Weak error analysis for the stochastic Allen–Cahn equation

IF 1.4 3区数学 Q2 MATHEMATICS, APPLIED

Stochastics and Partial Differential Equations-Analysis and Computations Pub Date : 2024-02-22 DOI:10.1007/s40072-024-00326-z

Dominic Breit, Andreas Prohl

引用次数: 0

Abstract

We prove strong rate resp. weak rate \({{\mathcal {O}}}(\tau )\) for a structure preserving temporal discretization (with \(\tau \) the step size) of the stochastic Allen–Cahn equation with additive resp. multiplicative colored noise in \(d=1,2,3\) dimensions. Direct variational arguments exploit the one-sided Lipschitz property of the cubic nonlinearity in the first setting to settle first order strong rate. It is the same property which allows for uniform bounds for the derivatives of the solution of the related Kolmogorov equation, and then leads to weak rate \({{\mathcal {O}}}(\tau )\) in the presence of multiplicative noise. Hence, we obtain twice the rate of convergence known for the strong error in the presence of multiplicative noise.

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随机艾伦-卡恩方程的弱误差分析

我们证明了在\(d=1,2,3\)维度上具有加法和乘法彩色噪声的随机Allen-Cahn方程的结构保持时间离散化（步长为\(\tau \)）的强率和弱率（{{\mathcal {O}}} (\tau )\）。直接变分论证利用了立方非线性在第一种情况下的单边立普齐兹特性来解决一阶强率问题。正是这一性质使得相关的科尔莫哥罗夫方程的解的导数有了统一的边界，进而在存在乘法噪声的情况下得到弱率（{{mathcal {O}} (\tau )\）。因此，我们得到的收敛率是存在乘法噪声时强误差的两倍。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Stochastics and Partial Differential Equations-Analysis and Computations Mathematics-Modeling and Simulation

CiteScore

2.70

自引率

13.30%

发文量

期刊介绍： Stochastics and Partial Differential Equations: Analysis and Computations publishes the highest quality articles presenting significantly new and important developments in the SPDE theory and applications. SPDE is an active interdisciplinary area at the crossroads of stochastic anaylsis, partial differential equations and scientific computing. Statistical physics, fluid dynamics, financial modeling, nonlinear filtering, super-processes, continuum physics and, recently, uncertainty quantification are important contributors to and major users of the theory and practice of SPDEs. The journal is promoting synergetic activities between the SPDE theory, applications, and related large scale computations. The journal also welcomes high quality articles in fields strongly connected to SPDE such as stochastic differential equations in infinite-dimensional state spaces or probabilistic approaches to solving deterministic PDEs.