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We consider the inverse source problem in the parabolic equation, where theunknown source possesses the semi-discrete formulation. Theoretically, we provethat the flux data from any nonempty open subset of the boundary can uniquelydetermine the semi-discrete source. This means the observed area can beextremely small, and that is why we call the data as sparse boundary data. Forthe numerical reconstruction, we formulate the problem from the Bayesiansequential prediction perspective and conduct the numerical examples whichestimate the space-time-dependent source state by state. To better demonstratethe performance of the method, we solve two common multiscale problems from twomodels with a long sequence of the source. The numerical results illustratethat the inversion is accurate and efficient.

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