Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
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2023
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| author | Klioba, Katharina Veraar, Mark |
| author_facet | Klioba, Katharina Veraar, Mark |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_00411 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs Klioba, Katharina Veraar, Mark Numerical Analysis Analysis of PDEs Functional Analysis Probability Primary: 65C30, Secondary: 47D06, 60H15, 60H35, 65J08, 65M12 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. |
| title | Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs |
| topic | Numerical Analysis Analysis of PDEs Functional Analysis Probability Primary: 65C30, Secondary: 47D06, 60H15, 60H35, 65J08, 65M12 |
| url | https://arxiv.org/abs/2303.00411 |