A New Total Variational Regularization Method For Nonlinear Inverse Problems With Its Application

Causality is a kind of internal connection between the objective things. Human beings have been exploring the causality actively in order to understand the world, and play the subjective initiative to transform the world. However, in fact, when people understand the world, the reason is difficult to obtain, which need to be derived from result. In the domain of mathematical physics, it is called the inverse problem. most of which are ill-posed. Therefore, the ill-posed inverse problem is a hotpot that must be solved at present.The inverse problem can be divided into two types: linear and nonlinear. This paper focuses on solving the nonlinear ill-posed inverse problems. Homotopy method can relieve the requirement of initial value, total variation regularization method can effectively reconstruct the discontinuity of the discontinuity of the problem, which is widely used in image processing. Therefore, in this paper, the inversion of the objective function is transformed into the minimum value of the corresponding Euler equation.Then we establish the fixed point homotopy equation and construct the basic iteration scheme, which is, however not stable. So we constructed a new total variation regularization method which combines the idea of total variation regularization and the Bregman penalty term together is proposed. Moreover, the iterative sequence is proved to converge to the minimum value of the objective function at the scale of Bregman distance.To solve the inverse problem of absorption parameters in fluorescence molecular tomography, we adopt the radiative transfer equation(RTE) to simulate the propagation process of photons in the organization, and use the total variation regularization to solve the inverse problem. The results of numerical simulation show that this method can effectively reconstruct the shape and the edge of the abnormal body. The calculated results is further compared with the results of Tikhonov regularization method. The comparison shows that the simulation of proposed algorithm are more close to the real value.