Accelerating Uncertainty Quantification for Nonlinear Inverse Scattering Problems With High Contrast Media by Direct Envelope Methods and Krylov Subspace Iterative Integral Equation Solvers

Uncertainty quantification related to nonlinear inverse scattering problems often involves the posterior covariance matrix that cannot be effectively estimated but still need to be considered as an important part of the nonlinear inverse scattering problem. The objective of this work is to show how to take advantage of a Bayesian framework to estimate the uncertainty of velocity reconstruction in the nonlinear inverse scattering imaging. The key ingredients are twofold. On the one hand, we rely on a Kalman Filter method, equipped with an optimization scheme, to solve the inverse problem. We use the distorted Born iterative method to formulate the sensitivity kernel. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method, in which the Green’s functions are based on integral equations. On the other hand, we use the direct envelope methods to provide the initial guess (large-scale smooth model) to overcome the nonlinearity in the inverse scattering problems. We first investigate the uncertainty of velocity imaging in high contrast media. Then we show by means of Lippmann-Schwinger-type equations that the uncertainty of multiparameter inverse scattering problems can be addressed. Numerical results dealing with the uncertainty of nonlinear inverse scattering problems in the case of both isotropic and anisotropic media highlight the important role played by uncertainty quantification.