Abstract:

The Landweber iterative regularization method is an effective approach for solving nonlinear ill-posed inverse problems. However, its convergence speed is often slow, which greatly limits its practical applications. An adaptive two-point gradient method with momentum (ATPGM) is proposed to accelerate the classic Landweber method. The main idea of the ATPGM is that the momentum is introduced into the gradient direction of the two-point gradient method, thus it can take full advantage of the previous gradients information, resulting in a highly fast convergence.The convergence and regularity of ATPGM are given. In numerical experiments, we test the numerical performance of ATPGM, based on one-dimensional and two-dimensional elliptic parameter identification problems. A comprehensive comparision between the Landweber iterative regularization method, the two-point gradient method (TPG) and ATPGM is also presented. The numerical results show that ATPGM performs little better in terms of iterations and running time. Numerical results also show that ATPGM is not sensitive to noise in terms of iterations.

Key words: two-point gradient, nonlinear ill-posed inverse problems, convergence analysis, momentum coefficient, adaptive