Identification of the Source for Full Parabolic Equations

TL;DR

This paper addresses the ill-posed problem of identifying a static source in full parabolic equations from noisy data, proposing a regularization method with proven stability and convergence, supported by numerical tests.

Contribution

It introduces a parametric regularization approach with a data-dependent parameter choice rule for source identification in parabolic equations.

Findings

Proves stability and convergence of the regularization method.

Establishes a H"older type bound for estimation error.

Numerical examples demonstrate effectiveness of the approach.

Abstract

In this work, we consider the problem of identifying the time independent source for full parabolic equations in $\mathbb{R}^n$ from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a H\"older type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.