On the cavity detection in a heat conductive medium from time-average boundary temperature measurement

Consider the shape identification of an inclusion in heat conductive medium from the time average measurement, which is modeled by an initial boundary value problem for a parabolic system with extra nonlocal measurement data specified on the outer boundary. For this nonlocal and nonlinear inverse problem for the two-dimensioned parabolic equation in a doubly-connected domain, the radius function describing the shape of inner boundary to be identified is defined as the minimizer of a regularizing cost functional. The existence of this minimizer is firstly proven in a suitable admissible set. Then we establish the convergence rate of the regularizing solution under alpha-posteriori choice strategy for the regularizing parameter. Finally the differentiability of the cost functional is proven, which provides a fundamental basis for gradient type iteration scheme. Based on the adjoint and sensitivity problem of the original problem which give the gradient of the cost functional, we propose a steepest descent iteration algorithm for finding the minimizer approximately. Numerical examples are presented to show the validity of our algorithm. (C) 2021 Elsevier B.V. All rights reserved.