Recursive linearization method for inverse medium scattering problems with complex mixture Gaussian error learning
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This paper is concerned with the numerical errors that have appeared in the calculation of inverse medium scattering problems (IMSPs). Optimization-based iterative methods are widely employed to solve IMSPs, which are computationally intensive due to a series of Helmholtz equations, and need to be solved numerically. Hence, rough approximations of Helmholtz equations can significantly speed up the iterative procedure. However, rough approximations will lead to instability and inaccurate estimation. Inspired by mixture Gaussian error construction used widely in the machine learning community, we model numerical errors brought by a rough forward solver as some complex mixture Gaussian (CMG) random variables. Based on this assumption, a new nonlinear optimization problem is derived by using the infinite-dimensional Bayes' inverse method. Then, we generalize the real valued expectation-maximization (EM) algorithm to our complex valued case to learn parameters in the CMG distribution. Next, we generalize the recursive linearization method (RLM) to a new iterative method named the mixture Gaussian recursive linearization method (MGRLM) which consists of two stages: (1) learn CMG; (2) solve IMSPs. Through the learning stage, numerical errors and some prior knowledge of the true scatterer have been incorporated into the proposed optimization problem. Hence, both the convergence speed and the resolution of the obtained result can be enhanced in the second stage. Finally, we provide two numerical examples to illustrate the effectiveness of the proposed method.
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