Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inverse Random Source and Cauchy Problems for Semi-Discrete Stochastic Parabolic Equations in Arbitrary Dimensions

Rodrigo Lecaros, Ariel A. Pérez, Manuel F. Prado

Abstract

In this paper, we study two types of inverse problems for space semi-discrete stochastic parabolic equations in arbitrary dimensions. The first problem concerns a semi-discrete inverse source problem, which involves determining the random source term of the white noise in the semi-discrete stochastic parabolic equation using observation data of the solution at the terminal time and on an arbitrary open spatial subdomain over a time interval. The second problem addresses a semi-discrete Cauchy inverse problem, which involves determining the solution of the stochastic parabolic equation in a space-time subdomain, from measurements of the solution and the trace of its discrete spatial derivative on an arbitrary open subset of the lateral boundary over a time interval. The key tools are three new global Carleman estimates for the semi-discrete stochastic parabolic operator, one for interior observations and two for boundary observations (homogeneous and nonhomogeneous Dirichlet conditions). Applying these Carleman estimates, we obtain Lipschitz and Hölder stability for the first and second inverse problems, respectively.

Inverse Random Source and Cauchy Problems for Semi-Discrete Stochastic Parabolic Equations in Arbitrary Dimensions

Abstract

In this paper, we study two types of inverse problems for space semi-discrete stochastic parabolic equations in arbitrary dimensions. The first problem concerns a semi-discrete inverse source problem, which involves determining the random source term of the white noise in the semi-discrete stochastic parabolic equation using observation data of the solution at the terminal time and on an arbitrary open spatial subdomain over a time interval. The second problem addresses a semi-discrete Cauchy inverse problem, which involves determining the solution of the stochastic parabolic equation in a space-time subdomain, from measurements of the solution and the trace of its discrete spatial derivative on an arbitrary open subset of the lateral boundary over a time interval. The key tools are three new global Carleman estimates for the semi-discrete stochastic parabolic operator, one for interior observations and two for boundary observations (homogeneous and nonhomogeneous Dirichlet conditions). Applying these Carleman estimates, we obtain Lipschitz and Hölder stability for the first and second inverse problems, respectively.

Paper Structure

This paper contains 13 sections, 16 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Semi-discrete inverse random source problem
  3. Proof of Theorem \ref{['firstresulttheorem']}.
  4. Semi-discrete Cauchy problem.
  5. Semi-discrete Carleman estimate with nonhomogeneous Dirichlet condition.
  6. Proof of Theorem \ref{['Teo:SecondResult']}.
  7. Comments and concluding remarks
  8. Proof of semi-discrete Carleman estimate with homogeneous Dirichlet condition (Theorem \ref{['theo:Carleman_datainterior']}).
  9. Conjugated operator
  10. An estimate for the cross-product
  11. Returning to the original variable
  12. Technical steps for the cross-product estimate
  13. Technical steps for return to the original variable for the case boundary data in the right-hand side

Key Result

Theorem 1.1

Let $a_{1i}\in L^\infty_{\mathcal{F}}(0,T;L_h^\infty(\mathcal{M}))$ for $i=1,\ldots,n$, and $a_2\in L_{\mathcal{F}}^{\infty}(0,T;L_h^{n^{\ast}}(\mathcal{M}))$ where $n^{\ast}$ satisfies Assume that $g_k \in L^2_{\mathcal{F}}(0,T;H_h^1(\mathcal{M}))$ for $k=1,2$, and that, for $i=1,\ldots,n$, Then, there exists a constant $\mathcal{C} > 0$, independent of $h$, such that

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 1.6: LOPD:2023
  • Proposition 1.7: BHLR:2010b
  • Remark 2.2
  • Example 3.2
  • Theorem 3.3
  • ...and 17 more