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An evolutionary form of a generalized Bayesian update method, which isstrictly derivative- free yet directed through an additive update term basedpurely on the statistical moments of the design variables, is proposed fornonlinear inverse problems in general and applied in particular to an opticalimaging problem, the ultrasound modulated optical tomography (UMOT). Theadditive update term, which bypasses most pitfalls of a conventional weight-based Bayesian update, results from a change of measures aimed at drivingappropriately derived observation-prediction error terms or increments of costfunctionals to zero-mean Brownian martingales. This constitutes a novelcharacterization corresponding to the extremization of the cost functional(s),where the design unknowns are represented as diffusion processes evolving withrespect to a continuously parameterized iteration variable. This leads to arecursive prediction-update algorithm to implement the search. The schemeoffers freedom from sample degeneracy and the accompanying divergence of theconventional weight-based Bayesian update schemes. We obtain the order ofconvergence of the conditioned process and also establish that the solutionsare stable against tolerable variations in the regularizing noise terms, evenas the original inverse problem remains severely ill-posed. Numerical evidenceon solutions to the UMOT problem also confirms substantive improvements in thereconstruction efficacy through the proposed method vis-\`a- vis a Gauss-Newtonapproach, especially where the regularized quasi-Newton direction has lowsensitivity to variations in the design unknowns.

A New Evolutionary Bayesian Approach Incorporating Additive Path Correction for Nonlinear Inverse Problems