Inverse problem of reconstructing the parabolic equation's initial value and the heat radiative coefficient

In this paper, the numerical method to reconstruct the radiative coefficient and initial condition simultaneously by measuring the domain temperature at a fixed time and the temperature of a subdomain all the time is studied. By least-square technique this inverse problem can be formulated into a variational problem and discretized into nonlinear programming problem with the cost function depending on the numerical solution of the corresponding direct problem of heat equation. We obtain the numerical solution of the direct problem by finite difference method and radial basis function (RBF)method respectively and derive the gradient formula for cost function, then implement the numerical reconstruction by quasi-Newton technique. In the case of the measuring data with noise, we use regularization method. Numerical results show that this method is available.