Regularization Techniques for Estimating the Source in a Complete Parabolic Equation in $\mathbb{R}^n$

TL;DR

This paper addresses the ill-posed problem of estimating the source in a complete parabolic equation using Fourier methods, proposing regularization techniques to stabilize solutions from noisy data, with theoretical and numerical validation.

Contribution

It introduces three novel regularization operators tailored for this inverse problem and provides a parameter choice rule along with error bounds.

Findings

Regularization operators improve stability of source estimation.

Theoretical error bounds demonstrate effectiveness.

Numerical examples confirm practical benefits.

Abstract

In this article, the problem of identifying the source term in transport processes given by a complete parabolic equation is studied mathematically from noisy measurements taken at an arbitrary fixed time. The problem is solved analytically with Fourier techniques and it is shown that this solution is not stable. Three single parameter families of regularization operators are proposed to dealt with the instability of the solution. Each of them is designed to compensate the factor that causes the instability of the inverse operator. Moreover, a rule of choice for the regularization parameter is included and a H\"older error bound type is obtained for each estimation. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies.