Using field theoretic techniques to go beyond mean-field solutions for inverse problems

Joshua Chang Mathematical Biosciences Institute, The Ohio State University

Abstract

Inverse problems of partial differential equations often require the use of regularization" tricks, particularly when they are ill-posed or ill-conditioned. Recent work has elucidated the connection between the commonly used techniques of Tikhonov regularization and Bayesian Gaussian random fields. This interpretation of regularization within the Bayesian framework suggests that one should use regularizing terms that are consistent with apriori knowledge of the desired field.

The resulting method, using the path-integral-formulation of random fields, is amenable to deterministic perturbative approximation. Using the path-integral formulation as a computational tool, one is able to quantify uncertainty in the solution to inverse problems. Such an approach is desirable compared to standard computational methods of inversion which typically involve the use of Markov-Chain Monte-Carlo methods. In this talk, I will present the path-integral method for inverse problems. Electrostatic inverse problems will be provided for illustrative purposes.
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