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Erschienen in:
Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

Numerical Methods for Stochastic Simulation: When Stochastic Integration Meets Geometric Numerical Integration

verfasst von : Assyr Abdulle

Erschienen in: Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology

Verlag: Springer International Publishing

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Abstract

In this paper we discuss a framework recently introduced to construct and analyze accurate stochastic integrators for the computation of expectation of functionals of a stochastic process for both finite time or long-time dynamics. Such accurate integrators are of interest for many applications in biology, chemistry or physics and are also often needed in multiscale stochastic simulations. We describe how ideas originating from geometric numerical integration or structure preserving methods for deterministic differential equations can help to design new integrators for weak approximation of stochastic differential equations or for long-time simulation of ergodic stochastic systems.

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Vorheriges Kapitel Multiscale Simulation of Stochastic Reaction-Diffusion Networks
Nächstes Kapitel Stability and Strong Convergence for Spatial Stochastic Kinetics
Fußnoten
1
See Sect. 1.1 for a precise definition.
 
2
Formally in backward error analysis one considers an infinite series \(\tilde{f}_{h}(x) = f(x) + h\tilde{f}_{1}(x) + h^{2}\tilde{f}_{2}(x) + \cdots \) such that \(\widetilde{X}(t_{n}) = X_{n}\) at t n = nh. But the infinite series for \(\tilde{f}_{h}\) is usually not convergent and one needs therefore to consider an appropriate truncation.
 
Literatur
1.
A. Abdulle, G. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales. J. Comput. Phys. 231(6), 2482–2497 (2012)MathSciNetCrossRefMATH
2.
A. Abdulle, D. Cohen, G. Vilmart, K.C. Zygalakis, High order weak methods for stochastic differential equations based on modified equations. SIAM J. Sci. Comput. 34(3), 1800–1823 (2012)MathSciNetCrossRefMATH
3.
A. Abdulle, W. E, B. Engquist, E. Vanden-Eijnden, The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)
4.
A. Abdulle, G. Vilmart, K.C. Zygalakis, High order numerical approximation of the invariant measure of ergodic SDEs. SIAM J. Numer. Anal. 52(4), 1600–1622 (2014)MathSciNetCrossRefMATH
5.
A. Abdulle, G. Vilmart, K.C. Zygalakis, Long time accuracy of Lie-Trotter splitting methods for Langevin dynamics. SIAM J. Numer. Anal. 53(1), 1–16 (2015)MathSciNetCrossRefMATH
6.
A. Abdulle, G. Pavliotis, U. Vaes, Spectral methods for multiscale stochastic differential equations. SIAM/ASA J. Uncertain. Quantif. (2016, to appear)
7.
N. Bou-Rabee, H. Owhadi, Long-run accuracy of variational integrators in the stochastic context. SIAM J. Numer. Anal. 48(1), 278–297 (2010)MathSciNetCrossRefMATH
8.
C.-E. Bréhier, Analysis of an HMM time-discretization scheme for a system of stochastic PDEs. SIAM J. Numer. Anal. 51(2), 1185–1210 (2013)MathSciNetCrossRefMATH
9.
C.-E. Bréhier, G. Vilmart, High-order integrator for sampling the invariant distribution of a class of parabolic stochastic PDEs with additive space-time noise. SIAM J. Sci. Comput. 38, A2283–A2306 (2016)MathSciNetCrossRefMATH
10.
Y. Cao, D.T. Gillespie, L. Petzold, The slow scale stochastic simulation algorithm. J. Chem. Phys. 122, 014116 (2005)CrossRef
11.
P. Chartier, E. Hairer, G. Vilmart, Numerical integrators based on modified differential equations. Math. Comp. 76(260), 1941–1953 (2007) (electronic)
12.
13.
W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations. Commun. Pure Appl. Math. 58(11), 1544–1585 (2005)
14.
W. E, D. Liu, E. Vanden-Eijnden, Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys. 221(1), 158–180 (2007)
15.
C.W. Gardiner, Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences, 2nd edn. (Springer, Berlin, 1985)
16.
D. Gillespie, Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem. 58, 35–55 (2007)CrossRef
17.
C. Graham, D. Talay, Stochastic simulation and Monte Carlo methods, in Stochastic Modelling and Applied Probability. Mathematical Foundations of Stochastic Simulation, vol. 68 (Springer, Heidelberg, 2013)
18.
E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31, 2nd edn. (Springer, Berlin, 2006)
19.
20.
P. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992)CrossRefMATH
21.
22.
23.
B. Leimkuhler, C. Matthews, Rational construction of stochastic numerical methods for molecular sampling. Appl. Math. Res. Express 2013, 34–56 (2013)MathSciNetMATH
24.
B. Leimkuhler, C. Matthews, Molecular dynamics, in Interdisciplinary Applied Mathematics, vol. 36 (Springer, Cham, 2015), pp. xxii+443
25.
B. Leimkuhler, C. Matthews, G. Stoltz, The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. IMA J. Numer. Anal. 36(1), 13–79 (2016)MathSciNetMATH
26.
T. Li, A. Abdulle, W. E, Effectiveness of implicit methods for stiff stochastic differential equations. Commun. Comput. Phys. 3(2), 295–307 (2008)
27.
J. Mattingly, A. Stuart, D. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101(2), 185–232 (2002)MathSciNetCrossRefMATH
28.
J.C. Mattingly, A.M. Stuart, M.V. Tretyakov, Convergence of numerical time-averaging and stationary measures via poisson equations. SIAM J. Numer. Anal. 48(2), 552–577 (2010)MathSciNetCrossRefMATH
29.
G. Milstein, Weak approximation of solutions of systems of stochastic differential equations. Theory Probab. Appl. 30(4), 750–766 (1986)MathSciNetCrossRef
30.
G. Milstein, M. Tretyakov, Stochastic Numerics for Mathematical Physics. Scientific Computing (Springer, Berlin, 2004)CrossRefMATH
31.
G.N. Milstein, M.V. Tretyakov, Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients. SIAM J. Numer. Anal. 43(3), 1139–1154 (2005) (electronic)
32.
33.
G. Pavliotis, A. Stuart, Multiscale Methods: Averaging and Homogenization. Text in Applied Mathematics, vol. 53 (Springer, New York, 2008)
34.
H. Risken, The Fokker-Planck Equation. Springer Series in Synergetics, vol. 18 (Springer, Berlin, 1989)
35.
G.O. Roberts, R.L. Tweedie, Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2(4), 341–363 (1996)MathSciNetCrossRefMATH
36.
A. Rößler, Second order Runge-Kutta methods for Itô stochastic differential equations. SIAM J. Numer. Anal. 47(3), 1713–1738 (2009)MathSciNetCrossRefMATH
37.
D. Talay, Efficient Numerical Schemes for the Approximation of Expectations of Functionals of the Solution of a SDE and Applications. Lecture Notes in Control and Information Sciences, vol. 61 (Springer, Berlin, 1984), pp. 294–313
38.
D. Talay, Second order discretization schemes of stochastic differential systems for the com- putation of the invariant law. Stochast. Stochast. Rep. 29(1), 13–36 (1990)CrossRefMATH
39.
E. Vanden-Eijnden, Numerical techniques for multiscale dynamical system with stochastic effects. Commun. Math. Sci. 1, 385–391 (2003)MathSciNetCrossRefMATH
40.
G. Vilmart, Postprocessed integrators for the high order integration of ergodic SDEs. SIAM J. Sci. Comput. 37(1), A201–A220 (2015)MathSciNetCrossRefMATH
41.
K.C. Zygalakis, On the existence and the applications of modified equations for stochastic differential equations. SIAM J. Sci. Comput. 33(1), 102–130 (2011)MathSciNetCrossRefMATH
Metadaten
Titel
Numerical Methods for Stochastic Simulation: When Stochastic Integration Meets Geometric Numerical Integration
verfasst von
Assyr Abdulle
Copyright-Jahr
2017
Buch
Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology

Print ISBN: 978-3-319-62626-0

Electronic ISBN: 978-3-319-62627-7

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-62627-7_4