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 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.