Computational identification of the lowest space-wise dependent coefficient of a parabolic equation

TL;DR

This paper presents a computational method for identifying the lowest spatially dependent coefficient in a parabolic equation using an iterative approach and final-time data, with proven monotonic convergence.

Contribution

It introduces a novel iterative algorithm for solving a nonlinear inverse problem to determine the lowest coefficient in a parabolic PDE, with convergence analysis and numerical validation.

Findings

The iterative process converges monotonically from above.

Numerical examples demonstrate the algorithm's effectiveness.

The method successfully recovers the coefficient in a 2D model problem.

Abstract

In the present work, we consider a nonlinear inverse problem of identifying the lowest coefficient of a parabolic equation. The desired coefficient depends on spatial variables only. Additional information about the solution is given at the final time moment, i.e., we consider the final redefinition. An iterative process is used to evaluate the lowest coefficient, where at each iteration we solve the standard initial-boundary value problem for the parabolic equation. On the basis of the maximum principle for the solution of the differential problem, the monotonicity of the iterative process is established along with the fact that the coefficient approaches from above. The possibilities of the proposed computational algorithm are illustrated by numerical examples for a model two-dimensional problem.