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

Abstract

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 a-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.