop, then the system's temperature is reduced and the algorithm continues. Simulated annealing is a stochastic algorithm that is guaranteed to converge if ran for an infinite number of iterations. It is one of the most robust global optimization algorithms, although it is also one of the slowest. (Be warned that simulated annealing can run for hours or even days!). http://identifiers.org/doi/10.1126/science.220.4598.671 S. Kirkpatrick, J., C.D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science. 220. 671 - 680. 1983. 2019-01-18 AZ COPASI random search Random search is an optimization method that attempts to find the optimum by testing the objective function's value on a series of combinations of random values of the adjustable parameters. The random values are generated complying with any boundaries selected by the user, furthermore, any combinations of parameter values that do not fulfill constraints on the variables are excluded. This means that the method is capable of handling bounds on the adjustable parameters and fulfilling constraints. For infinite number of iterations this method is guaranteed to find the global optimum of the objective function. In general one is interested in processing a very large number of iterations. 2019-01-18 AZ COPASI particle swarm http://identifiers.org/doi/10.1109/ICNN.1995.488968 The particle swarm optimization method suggested by Kennedy and Eberhart is inspired by a flock of birds or a school of fish searching for food. Each particle has a position Xi and a velocity Vi in the parameter space. Additionally, it remembers its best achieved objective value O and position Mi. Dependent on its own information and the position of its best neighbor (a random subset of particles of the swarm) a new velocity is calculated. With this information the position is updated. http://identifiers.org/doi/10.1109/ICNN.1995.488968 J. Kennedy and R. Eberhart. Particle Swarm Optimization. Proceedings of the Fourth IEEE International Conference on Neural Networks, Perth, Australia. 1942 - 1948. 1995. 2019-01-18 AZ COPASI genetic algorithm http://identifiers.org/isbn/9780750308953 GA The genetic algorithm (GA) is a computational technique that mimics evolution and is based on reproduction and selection. A GA is composed of individuals that reproduce and compete, each one is a potential solution to the (optimization) problem and is represented by a "genome" where each gene corresponds to one adjustable parameter. At each generation of the GA, each individual is paired with one other at random for reproduction. Two offspring are produced by combining their genomes and allowing for "cross-over", i.e., the two new individuals have genomes that are formed from a combination of the genomes of their parents. Also each new gene might have mutated, i.e. the parameter value might have changed slightly. At the end of the generation, the algorithm has double the number of individuals. Then each of the individuals is confronted with a number of others to count how many does it outperform (the number of wins is the number of these competitors that represent worse solutions than itself). All the individuals are ranked by their number of wins, and the population is again reduced to the original number of individuals by eliminating those which have worse fitness (solutions). http://identifiers.org/isbn/9780750308953 T. Bäck, D.B. Fogel, and Z. Michalewicz. Handbook of evolutionary computation. Oxford: IOP Publishing/Oxford University Press. 1997. 2019-01-18 AZ COPASI genetic algorithm SR http://identifiers.org/doi/10.1109/4235.873238 genetic algorithm with stochastic ranking http://identifiers.org/doi/10.1109/4235.873238 T. Runarsson and X. Yao. Stochastic ranking for constrained evolutionary optimization. EEE Transactions on Evolutionary Computation. 4. 284 - 294. 2000. 2019-01-18 AZ COPASI evolutionary programming http://identifiers.org/isbn/9780750308953 EP Evolutionary programming (EP) is a computational technique that mimics evolution and is based on reproduction and selection. An EP algorithm is composed of individuals that reproduce and compete, each one is a potential solution to the (optimization) problem and is represented by a "genome" where each gene corresponds to one adjustable parameter. At each generation of the EP, each individual reproduces asexually, i.e. divides into two individuals. One of these contains exactly the same "genome" as the parent while the other suffers some mutations (the parameter values of each gene change slightly). At the end of the generation, the algorithm has double the number of individuals. Then each of the individuals is confronted with a number of others to count how many does it outperform (the number of wins is the number of these competitors that represent worse solutions than itself). All the individuals are ranked by their number of wins, and the population is again reduced to the original number of individuals by eliminating those which have worse fitness (solutions). http://identifiers.org/isbn/9780750308953 T. Bäck, D.B. Fogel, and Z. Michalewicz. Handbook of evolutionary computation. Oxford: IOP Publishing/Oxford University Press. 1997. 2019-01-18 AZ COPASI evolutionary strategy http://identifiers.org/doi/10.1109/4235.873238 SRES evolutionary strategies with stochastic ranking Evolutionary Strategies with Stochastic Ranking (SRES) is similar to Evolutionary Programming. However, a parent has multiple offsprings during each generation. Each offspring will contain a recombination of genes with another parent and additional mutations. The algorithm assures that each parameter value will be within its boundaries. But constraints to the solutions may be violated. http://identifiers.org/doi/10.1109/4235.873238 T. Runarsson and X. Yao. Stochastic ranking for constrained evolutionary optimization. EEE Transactions on Evolutionary Computation. 4. 284 - 294. 2000. 2019-01-18 AZ COPASI truncated Newton http://identifiers.org/isbn/0122839528 The Truncated Newton method is a sophisticated variant of the Newton optimization method. The Newton optimization method searches for the minimum of a nonlinear function by following descent directions determined from the function's first and second partial derivatives. The Truncated Newton method does an incomplete (truncated) solution of a system of linear equations to calculate the Newton direction. This means that the actual direction chosen for the descent is between the steepest descent direction and the true Newton direction. http://identifiers.org/isbn/0122839528 P.E. Gill, W. Murray, and M.H. Wright. Practical Optimization. London, Academic Press. 1981. 2019-01-18 AZ steepest descent http://identifiers.org/doi/10.1109/ACSSC.1991.186507 Steepest descent is an optimization method that follows the direction of steepest descent on the hyper-surface of the objective function to find a local minimum. The direction of steepest descent is defined by the negative of the gradient of the objective function. http://identifiers.org/doi/10.1109/ACSSC.1991.186507 D.B. Fogel, L.J. Fogel, and J.W. Atmar. Meta-evolutionary programming. 25th Asiloma Conference on Signals, Systems and Computers. IEEE Computer Society, Asilomar . 540 - 545. 1992. 2019-01-18 AZ COPASI praxis http://identifiers.org/isbn/0486419983 Praxis is a direct search method that searches for the minimum of a nonlinear function without requiring (or attempting to calculate) derivatives of that function. Praxis was developed by Brent after the method proposed by Powell. The inspiration for Praxis was the well-known method of minimising each adjustable parameter (direction) at a time - the principal axes method. In Praxis directions are chosen that do not coincide with the principal axes, in fact if the objective function is quadratic then these will be conjugate directions, assuring a fast convergence rate. http://identifiers.org/isbn/0486419983 P.R. Brent. A new algorithm for minimizing a function of several variables without calculating derivatives. In Algorithms for minimization without derivatives, (Englewood Cliffs, NJ: Prentice-Hall, Inc.). 117 - 167. 1973. 2019-01-18 AZ COPASI NL2SOL http://identifiers.org/doi/10.1145/355958.355966 adaptive nonlinear least-squares algorithm The NL2SOL method is based on an adaptive nonlinear least-squares algorithm, devised by Dennis and colleagues. For problems with large number of residuals, this algorithm is known to be more reliable than Gauss-Newton or Levenberg-Marquardt method and more efficient than the secant or variable metric algorithms that are intended for general function minimization. http://identifiers.org/doi/10.1145/355958.355966 J. E. Dennis, D. M. Gay, and R. E. Welsch. An adaptive nonlinear least-squares algorithm. ACM Trans. Math. Softw.. 7(3). 348 - 368. 1981. 2019-01-18 AZ COPASI Nelder-Mead http://identifiers.org/doi/10.1093/comjnl/7.4.308 simplex method This method also known as the simplex method is due to Nelder and Mead. A simplex is a polytope of N+1 vertices in N dimensions. The objective function is evaluated at each vertex. Dependent on these calculated values a new simplex is constructed. The simplest step is to replace the worst point with a point reflected through the centroid of the remaining N points. If this point is better than the best current point, then we can try stretching exponentially out along this line. On the other hand, if this new point isn't much better than the previous value then we are stepping across a valley, so we shrink the simplex towards the best point. http://identifiers.org/doi/10.1093/comjnl/7.4.308 J. A. Nelder and R. Mead. A simplex method for function minimization. Computer Journal. 7. 308 - 313. 1965. 2019-01-18 AZ COPASI Levenberg-Marquardt http://identifiers.org/doi/10.1090/qam/10666 http://identifiers.org/doi/10.1137/0111030 Levenberg-Marquardt is a gradient descent method. It is a hybrid between the steepest descent and the Newton methods. Levenberg first suggested an improvement to the Newton method in order to make it more robust, i.e. to overcome the problem of non-convergence. His suggestion was to add a factor to the diagonal elements of the Hessian matrix of second derivatives when not close to the minimum (this can be judged by how positive definite the matrix is). The effect when this factor is large compared to the elements of Hessian is that the method then becomes the steepest descent method. Later Marquardt suggested that the factor should be multiplicative rather than additive and also defined a heuristic to make this factor increase or decrease. The method known as Levenberg-Marquardt is thus an adaptive method that effectively changes between the steepest descent to the Newton method. http://identifiers.org/doi/10.1090/qam/10666 K. Levenberg. A method for the solution of certain nonlinear problems in least squares. Quart. Appl. Math.. 2. 164 - 168. 1944. http://identifiers.org/doi/10.1137/0111030 D.W. Marquardt. An algorithm for least squares estimation of nonlinear parameters. SIAM Journal. 11. 431 - 441. 1963. 2019-01-18 AZ COPASI Hooke&Jeeves http://identifiers.org/doi/10.1145/321062.321069 Hooke and Jeeves method method of Hooke and Jeeves The method of Hooke and Jeeves is a direct search algorithm that searches for the minimum of a nonlinear function without requiring (or attempting to calculate) derivatives of the function. Instead it is based on a heuristic that suggests a descent direction using the values of the function calculated in a number of previous iterations. http://identifiers.org/doi/10.1145/321062.321069 R. Hooke and T. A. Jeeves. "Direct search" solution of numerical and statistical problems. Journal of the Association for Computing Machinery. 8. 212 - 229. 1961. 1 2019-01-18 AZ number of generations The parameter is a positive integer value to determine the number of generations the evolutionary algorithm shall evolve the population. 2019-01-18 AZ evolutionary algorithm parameter 2019-01-18 AZ population size The parameter is a positive integer value to determine the size of the population, i.e., the number of individuals that survive after each generation. 2019-01-18 AZ true evolutionary algorithm An optimisation algorithm that mimics evolution and is based on reproduction and selection. 2019-01-18 AZ true simulated annealing parameter 2019-01-18 AZ start temperature Initial temperature of the system. The higher the temperature, the larger the probability that a global optimum is found. Note that the temperature should be very high in the beginning of the method (the system should be above the "melting" temperature). This value has the same units as the objective function, so what represents "high" is different from problem to problem. 2019-01-18 AZ cooling factor Rate by which the temperature is reduced from one cycle to the next, given by the formula: Tnew=Told*"Cooling Factor". The simulated annealing algorithm works best if the temperature is reduced at a slow rate, so this value should be close to 1. true