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This study focuses on addressing the inverse source problem associated withthe parabolic equation. We rely on sparse boundary flux data as ourmeasurements, which are acquired from a restricted section of the boundary.While it has been established that utilizing sparse boundary flux data canenable source recovery, the presence of a limited number of observation sensorsposes a challenge for accurately tracing the inverse quantity of interest. Toovercome this limitation, we introduce a sampling algorithm grounded inLangevin dynamics that incorporates dynamic sensors to capture the fluxinformation. Furthermore, we propose and discuss two distinct sensor migrationstrategies. Remarkably, our findings demonstrate that even with only twoobservation sensors at our disposal, it remains feasible to successfullyreconstruct the high-dimensional unknown parameters.

Restoring the Discontinuous Heat Equation Source Using Sparse Boundary Data and Dynamic Sensors