Simultaneous reconstruction of the initial temperature and heat radiative coefficient

Given the measurement of temperature at a fixed time θ > 0 and the measurement of temperature in a subregion of the physical domain, we investigate the simultaneous reconstruction of the initial temperature and heat radiative coefficient in a heat conductive system. The stability of the inverse problem is first established, and then the numerical reconstruction is mainly studied. The reconstruction process is done by Tikhonov regularization with the regularizing terms being theL2-norms of gradients, and is carried out in such a way that the temperature solution of the heat equation matches its fixed time observation and its subregion observation optimally in the L2-norm sense. The continuous nonlinear optimization system will be discretized by the piecewise linear finite element method, and the existence of discrete minimizers and convergence of the finite element approximation are shown. The discrete finite element problem is solved by a nonlinear gradient method with an efficient nonlinear multigrid technique for accelerating the reconstruction process. Numerical experiments are given to demonstrate the efficiency of the proposed nonlinear multigrid gradient method for solving the inverse parabolic problem.