A homotopy method for nonlinear inverse problems

摘要

We develop a homotopy method for nonlinear inverse problems, where the forward problems are governed by some forms of differential equations. A Tikhonov-style regularization approach yields an optimization problem. Ordinary iterative methods may fail to solve this problem, due to their locally convergent properties. Then the fixed-point homotopy method is introduced to solving the normal equation of the optimization problem, and a new and globally convergent algorithm is constructed, which is highly effective in the aspects of speed of computation, ability of noise suppression and wide region of convergence. As a practical application, the method is used to solve the inverse problem of 2-D acoustic wave equation. We demonstrate the merits and effectiveness of our algorithm on two realistic model problems.