Iterative computational identification of a spacewise dependent the source in a parabolic equations

TL;DR

This paper presents iterative numerical methods for identifying the spatial dependence of the right-hand side in multidimensional parabolic equations using overdetermined data, with demonstrated effectiveness through numerical examples.

Contribution

It introduces two novel iterative approaches for solving inverse problems involving spatially dependent sources in parabolic PDEs, utilizing final and integral overdetermination data.

Findings

Methods successfully recover spatial source dependence in 2D models.

Numerical examples validate the accuracy and stability of the proposed algorithms.

Finite-element and implicit schemes effectively implement the methods.

Abstract

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to a finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time - the final overdetermination. More general problems are associated with some integral observation of the solution on time - the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spacial variables. Capabilities of proposed methods are illustrated by numerical examples for model two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation on space is used, the time discretization is based on fully implicit two-level schemes.