A gradient-based regularization algorithm for the Cauchy problem in steady-state anisotropic heat conduction

We study the numerical reconstruction of the missing thermal boundary data on a portion of the boundary occupied by an anisotropic solid in the case of the steady-state heat conduction equation from the knowledge of both the temperature and the normal heat flux (i.e. Cauchy data) on the remaining and accessible part of the boundary. This inverse problem is known to be ill-posed and hence a regularization procedure is required. Herein we develop a solver for this problem by exploiting two sources of regularization, namely the smoothing nature of the corresponding direct problems and a priori knowledge on the solution to the inverse problem investigated. Consequently, this inverse problem is reformulated as a control one which reduces to minimising a corresponding functional defined on a fractional Sobolev space on the inaccessible part of the boundary. This approach yields a gradient-based iterative algorithm that consists, at each step, of the resolution of two direct problems and three corresponding adjoint problems in accordance with the function space where the control is sought. The theoretical convergence of the algorithm is studied by deriving an iteration-dependent admissible range for the parameter. Numerical experiments are realized for the two-dimensional case by employing the finite-difference method, whilst the numerical solution is stabilised/regularized by stopping the iterative process based on three criteria. © 2022 Elsevier Ltd