Numerical reconstruction of heat fluxes

We consider the numerical reconstruction of heat flux reconstruction problem in some heat conduction system. In real applications, the temperature can be obtained by measurement while the heat flux is difficult to obtain. Thus it is of practical importance to reconstruct the heat flux from the measurement of temperature. This inverse reconstruction is an ill-posed problem.;We shall propose two alternative Tikhonov regularization formulations of the inverse problem. The resulting minimization problems are proved to be well-posed.;Then we use the finite element method to approximate the continuous minimization system arising from the Tikhonov regularization formulations. The solutions to the discrete minimization problems are demonstrated to exist uniquely, and converge to the solutions of original continuous minimization problems.;The conjugate gradient method is suggested to solve the discrete minimization problem. Some numerical experiments are carried out, which have clearly demonstrated the robustness and efficiency of the proposed reconstruction algorithms.;As it is well known in regard to Tikhonov regularizations, an important step to the success of the Tikhonov regularization approach relies on the selection of some reasonable regularization parameters. We shall propose some iterative methods to choose a suitable regularization parameter, which can be efficiently used in the considered inverse problem.;The iterative method for choosing regularization parameters can be extended to the solutions of some other more general inverse problems.