Research On The Numerical Methods Of Two Kinds Of Inverse Problem

In this thesis, we mainly discuss the following two different kinds of problems.First, we consider a class of severely ill-posed backward problems for linear evolution equations (corresponding to different operators). We use a convolution regularization method to obtain a stable approximate initial data from the noisy fi-nal data (when we use the convolution regularization method, we choose different convolution kernel). The convergence rates are obtained under an a priori and an a posteriori regularization parameter choice rule in which an a priori parameter choice is very direct but the a posteriori parameter choice is a new generalized discrepancy principle based on a modified version of Morozov’s discrepancy prin-ciple. The log-type convergence rate under the a priori regularization parameter choice rule and log log-type rate under the a posteriori regularization parameter choice rule are obtained. Two numerical examples are tested to support our theory results.Then, we consider a typical ill-posed inverse heat source problem, that is, we determine a space-dependent heat source term in a multi-dimensional heat equation from a pair of Cauchy data on a part of boundary. By a simple transformation, the inverse heat source problem is changed into a Cauchy problem of a homogenous heat conduction equation. We use the method of fundamental solutions (MFS) cou-pled with the Tikhonov regularization technique to solve the ill-conditioned linear system of equations resulted from the MFS discretization. The generalized cross-validation rule for determining the regularization parameter is used. Numerical results for four examples in1D,2D and3D cases show that the proposed method is effective and feasible.