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

Fangfang Dou, Peimin Lü, Yu Wang

TL;DR

This work addresses the ill-posed Cauchy problem for a forward stochastic parabolic equation by developing a Carleman-estimate framework that yields a Hölder-type conditional stability result. It then constructs a Tikhonov regularization functional and proves existence, uniqueness, and a convergence rate of the regularized solution under data perturbations, with the rate governed by the noise level via a parameter choice $\gamma=\delta^{2\alpha}$. To enable practical computation without solving adjoint problems, the authors adopt a kernel-based numerical scheme that expresses the solution as a kernel expansion using the deterministic fundamental solution, solved in one shot through a linear system and L-curve based regularization. Numerical experiments in 1D and 2D demonstrate stable reconstructions of lateral boundary data under noise, highlighting the method's effectiveness and computational efficiency for SPDE Cauchy problems.

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.

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

TL;DR

This work addresses the ill-posed Cauchy problem for a forward stochastic parabolic equation by developing a Carleman-estimate framework that yields a Hölder-type conditional stability result. It then constructs a Tikhonov regularization functional and proves existence, uniqueness, and a convergence rate of the regularized solution under data perturbations, with the rate governed by the noise level via a parameter choice . To enable practical computation without solving adjoint problems, the authors adopt a kernel-based numerical scheme that expresses the solution as a kernel expansion using the deterministic fundamental solution, solved in one shot through a linear system and L-curve based regularization. Numerical experiments in 1D and 2D demonstrate stable reconstructions of lateral boundary data under noise, highlighting the method's effectiveness and computational efficiency for SPDE Cauchy problems.

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.
Paper Structure (4 sections, 4 theorems, 100 equations, 9 figures)

This paper contains 4 sections, 4 theorems, 100 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Conditional Stability
  3. Regularization Method
  4. Numerical Approximations

Key Result

Theorem 2.1

(Hölder stability estimate). For any given $G'\subset\subset G$ and $\varepsilon \in (0,\frac{T}{2})$, there exists $\delta _{0} \in \left( 0,1\right)$, $\beta\in (0,1)$ and a constant $C >0$ such that if for $\delta \in \left( 0,\delta _{0}\right)$, then

Figures (9)

  • Figure 4.1: Approximated solution with different noisy levels of Example \ref{['1dex']}(a).
  • Figure 4.2: Numerical results by expectation of solutions of different numbers of sample paths.
  • Figure 4.3: Numerical results of Example \ref{['1dex']}(b) with different noisy level data $\delta$.
  • Figure 4.4: The relative error $E(x_1,x_2)$ for different $R$.
  • Figure 4.5: The relative error $E(x_1,x_2)$ for different $DT$.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Definition 1.1
  • Theorem 2.1
  • Lemma 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Example 4.1
  • Example 4.2
  • Example 4.3