Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements

TL;DR

This paper proves unique identifiability of semi-discrete sources in parabolic equations from sparse boundary data and demonstrates an efficient Bayesian numerical method for reconstructing the source in practical scenarios.

Contribution

It provides the first theoretical proof of source uniqueness with minimal boundary data and develops a Bayesian sequential prediction approach for numerical reconstruction.

Findings

Unique determination of semi-discrete source from sparse boundary measurements

Numerical method accurately reconstructs space-time-dependent sources

Method performs well on multiscale problems with long source sequences

Abstract

We consider the inverse source problem in the parabolic equation, where the unknown source possesses the semi-discrete formulation. Theoretically, we prove that the flux data from any nonempty open subset of the boundary can uniquely determine the semi-discrete source. This means the observed area can be extremely small, and that is why we call the data as sparse boundary data. For the numerical reconstruction, we formulate the problem from the Bayesian sequential prediction perspective and conduct the numerical examples which estimate the space-time-dependent source state by state. To better demonstrate the performance of the method, we solve two common multiscale problems from two models with a long sequence of the source. The numerical results illustrate that the inversion is accurate and efficient.