Landweber–Fridman algorithms for the Cauchy problem in steady-state anisotropic heat conduction

Abstract

We investigate the numerical reconstruction of the missing thermal boundary conditions on an inaccessible part of the boundary in the case of steady-state heat conduction in anisotropic solids from the knowledge of over-prescribed noisy data on the remaining accessible boundary. This inverse problem is tackled by employing a variational formulation that transforms it into an equivalent control problem; four such approaches are discussed thoroughly. The numerical implementation is realised for the 2D case via the boundary element method for perturbed Cauchy data, whilst the numerical solution is stabilised/regularised by stopping the iterative procedure according to Morozov’s discrepancy principle (Morozov, VA. On the solution of functional equations by the method of regularization. Doklady Mathematics 1966; 7: 414–417).