We consider numerical schemes for computing the linear response of steady-state averages with respect to a perturbation of the drift part of the stochastic differential equation. The schemes are based on the Girsanov change-of-measure theory in order to reweight trajectories with factors derived from a linearization of the Girsanov weights. The resulting estimator is the product of a time average and a martingale correlated to this time average. We investigate both its discretization and finite-time approximation errors. The designed numerical schemes are shown to be of a bounded variance with respect to the integration time which is desirable feature for long time simulations. We also show how the discretization error can be improved to second-order accuracy in the time step by modifying the weight process in an appropriate way.
Keywords: Non-equilibrium steady states, linear response, stochastic differential equations, likelihood ratio method, variance reduction
@article{M2AN_2021__55_S1_S593_0,
author = {Plech\'a\v{c}, Petr and Stoltz, Gabriel and Wang, Ting},
title = {Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {S593--S623},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {Suppl\'ement},
doi = {10.1051/m2an/2020050},
mrnumber = {4221323},
zbl = {07395682},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020050/}
}
TY - JOUR AU - Plecháč, Petr AU - Stoltz, Gabriel AU - Wang, Ting TI - Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - S593 EP - S623 VL - 55 IS - Supplément PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020050/ DO - 10.1051/m2an/2020050 LA - en ID - M2AN_2021__55_S1_S593_0 ER -
%0 Journal Article %A Plecháč, Petr %A Stoltz, Gabriel %A Wang, Ting %T Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P S593-S623 %V 55 %N Supplément %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020050/ %R 10.1051/m2an/2020050 %G en %F M2AN_2021__55_S1_S593_0
Plecháč, Petr; Stoltz, Gabriel; Wang, Ting. Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S593-S623. doi: 10.1051/m2an/2020050
[1] and , Stochastic Simulation: Algorithms and Analysis. Springer, New York, NY (2007). | MR | Zbl
[2] , , and , Computation of sensitivities for the invariant measure of a parameter dependent diffusion. Stochastics Part. Differ. Equ.: Anal. Comput. 6 (2018) 125–183. | MR | Zbl
[3] and , The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Relat. Fields 104 (1996) 43–60. | MR | Zbl
[4] , Ergodic properties of Markov processes. In: Open Quantum Systems II. Springer, Berlin-Heidelberg (2006) 1–39. | MR | Zbl
[5] , On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Probab. Theory Relat. Fields 60 (1982) 185–201. | MR | Zbl
[6] , Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. Z. Wahrsch. Verw. Gebiete 56 (1981) 469–505. | MR | Zbl
[7] and , Nonasymptotic mixing of the MALA algorithm. IMA J. Numer. Anal. 33 (2013) 80–110. | MR | Zbl
[8] and , Statistical Mechanics of Nonequilibrium Liquids. Cambridge University Press (2008). | Zbl
[9] , and , Error analysis of the transport properties of Metropolized schemes. ESAIM: Procs. 48 (2015) 341–363. | MR | Zbl
[10] and , Improving dynamical properties of metropolized discretizations of overdamped Langevin dynamics. Numer. Math. 136 (2017) 1–58. | MR | Zbl | DOI
[11] and , Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin-Heidelberg (2001). | MR | Zbl
[12] . Monte Carlo Methods in Financial Engineering. In: Vol. 53 of Stochastic Modelling and Applied Probability. Springer, New York, NY (2013). | MR | Zbl
[13] , Likelihood ratio gradient estimation for stochastic systems. Commun. ACM 33 (1990) 75–84. | DOI
[14] and , Likelihood ratio gradient estimation for steady-state parameters. Stochastic Syst. 9 (2019) 83–181. | MR | Zbl | DOI
[15] and , Yet another look at Harris’ ergodic theorem for Markov chains. In: Seminar on Stochastic Analysis, Random Fields and Applications VI, Springer, Berlin-Heidelberg (2011) 109–117. | MR | Zbl | DOI
[16] , Recurrence and invariant measures for degenerate diffusions. Ann. Probab. 15 (1987) 690–707. | MR | Zbl | DOI
[17] and , Numerical Solution of Stochastic Differential Equations. Vol. 23 of Stochastic Modelling and Applied Probability. Springer, Berlin-Heidelberg 23 (2013). | MR | Zbl
[18] , Weak backward error analysis for Langevin process. BIT Numer. Math. 55 (2015) 1057–1103. | MR | Zbl | DOI
[19] , and , The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. IMA J. Numer. Anal. 36 (2016) 13–79. | MR | Zbl
[20] and , Partial differential equations and stochastic methods in molecular dynamics. Acta Numer. 25 (2016) 681–880. | MR | Zbl | DOI
[21] , and , Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM J. Numer. Anal. 48 (2010) 552–577. | MR | Zbl | DOI
[22] and , Markov Chains and Stochastic Stability. Springer, London (2012). | Zbl | MR
[23] , Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin-Heidelberg (2013). | Zbl | MR
[24] , Stochastic Integration and Differential Equations. Vol. 21 of Stochastic Modelling and Applied Probability. Springer, Berlin-Heidelberg (2005). | MR | Zbl
[25] , and , Error analysis of modified Langevin dynamics. J. Stat. Phys. 164 (2016) 735–771. | MR | Zbl | DOI
[26] and , Simulation and the Monte Carlo Method. Wiley Series in Probability and Statistics. John Wiley & Sons Inc., Hoboken, NJ (2017). | MR | Zbl
[27] , Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Proc. Rel. Fields 8 (2002) 163–198. | MR | Zbl
[28] , Statistical Mechanics: Theory and Molecular Simulation. Oxford University Press (2010). | MR | Zbl
[29] and , Steady-state sensitivity analysis of continuous time Markov chains. SIAM J. Numer. Anal. 57 (2019) 192–217. | MR | Zbl | DOI
[30] and , Efficiency of the Girsanov transformation approach for parametric sensitivity analysis of stochastic chemical kinetics. SIAM/ASA J. Uncertainty Quant. 4 (2016) 1288–1322. | MR | Zbl | DOI
[31] and , On the validity of the Girsanov transformation method for sensitivity analysis of stochastic chemical reaction networks. Preprint: arXiv:1807.09935 (2018).
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