[Paper Review] Weak backward error analysis for Langevin process

Marie Kopec|arXiv (Cornell University)|Oct 9, 2013

Markov Chains and Monte Carlo Methods14 references4 citations

TL;DR

This paper presents a weak backward error analysis for implicit numerical schemes applied to stochastic Langevin equations. It demonstrates that the generator of the numerical solution approximates a modified Kolmogorov operator up to high-order terms in stepsize, implying the numerical invariant measure closely matches a modified invariant measure via asymptotic expansion. Furthermore, the scheme's dynamics are shown to be exponentially mixing up to negligible errors, ensuring long-time stability and accuracy.

ABSTRACT

We consider numerical approximations of stochastic Langevin equations by implicit methods. We show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the implicit scheme considered is exponentially mixing.

Motivation & Objective

• To analyze the long-time behavior of implicit numerical schemes for stochastic Langevin equations.
• To establish a weak backward error analysis framework for such schemes.
• To demonstrate that the numerical solution's invariant measure is close to a modified invariant measure derived from asymptotic expansion.
• To prove that the dynamic of the implicit scheme is exponentially mixing up to negligible terms.
• To provide theoretical justification for the accuracy and stability of implicit methods in simulating ergodic diffusion processes.

Proposed method

• The analysis uses the generator of the numerical scheme and compares it to a modified Kolmogorov equation with higher-order corrections in stepsize.
• Asymptotic expansion techniques are applied to derive the modified invariant measure corresponding to the perturbed generator.
• The method relies on weak convergence analysis to relate the numerical solution to the solution of a modified SDE.
• The proof leverages properties of the generator and perturbation theory to control error terms in the Kolmogorov equation.
• Exponential mixing is established by analyzing the spectral properties of the modified generator under suitable assumptions.
• The framework is applied to implicit time-stepping schemes, such as the implicit Euler method, for SDEs with drift and diffusion coefficients.

Experimental results

Research questions

• RQ1 How does the generator of an implicit numerical scheme for a Langevin SDE relate to a modified Kolmogorov operator?
• RQ2 To what extent is the numerical invariant measure close to a modified invariant measure obtained via asymptotic expansion?
• RQ3 Can the dynamic of the implicit scheme be shown to be exponentially mixing, up to negligible errors?
• RQ4 What is the order of accuracy of the weak backward error analysis for implicit schemes in the context of Langevin processes?
• RQ5 How do higher-order terms in the stepsize affect the long-time statistical properties of the numerical solution?

Key findings

• The generator of the numerical scheme coincides with that of a modified Kolmogorov equation up to high-order terms in the stepsize, ensuring high weak accuracy.
• The numerical invariant measure is arbitrarily close to a modified invariant measure derived through asymptotic expansion of the generator.
• The dynamic of the implicit scheme is exponentially mixing, which implies fast convergence to equilibrium and stability in long-time simulations.
• The analysis holds for general implicit schemes and applies to a wide class of stochastic Langevin equations with smooth coefficients.
• The results justify the use of implicit methods for simulating ergodic diffusions with controlled statistical error and long-time accuracy.
• The framework provides a theoretical basis for the reliability of implicit schemes in sampling from invariant measures of SDEs.