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Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

Sonja Cox, Arnulf Jentzen, Felix Lindner

TL;DR

The work addresses weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, focusing on the hyperbolic Anderson model. By splitting the discretisation error and employing a mild Itô formula, the authors establish essentially sharp weak convergence rates of order $1^{-}$ for exponential Euler-type time stepping, which is twice the rate typically seen for strong convergence. The framework handles Hilbert-space SPDEs with Lipschitz coefficients and multiplicative noise, and the main results extend to HA with rigorous multiplier and Sobolev-space estimates; numerical simulations in Python corroborate the theoretical rates. This provides a principled guidance for the design and analysis of weak approximations of hyperbolic SPDEs and suggests robustness of the mild-Itô methodology to other discretisations.

Abstract

In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. For this class of stochastic partial differential equations the weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild Itô formula. We complement our analytical findings by means of numerical simulations in Python for the decay of the weak approximation error for SPDEs for four different test functions.

Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

TL;DR

The work addresses weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, focusing on the hyperbolic Anderson model. By splitting the discretisation error and employing a mild Itô formula, the authors establish essentially sharp weak convergence rates of order for exponential Euler-type time stepping, which is twice the rate typically seen for strong convergence. The framework handles Hilbert-space SPDEs with Lipschitz coefficients and multiplicative noise, and the main results extend to HA with rigorous multiplier and Sobolev-space estimates; numerical simulations in Python corroborate the theoretical rates. This provides a principled guidance for the design and analysis of weak approximations of hyperbolic SPDEs and suggests robustness of the mild-Itô methodology to other discretisations.

Abstract

In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. For this class of stochastic partial differential equations the weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild Itô formula. We complement our analytical findings by means of numerical simulations in Python for the decay of the weak approximation error for SPDEs for four different test functions.

Paper Structure

This paper contains 17 sections, 22 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting
  3. Preliminaries
  4. Stochastic differential equations in Hilbert spaces
  5. Kolmogorov equations in Hilbert spaces
  6. Preparatory lemmas
  7. Weak convergence rates for temporal numerical approximations of semilinear stochastic wave equations
  8. Setting
  9. Basic results for the linear wave equation
  10. A priori bounds for the numerical approximations
  11. Upper bounds for the strong approximation errors
  12. Upper bounds for the weak approximation errors
  13. Weak convergence rates for temporal numerical approximations of the hyperbolic Anderson model
  14. Setting
  15. Preparatory lemmas
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $T, \vartheta \in ( 0 , \infty )$, $b_0,\, b_1 \in {\mathbbm R}$, $H=L^2((0,1);{\mathbbm R})$, let $( \Omega , \mathcal{F} , {\mathbbm P}, (\mathbbm{F}_t )_{ t \in [ 0 , T ] } )$ be a filtered probability space which fulfills the usual conditions, let $( W_t )_{ t \in [ 0 , T ] }$ be an $\mathop and let $Y^{N}\colon \{0,1,2,\ldots,N\} \times \Omega \rightarrow \mathbf{H}_0$, $N\in {\mathbbm N}

Figures (1)

  • Figure 1: Approximate simulations of the weak error estimator \ref{['eq:simulation_error']} for different test functions $\varphi(x)$, $x=(x^{(1)},x^{(2)})\in\mathbf H_0$, plotted against the used numbers of time steps $N\in\{2^3,2^4,\ldots,2^{11}\}$, employing $\mathbf N=2^{12}$ time steps for the reference value, $J=16$ spatial basis functions, and $M=5\cdot 10^5$ Monte Carlo runs (see the text and Listing \ref{['listing']} for details)

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['lem:kolmogorov']}.
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['lemma:Lip_est_standard']}
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['lem:sup_estimates']}.
  • Lemma 2.5
  • proof : Proof of Lemma \ref{['lemma:Schatten_hoelder']}.
  • ...and 28 more