[Paper Review] Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs

Katharina Klioba, Mark Veraar|arXiv (Cornell University)|Mar 1, 2023

Numerical methods for differential equations59 references4 citations

TL;DR

This paper establishes optimal pathwise uniform convergence rates for time discretisation schemes in semi-linear stochastic PDEs with non-parabolic (hyperbolic) operators and additive/multiplicative Gaussian noise. Using a Kato-type setting with Hilbert spaces $X$ and $Y$, it derives bounds for the uniform strong error $\mathbb{E}\left[\sup_{j}\|U(t_j)-U^j\|^{p}\right]^{1/p}$, achieving rates $\lesssim k\log(T/k)$ for linear equations with additive noise and $\lesssim \sqrt{k}\log(T/k)$ for nonlinear equations with multiplicative noise under contractive semigroups, with logarithmic factors removable for exponential Euler under (quasi)-contractive $S$. The results unify and extend prior work on Maxwell, Schrödinger, and wave equations.

ABSTRACT

In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly continuous semigroup $S$ on a Hilbert space $X$, and the focus is on non-parabolic problems. The main results are optimal bounds for the uniform strong error $$\mathrm{E}_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|U(t_j) - U^j\|^p\Big)^{1/p},$$ where $p \in [2,\infty)$, $U$ is the mild solution, $U^j$ is obtained from a time discretisation scheme, $k$ is the step size, and $N_k = T/k$. The usual schemes such as the exponential Euler, the implicit Euler, and the Crank-Nicolson method, etc. are included as special cases. Under conditions on the nonlinearity and the noise, we show - $\mathrm{E}_{k}^{\infty}\lesssim k \sqrt{\log(T/k)}$ (linear equation, additive noise, general $S$); - $\mathrm{E}_{k}^{\infty}\lesssim \sqrt{k} \sqrt{\log(T/k)}$ (nonlinear equation, multiplicative noise, contractive $S$); - $\mathrm{E}_{k}^{\infty}\lesssim k \sqrt{\log(T/k)}$ (nonlinear wave equation, multiplicative noise) for a large class of time discretisation schemes. The logarithmic factor can be removed if the exponential Euler method is used with a (quasi)-contractive $S$. The obtained bounds coincide with the optimal bounds for SDEs. Most of the existing literature is concerned with bounds for the simpler pointwise strong error $$\mathrm{E}_k:=\bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E} \|U(t_j) - U^{j}\|^p\bigg)^{1/p}.$$ Applications to Maxwell equations, Schrödinger equations, and wave equations are included. For these equations, our results improve and reprove several existing results with a unified method and provide the first results known for the implicit Euler and the Crank-Nicolson method.

Motivation & Objective

• To establish optimal convergence rates for time discretisation schemes in non-parabolic (hyperbolic) SPDEs with additive or multiplicative noise.
• To address the gap in the literature by focusing on the pathwise uniform strong error $\mathbb{E}\left[\sup_j \|U(t_j)-U^j\|^{p}\right]^{1/p}$, which better captures pathwise approximation quality than pointwise error.
• To unify and extend existing results for implicit Euler, Crank-Nicolson, and exponential Euler schemes across Maxwell, Schrödinger, and wave equations.
• To provide the first known convergence rates for implicit Euler and Crank-Nicolson schemes in the hyperbolic setting with pathwise uniform error control.
• To remove or minimize logarithmic factors in convergence bounds using the exponential Euler method under (quasi)-contractive semigroups.

Proposed method

• The analysis uses a Kato-type framework with two Hilbert spaces $X$ and $Y$ ($Y \hookrightarrow X$) to derive regularity for the mild solution and nonlinearities.
• The time discretisation scheme is formulated as $U^j = R_k U^{j-1} + k R_k F(U^{j-1}) + R_k G(U^{j-1}) \Delta W_j$, where $R_k$ approximates the semigroup $S(k)$.
• Key estimates rely on stochastic maximal $L^p$-regularity, stochastic convolution bounds, and Sobolev embeddings to control the nonlinearities $F$ and $G$ in the $Y$-space.
• The proof uses a priori bounds on the solution $U$ and its discrete approximation $U^j$, leveraging the contractivity and approximation order $\alpha$ of the scheme $R_k$ on $Y$.
• Logarithmic factors in convergence rates are removed via a refined analysis of the exponential Euler method, exploiting its (quasi)-contractive properties.
• The framework applies to abstract SPDEs with generators $A$ of $C_0$-semigroups, covering wave, Maxwell, and Schrödinger equations via appropriate choice of $X$, $Y$, and $A$.

Experimental results

Research questions

• RQ1 What are the optimal pathwise uniform convergence rates for time discretisation schemes in non-parabolic SPDEs with additive or multiplicative noise?
• RQ2 Can the logarithmic factor in convergence bounds be removed for specific schemes like exponential Euler under (quasi)-contractive semigroups?
• RQ3 How do the convergence rates for implicit Euler and Crank-Nicolson schemes compare in the hyperbolic setting with pathwise uniform error control?
• RQ4 To what extent do the results unify and extend prior findings for Maxwell, Schrödinger, and wave equations?
• RQ5 Can the Kato-type framework with two spaces $X$ and $Y$ yield sharp bounds for the uniform strong error in the absence of parabolic regularization?

Key findings

• For linear SPDEs with additive noise and general $C_0$-semigroup $S$, the uniform strong error satisfies $\mathbb{E}\left[\sup_j \|U(t_j)-U^j\|^{p}\right]^{1/p} \lesssim k \log(T/k)$.
• For nonlinear SPDEs with multiplicative noise and contractive semigroup $S$, the uniform strong error is bounded by $\lesssim \sqrt{k} \log(T/k)$.
• For the nonlinear wave equation with multiplicative noise, the uniform strong error is bounded by $\lesssim k \log(T/k)$.
• The logarithmic factor can be removed for the exponential Euler method when $S$ is (quasi)-contractive, yielding $\lesssim k$ convergence.
• For implicit Euler and Crank-Nicolson schemes, the convergence rate is $k^{\delta/2}$ and $k^{\min(2\delta/3,1)}$ respectively, with $\delta \in (1,2]$ controlling noise regularity.
• The results recover and improve upon known bounds for SDEs and provide the first pathwise uniform convergence rates for implicit Euler and Crank-Nicolson in the hyperbolic SPDE setting.