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Reconstruction of source function in a parabolic equation using partial boundary measurements

T. Sharma, L. Beilina, K. Sakthivel

TL;DR

This work addresses the inverse problem of identifying the space-dependent source $F(x)$ in the parabolic equation $a(x)\frac{\partial u}{\partial t}-\Delta u = F(x)G(x,t)$ from partial boundary measurements. It adopts a Lagrangian-based variational approach with a regularized functional $J_\gamma(u,F)$ and derives the gradient $J'_\gamma(u,F)(x) = -\int_0^T G(x,t)\lambda(x,t)\,dt + \gamma(F-F_0)(x)$ via an adjoint system, establishing Fréchet differentiability and existence of minimizers. The authors prove a local stability estimate for the inverse problem and implement a Crank–Nicolson finite-difference discretization with a conjugate gradient algorithm that uses exact line search and an adaptive $\gamma$, validating the method on 2D and 3D tests with noisy data. Numerical experiments demonstrate accurate localization of $F(x)$ under modest noise, including discontinuous sources, confirming the practicality of the approach for partial-boundary parabolic inverse source problems. Overall, the paper provides a rigorous variational framework, provable regularity properties, and an effective algorithm for reconstructing spatial sources from incomplete boundary information.

Abstract

In this paper, we present the analytical and numerical study of the optimization approach for determining the space-dependent source function in the parabolic inverse source problem using partial boundary measurements. The Lagrangian approach for the solution of the optimization problem is presented, and optimality conditions are derived. The proof of the Fréchet differentiability of the regularized Tikhonov functional and the existence result for the solution of the inverse source problem are established. A local stability estimate for the unknown source term is also presented. The numerical examples justify the theoretical investigations using the conjugate gradient method (CGM) in 2D and 3D tests with noisy data.

Reconstruction of source function in a parabolic equation using partial boundary measurements

TL;DR

This work addresses the inverse problem of identifying the space-dependent source in the parabolic equation from partial boundary measurements. It adopts a Lagrangian-based variational approach with a regularized functional and derives the gradient via an adjoint system, establishing Fréchet differentiability and existence of minimizers. The authors prove a local stability estimate for the inverse problem and implement a Crank–Nicolson finite-difference discretization with a conjugate gradient algorithm that uses exact line search and an adaptive , validating the method on 2D and 3D tests with noisy data. Numerical experiments demonstrate accurate localization of under modest noise, including discontinuous sources, confirming the practicality of the approach for partial-boundary parabolic inverse source problems. Overall, the paper provides a rigorous variational framework, provable regularity properties, and an effective algorithm for reconstructing spatial sources from incomplete boundary information.

Abstract

In this paper, we present the analytical and numerical study of the optimization approach for determining the space-dependent source function in the parabolic inverse source problem using partial boundary measurements. The Lagrangian approach for the solution of the optimization problem is presented, and optimality conditions are derived. The proof of the Fréchet differentiability of the regularized Tikhonov functional and the existence result for the solution of the inverse source problem are established. A local stability estimate for the unknown source term is also presented. The numerical examples justify the theoretical investigations using the conjugate gradient method (CGM) in 2D and 3D tests with noisy data.

Paper Structure

This paper contains 16 sections, 10 theorems, 91 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Analysis of the Direct Problem and Inverse Problem
  3. Function Spaces
  4. Stability Analysis for the Direct Problem
  5. Lagrangian Approach for the Solution of ISP
  6. Stability Analysis for the Adjoint Problem
  7. Gradient of the Functional
  8. Existence of Minimizer for the Functional
  9. Stability Analysis for the Inverse Problem
  10. Numerical Reconstruction of Source Term
  11. Finite Difference Discretization
  12. Optimization Algorithms
  13. Numerical Examples
  14. Reconstruction of the Source Function $F$ in 2D
  15. Reconstruction of the Source Function $F$ in 3D
  16. ...and 1 more sections

Key Result

Theorem 2.1

Assume that the conditions of Assumption assm on the functions $a(x)$, $F(x)$ and $G(x,t)$ hold. Then there exists a unique weak solution $u \in L^2(0,T; H^1(\Omega))$, $\frac{\partial u}{\partial t}\in L^2(0,T; H^1(\Omega)^*)$ of the problem (1), which satisfies the following energy estimates: where $C_1(a,T) = \left(1+\frac{T}{a_\min}e^{\frac{T}{a_\min}}\right).$ Moreover, the weak solution $u$

Figures (7)

  • Figure 1: Exact $F$ and reconstructed function $F^m$ at the iteration $m= 40$ of CGA. Computations are performed on the mesh with $h = 2^{-6}$.
  • Figure 2: Exact $F$ and reconstructed function $F^m$ at the iteration $m= 40$ of CGA. Computations are performed on the mesh with $h = 2^{-6}$.
  • Figure 3: Exact $F$ and reconstructed function $F^m$ at the iteration $m= 40$ of CGA. Computations are performed on the mesh with $h = 2^{-6}$.
  • Figure 4: Exact $F$ and reconstructed function $F^m$ at the iteration $m= 40$ of CGA. Computations are performed on the mesh with $h = 2^{-6}$.
  • Figure 5: Exact $F$ and reconstructed function $F^m$ at the iteration $m= 40$ of CGA. Computations are performed on the mesh with $h = 2^{-6}$.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 12 more