Stability and regularization for ill-posed Cauchy problem of a stochastic parabolic differential equation

2023/08/30 by Fangfang Dou, Dou, Fangfang, Peimin Lu +2 · 2 citations
Mathematics · #35R30 #60H15 #65N21 #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2308.15741

openalex publication_date 2023/08/30 · openalex created_date 2023/09/02 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we derive the conditional stability and convergence rate of the Tikhonov regularization method for the aforementioned ill-posed Cauchy problem. To complement our theoretical analysis, we employ kernel-based learning theory to implement the completed Tikhonov regularization method for several numerical examples.

Cited by

• Lipschitz stability for an inverse problem of a fully-discrete stochastic hyperbolic equation
• Inverse problem for semidiscrete stochastic parabolic operators in arbitrary dimensions

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