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Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Bin Wu, Ying Wang, Zewen Wang

Abstract

In this paper, we study discrete Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation. As applications of these discrete Carleman estimates, we apply them to study two inverse problems for the spatial semi-discrete stochastic parabolic equations, including a discrete inverse random source problem and a discrete Cauchy problem. We firstly establish two Carleman estimates for a one-dimensional semi-discrete stochastic parabolic equation, one for homogeneous boundary and the other for non-homogeneous boundary. Then we apply these two estimates separately to derive two stability results. The first one is the Lipschitz stability for the discrete inverse random source problem. The second one is the Hölder stability for the discrete Cauchy problem.

Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Abstract

In this paper, we study discrete Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation. As applications of these discrete Carleman estimates, we apply them to study two inverse problems for the spatial semi-discrete stochastic parabolic equations, including a discrete inverse random source problem and a discrete Cauchy problem. We firstly establish two Carleman estimates for a one-dimensional semi-discrete stochastic parabolic equation, one for homogeneous boundary and the other for non-homogeneous boundary. Then we apply these two estimates separately to derive two stability results. The first one is the Lipschitz stability for the discrete inverse random source problem. The second one is the Hölder stability for the discrete Cauchy problem.
Paper Structure (13 sections, 6 theorems, 154 equations)

This paper contains 13 sections, 6 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Discrete settings and main results
  3. Discrete settings
  4. Main results
  5. Discrete Carleman estimates
  6. Proof of Theorem 2.3
  7. Estimates involving the terms of ${\rm d}Y_h$
  8. Estimates involving the terms of ${\rm d}t$
  9. Positive lower bounds for the terms of $|Y_h|^2$ and $|\mathbf D_h^+ Y_h|^2$
  10. The remainder of the proof of Theorem 2.3
  11. Proof of Theorem 2.4
  12. Proof of Theorem 2.5
  13. Proof of Theorem 2.6

Key Result

Lemma 2.1

Let $u$ and $v$ be discrete functions defined on $\overline G_h$. Then for the averaging operators and difference operators, we have the following identities:

Theorems & Definitions (6)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6