Finite Element Approximation of Lyapunov Equations Related to Parabolic Stochastic PDEs

IF 1.6 2区数学 Q2 MATHEMATICS, APPLIED

Applied Mathematics and Optimization Pub Date : 2025-05-03 DOI:10.1007/s00245-025-10260-8

Adam Andersson, Annika Lang, Andreas Petersson, Leander Schroer

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引用次数: 0

Abstract

A numerical analysis for the fully discrete approximation of an operator Lyapunov equation related to linear stochastic partial differential equations (SPDEs) driven by multiplicative noise is considered. The discretization of the Lyapunov equation in space is given by finite elements and in time by a semiimplicit Euler scheme. The main result is the derivation of the rate of convergence in operator norm. Moreover, it is shown that the solution of the equation provides a representation of a quadratic and path dependent functional of the SPDE solution. This fact yields a deterministic numerical method to compute such functionals. As a secondary result, weak error rates are established for a fully discrete finite element approximation of the SPDE with respect to this functional. This is obtained as a consequence of the approximation analysis of the Lyapunov equation. It is the first weak convergence analysis for fully discrete finite element approximations of SPDEs driven by multiplicative noise that obtains double the strong rate of convergence, especially for path dependent functionals and smooth spatial noise. Numerical experiments illustrate the results empirically, and it is demonstrated that the deterministic method has advantages over Monte Carlo sampling in a stability context.

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抛物型随机偏微分方程Lyapunov方程的有限元逼近

研究了乘性噪声驱动下线性随机偏微分方程算子Lyapunov方程的全离散逼近问题。李雅普诺夫方程在空间上的离散由有限元给出，在时间上的离散由半隐式欧拉格式给出。主要结果是算子范数的收敛速率的推导。此外，还证明了该方程的解提供了SPDE解的二次和路径相关泛函的表示。这一事实产生了计算这种泛函的确定性数值方法。作为次要结果，建立了相对于该泛函的SPDE的完全离散有限元近似的弱错误率。这是李亚普诺夫方程近似分析的结果。本文首次对乘性噪声驱动下的全离散SPDEs有限元近似进行弱收敛分析，得到了两倍的强收敛速率，特别是对于路径相关泛函和光滑空间噪声。数值实验对结果进行了实证验证，结果表明确定性方法在稳定性情况下优于蒙特卡罗采样方法。

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来源期刊

Applied Mathematics and Optimization 数学-应用数学

CiteScore

3.30

自引率

5.60%

发文量

103

审稿时长

>12 weeks

期刊介绍： The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.