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This paper is concerned with the modeling errors appeared in the numericalmethods of inverse medium scattering problems (IMSP). Optimization basediterative methods are wildly employed to solve IMSP, which are computationallyintensive due to a series of Helmholtz equations need to be solved numerically.Hence, rough approximations of Helmholtz equations can significantly speed upthe iterative procedure. However, rough approximations will lead to instabilityand inaccurate estimations. Using the Bayesian inverse methods, we incorporatethe modelling errors brought by the rough approximations. Modelling errors areassumed to be some complex Gaussian mixture (CGM) random variables, and inaddition, well-posedness of IMSP in the statistical sense has been establishedby extending the general theory to involve CGM noise. Then, we generalize thereal valued expectation-maximization (EM) algorithm used in the machinelearning community to our complex valued case to learn parameters in the CGMdistribution. Based on these preparations, we generalize the recursivelinearization method (RLM) to a new iterative method named as Gaussian mixturerecursive linearization method (GMRLM) which takes modelling errors intoaccount. Finally, we provide two numerical examples to illustrate theeffectiveness of the proposed method.

Recursive linearization method for inverse medium scattering problems with complex mixture Gaussian error learning