]>
Kinetic Simulation Algorithm Ontology (KiSAO)
http://identifiers.org/pubmed/22027554
http://www.biomodels.net/kisao/
2.3.7
Artistic License 2.0
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.
This is a core version which contains all but deprecated classes.
en
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.
is similar to
2011-06-10
A relationship indicating that two entities are similar to each other.
AZ
uses
2011-06-10
A relation between algorithms indicating that one algorithm uses another one (for example, if the algorithm switches between several algorithms).
AZ
is generalization of
2011-06-10
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 ).
AZ
is used by
2011-07-19
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'.
AZ
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).
modelling and simulation algorithm
2008-05-26
true
Algorithm used to instantiate a simulation from a mathematical model.
dk
modeling and simulation algorithm
weighted stochastic simulation algorithm
24JAN2009
http://identifiers.org/pubmed/19045316
NLN
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
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.
EXACT
weighted SSA
Gillespie first reaction algorithm
2007-11-09
http://identifiers.org/doi/10.1016/0021-9991(76)90041-3
http://identifiers.org/doi/10.1021/j100540a008
Cain
Gillespie's first reaction method
NLN
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.
EXACT
Gillespie's first reaction method
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.
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).
multi-state agent-based simulation method
http://identifiers.org/pubmed/9628844
Morton-Firth
StochSim
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).
EXACT
Morton-Firth
CVODE
2007-11-30
citeulike:1832863
code value ordinary differential equation solver
http://identifiers.org/doi/10.1145/1089014.1089020
BioNetGen
JSim
SBML-SAT
SUNDIALS
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].
VCell
VODE
VODEPK
code value ordinary differential equation solver
dk
RELATED
VODEPK
EXACT
code value ordinary differential equation solver
RELATED
VODE
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.
citeulike:1832863
Cohen S, Hindmarsh C. Cvode, A Stiff/nonstiff Ode Solver In C. Computers in Physics, Vol. 10 (2), pages 138-143 (1996).
PVODE
http://identifiers.org/doi/10.1145/1089014.1089020
http://identifiers.org/doi/10.1177/109434209901300405
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].
SUNDIALS
parallel code value ordinary differential equation solver
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).
EXACT
parallel code value ordinary differential equation solver
StochSim nearest-neighbour algorithm
http://identifiers.org/pubmed/11395441
Stochsim 1.2 and more recent versions
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).
Elf and Ehrenberg method
http://identifiers.org/doi/10.1049/sb:20045021
Elf algorithm
MesoRD
NSM
SmartCell
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).
next-subvolume method
EXACT
NSM
EXACT
Elf algorithm
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).
EXACT
next-subvolume method
Gibson-Bruck next reaction algorithm
2007-11-10
http://identifiers.org/doi/10.1021/jp993732q
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.
Cain
E-Cell
Gibson and Bruck algorithm
Gibson-Bruck's next reaction algorithm
Gillespie-Gibson stochastic simulation algorithm
SSA-GB
SmartCell
VCell
dk
next reaction method
EXACT
Gibson-Bruck's next reaction algorithm
EXACT
Gillespie-Gibson stochastic simulation algorithm
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).
EXACT
Gibson and Bruck algorithm
EXACT
SSA-GB
EXACT
next reaction method
slow-scale stochastic simulation algorithm
http://identifiers.org/pubmed/15638651
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''.
slow-scale stochastic SSA
ssSSA
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).
EXACT
slow-scale stochastic SSA
EXACT
ssSSA
Gillespie direct algorithm
2007-11-10
http://identifiers.org/doi/10.1016/0021-9991(76)90041-3
http://identifiers.org/doi/10.1021/j100540a008
BetaWB
BioNetGen
ByoDyn
Cain
DM
Doob-Gillespie method
Gillespie's direct method
SSA
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.
dk
iBioSim
stochastic simulation algorithm
DM
EXACT
Doob-Gillespie method
EXACT
EXACT
Gillespie's direct method
EXACT
SSA
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).
EXACT
stochastic simulation algorithm
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.
Euler forward method
2007-11-10
http://identifiers.org/isbn/052143064X
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.
VCell
dk
explicit Euler method
explicit Gaussian first order Runge-Kutta
iBioSim
http://identifiers.org/isbn/052143064X
Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes. Cambridge University Press, New York, 2nd edition (1992).
EXACT
explicit Gaussian first order Runge-Kutta
EXACT
explicit Euler method
Euler backward method
2007-11-02
http://identifiers.org/isbn/052143064X
GSL
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.
dk
implicit Euler method
implicit Gaussian first order Runge-Kutta
EXACT
implicit Gaussian first order Runge-Kutta
EXACT
implicit Euler method
http://identifiers.org/isbn/052143064X
Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001).
explicit fourth-order Runge-Kutta method
4
2007-11-12
http://identifiers.org/isbn/0-471-91046-5
ERK4
GSL
JSim
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).
VCell
dk
http://identifiers.org/isbn/0-471-91046-5
Butcher JC. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987).
EXACT
RK4
EXACT
Runge-Kutta method
ERK4
EXACT
Rosenbrock method
2007-11-12
http://identifiers.org/doi/10.1093/comjnl/5.4.329
http://identifiers.org/isbn/052143064X
E-Cell
Kaps-Rentrop 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.
dk
generalized fourth order Runge-Kutta method
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).
EXACT
Kaps-Rentrop method
EXACT
generalized fourth order Runge-Kutta method
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).
sorting stochastic simulation algorithm
http://identifiers.org/doi/10.1016/j.compbiolchem.2005.10.007
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.
sorting SSA
sorting direct method
EXACT
sorting direct method
EXACT
sorting SSA
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).
tau-leaping method
1
2008-07-08
http://identifiers.org/doi/10.1063/1.1378322
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.
ByoDyn
Cain
NLN
SmartCell
tauL
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.
EXACT
tauL
Poisson tau-leaping method
explicit tau-leaping method with basic pre-leap check
http://identifiers.org/doi/10.1063/1.1378322
ByoDyn
Explicit tau-leaping method with basic pre leap check.
explicit tau-leaping
explicit tau-leaping method with basic preleap check
poisson tau-leaping
EXACT
explicit tau-leaping method with basic preleap check
EXACT
explicit tau-leaping
EXACT
poisson tau-leaping
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.
implicit tau-leaping method
1
2007-10-12
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.
dk
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).
trapezoidal tau-leaping method
2007-10-16
citeulike:1755561
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.
dk
trapezoidal implicit tau-leaping method
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).
EXACT
trapezoidal implicit tau-leaping method
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
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.
dynamic Monte Carlo
dynamic Monte Carlo method
kinetic Monte Carlo
kinetic Monte Carlo method
n-fold way
EXACT
dynamic Monte Carlo method
EXACT
dynamic Monte Carlo
EXACT
kinetic Monte Carlo
EXACT
kinetic Monte Carlo method
BKL
EXACT
EXACT
n-fold way
EXACT
KMC
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).
DMC
EXACT
Smoluchowski equation based method
2007-10-29
true
urn:issn:0942-9352
Method based on the Smoluchowski equation.
dk
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).
Brownian diffusion Smoluchowski method
1
1
1
1
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.
Smoldyn
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.
EXACT
GFRD
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).
EXACT
Green's function reaction dynamics
Runge-Kutta based method
2007-11-12
http://identifiers.org/isbn/0-471-91046-5
true
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.
ByoDyn
dk
modified Euler method
http://identifiers.org/isbn/0-471-91046-5
Butcher JC. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987).
RELATED
modified Euler method
deterministic cellular automata update algorithm
2007-11-30
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.
dk
http://identifiers.org/isbn/978-0252000232
Burks, Arthur W (Editor). Essays on Cellular Automata. University of Illinois Press (1970).
LSODE
2007-11-16
citeulike:1774586
http://identifiers.org/doi/10.1145/1218052.1218054
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.
Livermore solver for ordinary differential equations
dk
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).
EXACT
Livermore solver for ordinary differential equations
binomial tau-leaping method
1
2007-10-16
binomial tau-leap spatial stochastic simulation algorithm
http://identifiers.org/doi/10.1063/1.2771548
BtauL
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.
binomial tau-leap spatial stochastic simulation algorithm
dk
BtauL
EXACT
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).
EXACT
binomial tau-leap spatial stochastic simulation algorithm
Gillespie multi-particle method
http://identifiers.org/pubmed/16731694
Combination of the multiparticle method for diffusion [http://identifiers.org/biomodels.kisao/KISAO_0000334] and the SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029].
GMP
Gillespie's multi-particle method
particle-based spatial stochastic method
EXACT
particle-based spatial stochastic method
EXACT
Gillespie's multi-particle method
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).
EXACT
GMP
Stundzia and Lumsden method
http://identifiers.org/doi/10.1006/jcph.1996.0168
RD SSA
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.
reaction-diffusion stochastic simulation algorithm
EXACT
RD SSA
EXACT
reaction-diffusion stochastic simulation algorithm
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).
estimated midpoint tau-leaping method
http://identifiers.org/doi/10.1063/1.1378322
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.
explicit tau-leaping method with estimated-mid point technique
EXACT
explicit tau-leaping method with estimated-mid point technique
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.
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.
nonnegative Poisson tau-leaping method
1
http://identifiers.org/doi/10.1063/1.1992473
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].
modified poisson tau-leaping
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).
Fehlberg method
http://identifiers.org/doi/10.1007/BF02241732
http://identifiers.org/pubmed/14990450
E-Cell
GSL
JSim
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.
VCell
iBioSim
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).
EXACT
Runge-Kutta-Fehlberg method
EXACT
RKF45
Dormand-Prince method
2007-11-12
http://identifiers.org/doi/10.1016/0771-050X(80)90013-3
http://identifiers.org/pubmed/14990450
DOPRI
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.
ECell3
GSL
JSim
Matlab
Prince-Dormand method
dk
iBioSim
EXACT
Prince-Dormand method
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).
LSODA
1
1
1
1
2007-11-30
http://identifiers.org/doi/10.1137/0904010
http://identifiers.org/isbn/978-0444866073
http://www.nea.fr/abs/html/uscd1227.html
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.
Livermore solver for ordinary differential equations with automatic method switching
dk
EXACT
Livermore solver for ordinary differential equations with automatic method switching
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://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).
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.
LSODAR
2007-10-27
http://identifiers.org/isbn/978-0444866073
http://www.nea.fr/abs/html/uscd1228.html
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.
Livermore solver for ordinary differential equations with automatic method switching and root finding
dk
ordinary differential equation solver for stiff or non-stiff systems with root finding
EXACT
ordinary differential equation solver for stiff or non-stiff systems with root finding
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).
EXACT
Livermore solver for ordinary differential equations with automatic method switching and root finding
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.
LSODI
http://identifiers.org/doi/10.1145/1218052.1218054
http://identifiers.org/isbn/978-0444866073
LSODI solves systems given in linearly implicit form, including differential-algebraic systems.
Livermore solver for ordinary differential equations, implicit version
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.
EXACT
Livermore solver for ordinary differential equations, implicit version
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).
LSODIS
2007-11-30
http://identifiers.org/isbn/978-0444866073
http://www.nea.fr/abs/html/uscd1225.html
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.
Livermore solver for ordinary differential equations, implicit sparse version
dk
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).
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.
EXACT
Livermore solver for ordinary differential equations, implicit sparse version
LSODPK
2008-07-08
http://identifiers.org/isbn/978-0444866073
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.
Livermore solver for ordinary differential equations for stiff and nonstiff systems with krylov corrector iteration
NLN
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.
EXACT
Livermore solver for ordinary differential equations for stiff and nonstiff systems with krylov corrector iteration
Livermore solver
2008-07-08
true
Method to solve ordinary differential equations developed at the Lawrence Livermore National Laboratory.
NLN
sub-volume stochastic reaction-diffusion algorithm
2008-07-08
true
NLN
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.
modelling and simulation algorithm characteristic
true
AZ
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).
modeling and simulation algorithm characteristic
type of variable
true
AZ
Type of variables used for the simulation.
type of system behaviour
true
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
type of progression time step
true
AZ
Type of time steps used by the algorithm.
spatial description
2008-07-08
Algorithm, possessing this characteristic, takes into account the location of the reacting components.
NLN
deterministic system behaviour
2008-07-08
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.
NLN
stochastic system behaviour
2008-07-08
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.
NLN
discrete variable
2008-07-08
Algorithm, possessing this characteristic, allows values of system's variables to change by discrete (integral) amounts.
NLN
continuous variable
2008-07-08
Algorithm, possessing this characteristic, allows the values of a system's variables to change by continuous (non-integral) amounts.
NLN
progression with adaptive time step
2008-07-08
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.
NLN
progression with fixed time step
2008-07-08
Algorithm, possessing this characteristic, uses time steps of constant length to update the state of a system during the whole simulation.
NLN
modelling and simulation algorithm parameter
true
AZ
Parameter that can be used in the simulation experiment settings.
modeling and simulation algorithm parameter
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.
EXACT
RTOL
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).
LSODA maximum non-stiff order
Adams max 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].
Adams max order
NARROW
LSODA maximum stiff order
BDF max 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.
BDF max order
NARROW
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
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.
epsilon
tolerance
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.
EXACT
epsilon
minimum reactions per leap
'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.
threshold
Pahle hybrid method
1
1
http://identifiers.org/doi/10.1093/bioinformatics/btl485
AZ
COPASI
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
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.
Livermore solver for ordinary differential equations given in implicit form, with block-tridiagonal Jacobian treatment
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.
EXACT
Livermore solver for ordinary differential equations given in implicit form, with block-tridiagonal Jacobian treatment
LSODES
http://identifiers.org/isbn/978-0444866073
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.
Livermore solver for ordinary differential equations with general sparse Jacobian matrix
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.
EXACT
Livermore solver for ordinary differential equations with general sparse Jacobian matrix
LSODKR
http://identifiers.org/isbn/978-0444866073
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.
Livermore solver for ordinary differential equations, with preconditioned Krylov iteration methods for the Newton correction linear systems, and with root finding.
EXACT
Livermore solver for ordinary differential equations, with preconditioned Krylov iteration methods for the Newton correction linear systems, and with root finding.
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.
type of solution
true
A characteristic describing the type of the solution produced by the method, specifically whether it is exact or approximate.
AZ
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.
type of method
true
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
AZ
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).
implicit method type
AZ
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).
Gillespie-like method
true
Stochastic simulation algorithm using an approach alike the one described in Gillespie's papers of 1976 and 1977.
error control parameter
true
AZ
Parameter controlling method accuracy.
method switching control parameter
true
AZ
Parameters describing threshold conditions for algorithms that switch between different methods.
granularity control parameter
true
AZ
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
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
nonnegative tau-leaping second control parameter
partitioning control parameter
true
Parameter describing partitioning of the system.
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.
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
Smoldyn
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.
virtual box side length
Smoldyn
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].
Euler method
1
http://identifiers.org/isbn/052143064X
AZ
ByoDyn
JSim
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).
NFSim agent-based simulation method
2011-04-07
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.
AZ
NFSim
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.
cellular automata update method
2011-04-07
http://identifiers.org/doi/10.1103/RevModPhys.55.601
AZ
CA
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''.
cellular automata
cellular spaces
cellular structures
homogeneous structures
iterative arrays
tessellation automata
tessellation structures
EXACT
cellular spaces
EXACT
tessellation automata
EXACT
homogeneous structures
Wolfram, Stephen (1983) Statistical mechanics of cellular automata. Reviews of Modern Physics 55 (3): 601-644.
http://identifiers.org/doi/10.1103/RevModPhys.55.601
CA
EXACT
EXACT
iterative arrays
EXACT
cellular structures
EXACT
tessellation structures
EXACT
cellular automata
hard-particle molecular dynamics
2011-05-05
http://identifiers.org/doi/10.1016/j.jcp.2004.08.014
A collision-driven molecular dynamics algorithm for a system of non-spherical particles.
AZ
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
first-passage Monte Carlo algorithm
2011-05-05
http://identifiers.org/doi/10.1103/PhysRevLett.97.230602
AED DKMC
AED diffusion kinetic Monte Carlo method
AZ
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.
asynchronous event-driven diffusion Monte Carlo
AED diffusion kinetic Monte Carlo method
EXACT
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.
EXACT
asynchronous event-driven diffusion Monte Carlo
AED DKMC
EXACT
Gill method
4
2011-05-09
http://identifiers.org/doi/10.1017/S0305004100026414
AZ
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.
Gill's method
Runge-Kutta-Gill method
EXACT
Runge-Kutta-Gill method
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.
EXACT
Gill's method
Metropolis Monte Carlo algorithm
2011-05-09
http://identifiers.org/doi/10.1063/1.1699114
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.
AZ
CompuCell3D
Metropolis algorithm
Metropolis-Hastings algorithm
EXACT
Metropolis-Hastings algorithm
EXACT
Metropolis algorithm
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.
Adams-Bashforth method
2011-05-09
http://identifiers.org/isbn/978-3-540-56670-0
AZ
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.
explicit Adams 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
EXACT
explicit Adams method
Adams-Moulton method
2011-05-09
http://identifiers.org/isbn/978-3-540-56670-0
AZ
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.
VCell
implicit Adams 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
EXACT
implicit Adams method
multistep method
2011-05-09
http://identifiers.org/isbn/978-3-540-56670-0
true
A numerical method for differential equations which is based on several values of the solution.
AZ
multi-value 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
EXACT
multi-value method
KINSOL
2011-05-09
http://identifiers.org/doi/10.1137/0911026
http://identifiers.org/doi/10.1145/1089014.1089020
http://identifiers.org/doi/10.4249/scholarpedia.2860
AZ
FKINSOL
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.
NKSOL
Newton-Krylov solver for nonlinear algebraic systems
SUNDIALS
FKINSOL
RELATED
Newton-Krylov solver for nonlinear algebraic systems
RELATED
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/10.4249/scholarpedia.2860
Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860.
NKSOL
RELATED
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.
IDA
2011-05-09
http://identifiers.org/doi/10.1137/0915088
http://identifiers.org/doi/10.1145/1089014.1089020
http://identifiers.org/doi/10.4249/scholarpedia.2860
AZ
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.
SUNDIALS
VCell
implicit differential-algebraic solver
solver for differential-algebraic equation systems
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.
EXACT
implicit differential-algebraic solver
RELATED
solver for differential-algebraic equation systems
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.
Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860.
IDA
http://identifiers.org/doi/10.4249/scholarpedia.2860
Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860.
finite volume method
2011-05-09
http://identifiers.org/isbn/0898715342
AZ
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.
VCell
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.
EXACT
FVM
Euler-Maruyama method
2011-05-09
http://identifiers.org/doi/10.1098/rspa.2003.1247
AZ
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.
stochastic Euler scheme
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.
EXACT
stochastic Euler scheme
Milstein method
2011-05-10
http://identifiers.org/isbn/079233213X
AZ
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.
backward differentiation formula
2011-05-10
http://identifiers.org/isbn/0136266061
AZ
BDF
ByoDyn
GSL
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.
iBioSim
EXACT
Gear method
EXACT
Gear's method
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
Adams method
2011-05-10
http://identifiers.org/isbn/978-3-540-56670-0
AZ
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).
ByoDyn
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
Merson method
4
2011-05-10
http://nla.gov.au/nla.cat-vn870866
A five-stage Runge-Kutta method with fourth-order accuracy.
AZ
JSim
KM
Kutta-Merson method
Merson's method
Runge-Kutta-Merson method
EXACT
Kutta-Merson method
EXACT
KM
EXACT
Runge-Kutta-Merson method
EXACT
Merson's method
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.
Hammer-Hollingsworth method
2011-05-10
urn:issn:0891-6837
AZ
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.
Lobatto method
2011-05-10
urn:issn:1088-6842(e)
AZ
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.
implicit Runge-Kutta method based on Lobatto quadrature
EXACT
implicit Runge-Kutta method based on Lobatto quadrature
urn:issn:1088-6842(e)
J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comp., v. 18, 1964, pp. 233-244.
Butcher-Kuntzmann method
2011-05-10
http://identifiers.org/doi/10.1002/zamm.19660460519
urn:issn:1088-6842(e)
AZ
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.
Gauss method
urn:issn:1088-6842(e)
Butcher, J. C. (1964) Implicit Runge-Kutta processes. Math. Comput. 18, 50-64.
EXACT
Gauss method
http://identifiers.org/doi/10.1002/zamm.19660460519
F. Ceschin, J. Kuntzmann, Problèmes différentiels de conditions initiales. Paris 1963. Dunod.
Heun method
2
2011-05-10
http://identifiers.org/isbn/978-3-540-56670-0
AZ
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.
EXACT
Heun's 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
embedded Runge-Kutta method
2011-05-10
http://identifiers.org/isbn/978-3-540-56670-0
true
AZ
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.
embedded RK
EXACT
embedded RK
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
Zonneveld method
4
3
2011-05-10
http://trove.nla.gov.au/work/21424455
AZ
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
Radau method
2011-05-11
urn:issn:1088-6842(e)
AZ
Implicit Runge-Kutta methods based on Radau quadrature.
JSim
implicit Runge-Kutta method 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.
EXACT
implicit Runge-Kutta method based on Radau quadrature
Verner method
6
5
2011-05-10
http://identifiers.org/doi/10.1137/0728027
AZ
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.
Verner's method
http://identifiers.org/doi/10.1137/0728027
J.H. Verner, Some Runge-Kutta formula pairs, SIAM J. Numer. Anal 28 (1991) 496-511
EXACT
Verner's method
Lagrangian sliding fluid element algorithm
2011-05-11
http://identifiers.org/pubmed/1449234
AZ
BTEX
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.
JSim
LSFEA
blood-tissue exchange method
EXACT
blood-tissue exchange method
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.
EXACT
LSFEA
BTEX
EXACT
finite difference method
2011-05-11
http://identifiers.org/isbn/0898715342
AZ
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.
EXACT
FDM
MacCormack method
2011-05-11
http://identifiers.org/doi/10.2514/2.6901
AZ
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.
JSim
http://identifiers.org/doi/10.2514/2.6901
MacCormack, R. W., The Effect of viscosity in hypervelocity impact cratering, AIAA Paper, 69-354 (1969).
Crank-Nicolson method
2011-05-11
http://identifiers.org/doi/10.1007/BF02127704
AZ
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.
method of lines
2011-05-11
http://identifiers.org/isbn/0126241309
AZ
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.
EXACT
MOL
EXACT
NUMOL
EXACT
method of lines
http://identifiers.org/isbn/0126241309
Schiesser, W. E. (1991). The Numerical Method of Lines. Academic Press.
EXACT
NMOL
type of domain geometry handling
true
AZ
S-System power-law canonical differential equations solver
2011-05-20
http://identifiers.org/doi/10.1137/0727042
AZ
E-Cell
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.
ESSYNS GMA
EXACT
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.
lattice gas automata
2011-05-23
http://identifiers.org/isbn/3-540-66973-6
AZ
LGA
LGCA
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.
lattice gas cellular automata
EXACT
LGA
http://identifiers.org/isbn/3-540-66973-6
Dieter A. Wolf-Gladrow (2000). Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer.
EXACT
lattice gas cellular automata
EXACT
LGCA
enhanced Greens function reaction dynamics
2011-05-23
http://identifiers.org/doi/10.1007/978-3-540-88562-7_3
AZ
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.
eGFRD
enhanced Greens function reaction dynamics
EXACT
eGFRD
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.
EXACT
enhanced Greens function reaction dynamics
E-Cell multi-algorithm simulation method
2011-05-23
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.
AZ
E-Cell
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.
Gauss-Legendre Runge-Kutta method
2011-05-26
http://identifiers.org/isbn/960-8457-54-8
AZ
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.
iBioSim
EXACT
Open Formula
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.
Monte Carlo method
2011-05-26
http://identifiers.org/doi/10.2307/2280232
true
AZ
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.
EXACT
MC
BioRica hybrid method
2011-05-26
http://identifiers.org/pubmed/19425152
AZ
BioRica
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.
Cash-Karp method
2011-05-26
http://identifiers.org/doi/10.1145/79505.79507
AZ
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.
Cain
GSL
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.
hybridity
2011-05-27
AZ
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.
equation-free probabilistic steady-state approximation
http://identifiers.org/doi/10.1063/1.2131050
2011-06-02
AZ
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.
nested stochastic simulation algorithm
2011-06-02
http://identifiers.org/pubmed/16321076
AZ
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.
nested SSA
EXACT
nested SSA
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.
minimum fast/discrete reaction occurrences number
http://identifiers.org/doi/10.1063/1.2131050
2011-06-02
AZ
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.
number of samples
http://identifiers.org/doi/10.1063/1.2131050
2011-06-02
AZ
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.
maximum discrete number
http://identifiers.org/doi/10.1063/1.2131050
2011-06-02
AZ
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.
minimum fast rate
http://identifiers.org/doi/10.1063/1.2131050
2011-06-02
AZ
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.
constant-time kinetic Monte Carlo algorithm
2011-06-03
http://identifiers.org/pubmed/18513044
AZ
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.
EXACT
SSA-CR
R-leaping algorithm
2011-06-03
http://identifiers.org/pubmed/16964997
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.
AZ
R-leap method
EXACT
R-leap method
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.
exact R-leaping algorithm
2011-06-03
http://identifiers.org/pubmed/2852436
AZ
ER-leap method
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.
exact R-leap method
exact accelerated stochastic simulation algorithm
ER-leap method
EXACT
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.
EXACT
exact R-leap method
EXACT
exact accelerated stochastic simulation algorithm
ER-leap initial leap
2011-06-03
http://identifiers.org/pubmed/2852436
AZ
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.
EXACT
L
accelerated stochastic simulation algorithm
2011-06-03
true
AZ
An algorithm, which accelerates SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] either at the expense of its accuracy or exact.
accelerated SSA
EXACT
accelerated SSA
multiparticle lattice gas automata
2011-06-03
http://identifiers.org/doi/10.1142/S0129183194000052
AZ
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].
multiparticle lattice gas cellular automata
EXACT
multiparticle lattice gas cellular automata
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.
generalized stochastic simulation algorithm
2011-06-03
true
AZ
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.
D-leaping method
2011-06-03
http://identifiers.org/doi/10.1016/j.jcp.2009.05.004
AZ
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.
finite element method
2011-06-07
http://identifiers.org/doi/10.1109/MAP.2007.376627
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.
AZ
FEA
FEM
finite element analysis
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.
EXACT
finite element analysis
EXACT
FEA
EXACT
FEM
h-version of the finite element method
2011-06-07
http://identifiers.org/doi/10.1016/0168-874X(94)90003-5
AZ
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].
h-FEM
h-method
EXACT
h-method
EXACT
h-FEM
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.
p-version of the finite element method
2011-06-07
http://identifiers.org/doi/10.1137/0718033
AZ
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].
p-FEM
p-method
EXACT
p-FEM
EXACT
p-method
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.
h-p version of the finite element method
2011-06-07
http://identifiers.org/doi/10.1007/BF00272624
AZ
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.
hp-FEM
hp-method
EXACT
hp-FEM
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.
EXACT
hp-method
mixed finite element method
2011-06-07
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.
AZ
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.
level set method
2011-06-07
http://identifiers.org/doi/10.1016/0021-9991(88)90002-2
AZ
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.
LSM
level-set method
EXACT
LSM
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.
EXACT
level-set method
generalized finite element method
2011-06-07
http://identifiers.org/doi/10.1142/S0219876204000083
AZ
GFEM
PUM
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.
partition of unity method
EXACT
PUM
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.
EXACT
partition of unity method
EXACT
GFEM
h-p cloud method
2011-06-09
http://identifiers.org/doi/10.1002/(SICI)1098-2426(199611)12:6<673::AID-NUM3>3.0.CO;2-P
A meshless method, which uses a partition of unity to construct the family of h-p cloud functions.
AZ
h-p clouds
method of clouds
EXACT
h-p clouds
EXACT
method of clouds
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.
mesh-based geometry handling
AZ
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.
meshless geometry handling
AZ
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.
extended finite element method
2011-06-09
http://identifiers.org/doi/10.1007/s00466-002-0391-2
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.
AZ
X-FEM
XFEM
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).
EXACT
X-FEM
EXACT
XFEM
method of finite spheres
2011-06-09
http://identifiers.org/doi/10.1007/s004660050481
AZ
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.
EXACT
MFS
probability-weighted dynamic Monte Carlo method
2011-06-09
http://identifiers.org/doi/10.1021/jp011404w
AZ
PW-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.
probability-weighted DMC
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.
EXACT
PW-DMC
EXACT
probability-weighted DMC
multinomial tau-leaping method
2011-06-09
http://identifiers.org/pubmed/17343434
AZ
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.
EXACT
MtauL
hybrid method
2011-06-09
true
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.
AZ
generalized minimal residual algorithm
2011-06-10
http://identifiers.org/doi/10.1137/0907058
AZ
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.
GMRES
EXACT
GMRES
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).
Krylov subspace projection method
2011-06-10
http://identifiers.org/isbn/0898715342
AZ
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.
EXACT
Krylov subspace method
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.
DASPK
2011-06-10
http://identifiers.org/doi/10.1137/0915088
AZ
DDASPK
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.
SDASPK
differential algebraic system solver with Krylov preconditioning
EXACT
differential algebraic system solver with Krylov preconditioning
NARROW
SDASPK
DDASPK
NARROW
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.
DASSL
2011-06-10
http://www.nea.fr/abs/html/nesc9918.html
AZ
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.
DDASSL
SDASSL
differential algebraic system solver
DDASSL
NARROW
NARROW
SDASSL
http://www.nea.fr/abs/html/nesc9918.html
Linda R. Petzold: A Description of DASSL: A Differential/Algebraic System Solver, SAND82-8637 (September 1982).
EXACT
differential algebraic system solver
conjugate gradient method
2011-06-10
urn:issn:0091-0635
AZ
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
biconjugate gradient method
2011-06-10
http://identifiers.org/doi/10.1007/BFb0080109
AZ
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.
BCG
EXACT
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.
BiCG
EXACT
Bi-CG
EXACT
implicit-state Doob-Gillespie algorithm
2011-06-13
http://identifiers.org/isbn/3-540-76636-7 978-3-540-76636-0
AZ
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.
rule-based simulation method
2011-06-13
true
AZ
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 are methods, used to simulated such models.
Adams predictor-corrector method
2011-06-16
urn:asin:B0000EFQ8B
AZ
GSL
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.
NDSolve method
2011-06-16
http://identifiers.org/isbn/978-1-57955-058-5
AZ
Mathematica
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.
symplecticness
2011-06-16
http://identifiers.org/doi/10.1017/S0962492900002282
AZ
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
partitioned Runge-Kutta method
2
2011-06-16
AZ
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.
PRK
SPRK
http://identifiers.org/doi/10.1007/BF01389456
symplectic partitioned Runge-Kutta method
EXACT
symplectic partitioned Runge-Kutta method
P. Rentrop (1985): Partitioned Runge-Kutta methods with stiffness detection and stepsize control. Numer. Math., Vol. 47, p.545-564.
http://identifiers.org/doi/10.1007/BF01389456
EXACT
PRK
EXACT
SPRK
partial differential equation discretization method
2011-06-27
true
A method which solves partial differential equations by discretizing them, i.e. approximating them by equations that involve a finite number of unknowns.
AZ
type of problem
2011-06-29
true
A characteristic describing the type of the problems which can be solved by the algorithm.
AZ
stochastic differential equation problem
AZ
SDE problem
EXACT
SDE problem
partial differential equation problem
AZ
PDE problem
EXACT
PDE problem
differential-algebraic equation problem
AZ
DAE
DAE
EXACT
ordinary differential equation problem
AZ
ODE problem
EXACT
ODE problem
delay differential equation problem
AZ
DDE problem
linearity of equation
2011-07-19
AZ
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.
one-step method
2011-06-30
http://identifiers.org/isbn/978-3-540-56670-0
true
A numerical method for differential equations which uses one starting value at each step.
AZ
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 midpoint rule
2
2011-06-30
http://identifiers.org/isbn/978-3-540-56670-0
AZ
GSL
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)].
implicit Gaussian second order Runge-Kutta method
EXACT
implicit Gaussian 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.
Bulirsch-Stoer algorithm
2011-07-01
http://identifiers.org/doi/10.1007/BF02165234
AZ
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.
EXACT
GBS
EXACT
Gragg-Bulirsch-Stoer algorithm
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.
Richardson extrapolation based method
2011-07-01
http://identifiers.org/doi/10.1098/rsta.1911.0009
true
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.
AZ
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.
midpoint method
2
2011-07-01
http://identifiers.org/isbn/978-3-540-56670-0
AZ
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.
modified midpoint method
2011-07-01
http://identifiers.org/isbn/052143064X
AZ
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.
EXACT
Gragg's method
http://identifiers.org/isbn/052143064X
Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001).
Bader-Deuflhard method
2011-07-01
http://identifiers.org/doi/10.1007/BF01418331
AZ
GSL
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.
semi-implicit midpoint rule
2011-07-01
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].
AZ
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.
scaled preconditioned generalized minimal residual method
2011-07-18
http://identifiers.org/doi/10.1137/0915088
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].
AZ
SPGMR
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.
EXACT
SPGMR
minimal residual method
2011-07-18
http://identifiers.org/doi/10.1137/0712047
AZ
MINRES
The 'minimal residual method' is an algorithm for the numerical solution of indefinite symmertic systems of linear equations.
EXACT
MINRES
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.
quasi-minimal residual method
2011-07-18
http://identifiers.org/doi/10.1007/BF01385726
AZ
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).
EXACT
QMR
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.
biconjugate gradient stabilized method
http://identifiers.org/doi/10.1137/0913035
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.
Bi-CGSTAB
BiCGSTAB
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
ingenious conjugate gradients-squared method
2011-07-18
http://identifiers.org/doi/10.1137/0910004
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.
AZ
CGS
CGS
EXACT
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.
quasi-minimal residual variant of biconjugate gradient stabilized method
2011-07-19
http://identifiers.org/doi/10.1137/0915023
AZ
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.
EXACT
QMRCGSTAB
improved biconjugate gradient method
2011-07-19
true
AZ
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.
transpose-free quasi-minimal residual algorithm
2011-07-19
http://identifiers.org/doi/10.1137/0914029
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.
AZ
TFQMR
EXACT
TFQMR
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.
preconditioning technique
2011-07-19
http://identifiers.org/isbn/0898715342
AZ
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.
iterative method for linear system
2011-07-19
http://identifiers.org/isbn/0898715342
true
AZ
http://identifiers.org/isbn/0898715342
Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM.
homogeneousness of equation
2011-07-19
AZ
Homogeneous equations are of the form Ly = 0, where the differential operator L is a linear operator and y is the unknown function.
symmetricity of matrix
2011-07-19
AZ
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.
type of differential equation
2011-07-19
true
AZ
steady state method
2012-01-17
true
A method looking for a steady state of a dynamic system.
AZ
Requested by Frank T. Bergmann on Sunday, November 27, 2011 4:45:30 PM.
Newton-type method
2012-01-18
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].
AZ
Requested by Frank T. Bergmann on Sunday, November 27, 2011 4:45:30 PM.
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.
ordinary Newton method
2012-01-18
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,...
AZ
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.
simlified Newton method
2012-01-18
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[0])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,...
AZ
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.
Newton-like method
2012-01-18
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[0], or by some approximation so that: M'(x[0])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,...
AZ
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.
inexact Newton method
2012-01-18
http://identifiers.org/isbn/9783540210993
AZ
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[0], the iterates are defined as x[k+1]=xi[k+1] for i=imax[k], k=0,1,...
iterative Newton method
truncated Newton method
EXACT
truncated Newton method
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.
EXACT
iterative Newton method
exact Newton method
2012-01-18
http://identifiers.org/isbn/9783540210993
AZ
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.
direct Newton method
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.
maximum number of steps
2012-01-18
AZ
Maximum number of steps the 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] algorithm is allowed to take.
partial least squares regression method
1
1
2012-01-18
http://identifiers.org/biomodels.kisao/10.1007/BFb0062108
AZ
Multivariate regression method based on estimated latent variables. Related to Principal Component Analysis (PCA) and Principal Component Regression (PCR).
PLSR
PLSR method
Requested by Kristin Tøndel on Thursday, October 13, 2011 10:46:04 AM.
EXACT
PLSR
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.
EXACT
PLSR method
hierarchical cluster-based partial least squares regression method
1
2012-01-18
http://identifiers.org/doi/10.1186/1752-0509-5-90
AZ
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.
Requested by Kristin Tøndel on Thursday, October 13, 2011 11:13:17 AM.
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.
N-way partial least squares regression method
1
2012-01-18
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
AZ
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).
N-PLS
N-way PLSR
N-way partial least squares method
Requested by Kristin Tøndel on Thursday, October 13, 2011 11:33:06 AM.
EXACT
N-way partial least squares method
http://www.models.life.ku.dk/nwaytoolbox
N-way toolbox website
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.
EXACT
N-PLS
EXACT
N-way PLSR
metamodelling method
2012-01-18
http://identifiers.org/doi/10.1186/1752-0509-5-90
true
AZ
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.
number of partial least squares components
2012-01-18
http://identifiers.org/biomodels.kisao/10.1007/BFb0062108
AZ
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.
type of validation
2012-01-18
AZ
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.
number of N-way partial least squares regression factors
2012-01-18
AZ
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
number of factors
partial least squares regression-like method
1
2012-01-18
true
AZ
Method for building regression models between independent and dependent variables.
mean-centring of variables
2012-01-18
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.
AZ
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.
standardising of variables
2012-01-18
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.
AZ
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.
number of clusters
2012-01-18
http://identifiers.org/doi/10.1186/1752-0509-5-90
AZ
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.
matrix for clusterization
2012-01-18
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].
AZ
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.
clusterization parameter
true
AZ
Parameter used by algorithms performing clusterization.
variables preprocessing parameter
true
AZ
IDA-like method
1
2012-05-24
http://identifiers.org/doi/10.1145/1089014.1089020
true
AZ
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.
CVODE-like method
1
2012-05-24
http://identifiers.org/doi/10.1145/1089014.1089020
true
AZ
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.
Higham-Hall method
2012-09-25
http://identifiers.org/doi/10.1016/0377-0427(90)90192-3
AZ
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.
embedded Runge-Kutta 5(4) method
4
5
2012-09-26
true
AZ
An embedded Runge-Kutta integrator of order 5(4).
RK5(4)
Dormand-Prince 8(5,3) method
5
8
2012-09-26
http://identifiers.org/isbn/978-3-540-56670-0
AZ
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
flux balance analysis
2012-11-29
http://identifiers.org/doi/10.1038/nbt.1614
AZ
FBA
Flux balance analysis is a mathematical approach for analyzing the flow of metabolites through a metabolic network.
Requested by Frank T. Bergmann on Thursday, November 29, 2012 9:51:58 AM.
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
COAST
1
2013-01-28
http://identifiers.org/pubmed/17100426
AZ
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.
Requested by Mark Moeller on Friday, January 25, 2013 11:11:30 AM.
controllable approximative stochastic reaction algorithm
http://identifiers.org/pubmed/17100426
Holger Wagner, Mark Moeller, and Klaus Prank (2006) COAST: Controllable approximative stochastic reaction algorithm. J. Chem. Phys. 125, 174104
logical model simulation method
2013-01-28
http://www.identifiers.org/pubmed/22141422
AZ
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
synchronous logical model simulation method
2013-01-28
AZ
Qualitative (logical) models specify the evolution rules of their components. In the case of a synchronous updating all enabled transitions are processed simultaneously.
asynchronous logical model simulation method
2013-01-28
AZ
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).
type of updating policy
2013-01-28
true
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].
AZ
random updating policy
2013-01-28
AZ
An updating policy that chooses a transition randomly.
ordered updating policy
2013-01-28
AZ
An updating policy that chooses a transition in a definite way.
constant updating policy
2013-01-28
AZ
An updating policy that chooses a transition in a constant way.
prioritized updating policy
2013-01-28
AZ
An updating policy that chooses a transition in a prioritized way.
maximum step size
2013-07-05
AZ
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.
Requested by Andrew Miller on Tuesday, June 4, 2013 2:33:51 AM.
maximal timestep method
2014-04-25
http://identifiers.org/pubmed/14990466
AZ
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].
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.
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.
maximal timestep
2014-04-25
http://identifiers.org/pubmed/14990466
AZ
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.
optimization algorithm
2015-04-23
http://en.wikipedia.org/wiki/Mathematical_optimization#Optimization_algorithms
http://identifiers.org/isbn/0691102872
true
AZ
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.
optimization method
http://identifiers.org/isbn/0691102872
Brinkhuis J, Tikhomirov V. Optimization: Insights and Applications
Princeton Series in Applied Mathematics (illustrated), Princeton University Press, 2005.
local optimization algorithm
2015-04-23
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.
AZ
local optimiation method
global optimization algorithm
2015-04-23
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.
AZ
global optimization method
Bayesian inference algorithm
2015-04-23
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.
AZ
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.
true