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

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 the reason we call it 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 method's performance, we solve two common multiscale problems from two models with a long source sequence. The numerical results illustrate that the inversion is accurate and efficient.

Automated summary

This paper investigates the inverse source problem in the parabolic equation with a semi-discrete unknown source. The authors theoretically prove that the flux data from any nonempty open subset of the boundary can uniquely determine the semi-discrete source. For numerical reconstruction, they propose a Bayesian sequential prediction approach and conduct numerical examples to estimate the space-time-dependent source state. The results demonstrate the accuracy and efficiency of the inversion.