Point sources identification problems with pointwise overdetermination

TL;DR

This paper investigates the inverse problem of identifying point sources in heat and mass transfer models, focusing on well-posedness, uniqueness, and measurement requirements, with implications for numerical solution methods.

Contribution

It provides new conditions for existence and uniqueness of solutions, examples of non-uniqueness, and estimates on measurement numbers needed for source identification.

Findings

Conditions for solution existence and uniqueness

Examples demonstrating non-uniqueness

Estimates on measurements for source identification

Abstract

This article is devoted to inverse problems of recovering point sources in mathematical models of heat and mass transfer. The main attention is paid to well-posedness questions of these inverse problems with pointwise overdetermination conditions. We present conditions for existence and uniqueness of solutions to the problem, display non-uniqueness examples, and, in model situations, we give estimates on the number of measurements that allow completely identify sources and their locations. The results rely on asymptotic representations of Green functions of the corresponding elliptic problems with a parameter. They can be used in constructing new numerical algorithms for determining a solution.