A Regularization Operator for the Source Approximation of a Transport Equation

TL;DR

This paper introduces a parametric regularization operator for identifying time-dependent sources in a transport equation from noisy data, providing a method to select regularization parameters based on noise level, with proven error bounds and numerical validation.

Contribution

The paper proposes a novel family of regularization operators and a parameter selection method for source identification in transport equations, addressing ill-posedness with theoretical error bounds.

Findings

The regularization method converges and is stable in numerical experiments.

A H"older type error bound is established for the source approximation.

Parameter selection adapts to noise levels, improving solution accuracy.

Abstract

Source identification problems have multiple applications in engineering such as the identification of fissures in materials, determination of sources in electromagnetic fields or geophysical applications, detection of contaminant sources, among others. In this work we are concerned with the determination of a time-dependent source in a transport equation from noisy data measured at a fixed position. By means of Fourier techniques can be shown that the problem is ill-posed in the sense that the solution exists but it does not vary continuously with the data. A number of different techniques were developed by other authors to approximate the solution. In this work, we consider a family of parametric regularization operators to deal with the ill-posedness of the problem. We proposed a manner to select the regularization parameter as a function of noise level in data in order to obtain a regularized solution that approximate the unknown source. We find a H\"older type bound for the error of the approximated source when the unknown function is considered to be bounded in a given norm. Numerical examples illustrate the convergence and stability of the method.