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An inverse Cauchy problem of a stochastic hyperbolic equation

Fangfang Dou, Peimin Lü

Abstract

In this paper, we investigate an inverse Cauchy problem for a stochastic hyperbolic equation. A Lipschitz type observability estimate is established using a pointwise Carleman identity. By minimizing the constructed Tikhonov-type functional, we obtain a regularized approximation to the problem. The properties of the approximation are studied by means of the Carleman estimate and Riesz representation theorem. Leveraging kernel-based learning theory, we simulate numerical algorithms based on the proposed regularization method. These reconstruction algorithms are implemented and validated through several numerical experiments, demonstrating their feasibility and accuracy.

An inverse Cauchy problem of a stochastic hyperbolic equation

Abstract

In this paper, we investigate an inverse Cauchy problem for a stochastic hyperbolic equation. A Lipschitz type observability estimate is established using a pointwise Carleman identity. By minimizing the constructed Tikhonov-type functional, we obtain a regularized approximation to the problem. The properties of the approximation are studied by means of the Carleman estimate and Riesz representation theorem. Leveraging kernel-based learning theory, we simulate numerical algorithms based on the proposed regularization method. These reconstruction algorithms are implemented and validated through several numerical experiments, demonstrating their feasibility and accuracy.

Paper Structure

This paper contains 2 sections, 2 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Observability estimate

Key Result

Theorem 2.1

Let Conditions condition of d and condition2 be satisfied. For any solution of the equation system1, we have where

Theorems & Definitions (5)

  • Remark 1.1
  • Theorem 2.1
  • Proposition 2.1
  • Remark 2.1
  • proof