Adjoint-weighted variational formulation for a direct computational solution of an inverse heat conduction problem

We consider the direct (i.e. non-iterative) solution of the inverse problem of heat conduction for which at least two interior temperature fields are available. The strong form of the problem for the single, unknown, thermal conductivity field is governed by two partial differential equations of pure advective transport. The given temperature fields must satisfy a compatibility condition for the problem to have a solution. We introduce a novel variational formulation, the adjoint-weighted equation (AWE), for solving the two-field problem. In this case, the gradients of two given temperature fields must be linearly independent in the entire domain, a weaker condition than the compatibility required by the strong form. We show that the solution of the AWE formulation is equivalent to that of the strong form when both are well posed. We prove that the Galerkin discretization of the AWE formulation leads to a stable, convergent numerical method that has optimal rates of convergence. We show computational examples that confirm these optimal rates. The AWE formulation shows good numerical performance on problems with both smooth and rough coefficients and solutions.