Determination and reconstruction of a semilinear term from point measurements

TL;DR

This paper introduces a new method for reconstructing a semilinear term in an elliptic equation from minimal boundary measurements, including a single point measurement, and provides a numerical algorithm validated by experiments.

Contribution

It develops a flexible theoretical framework and a novel reconstruction algorithm for inverse elliptic problems using minimal measurement data.

Findings

Effective reconstruction of nonlinear source from single point data

Relaxed data requirements compared to previous methods

Numerical experiments confirm algorithm's accuracy

Abstract

In this article we study the inverse problem of determining a semilinear term appearing in an elliptic equation from boundary measurements. Our main objective is to develop flexible and general theoretical results that can be used for developing numerical reconstruction algorithm for this inverse problem. For this purpose, we develop a new method, based on different properties of solutions of elliptic equations, for treating the determination of the semilinear term as a source term from a point measurement of the solutions. This approach not only allows us to make important relaxations on the data used so far for solving this class of inverse problems, including general Dirichlet excitation lying in a space of dimension one and measurements located at one point on the boundary of the domain, but it also allows us to derive a novel algorithm for the reconstruction of the semilinear term. The effectiveness of our algorithm is corroborated by extensive numerical experiments. Notably, as demonstrated by the theoretical analysis, we are able to effectively reconstruct the unknown nonlinear source term by utilizing solely the information provided by the measurement data at a single point.