A TIME-DEPENDENT DIRECT SAMPLING METHOD FOR RECOVERING MOVING POTENTIALS IN A HEAT EQUATION

摘要

We are concerned with a numerical reconstruction of the moving potential/absorption coefficient in a heat conduction process when only a single set of boundary measurements of the thermal reflection is available. We propose an efficient direct sampling method (DSM) to locate moving extended objects, represented by time-dependent potentials in a heat equation, and track the trajectories of the moving objects. This appears to be the first DSM for recovering and tracking moving inhomogeneous inclusions in a time-dependent PDE system. Our new method is essentially different from the existing DSMs for solving various stationary or time-harmonic inverse problems but still preserves several important features: it is robust against the noise in the data, easy to implement, and inexpensive computationally. Mathematical justifications are provided to verify the validity of this new method, and insightful mathematical analysis is performed to understand the behavior of the key probing functions proposed. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the method. The DSM provides a new promising numerical strategy for the ill-posed inverse problem of recovering time-dependent moving inhomogeneous media.