Sparse Variational Bayesian Approximations for Nonlinear Inverse Problems: applications in nonlinear elastography

TL;DR

This paper introduces a Bayesian variational approach for efficiently solving high-dimensional nonlinear inverse problems, specifically applied to nonlinear elastography, enabling reduced computational cost and improved parameter estimation accuracy.

Contribution

It develops a variational Bayesian framework that reduces dimensionality and computational effort in nonlinear inverse problems, validated through elastography applications.

Findings

Effective dimensionality reduction of unknown parameters

Accurate approximation of posterior distributions

Validated approach with importance sampling

Abstract

This paper presents an efficient Bayesian framework for solving nonlinear, high-dimensional model calibration problems. It is based on a Variational Bayesian formulation that aims at approximating the exact posterior by means of solving an optimization problem over an appropriately selected family of distributions. The goal is two-fold. Firstly, to find lower-dimensional representations of the unknown parameter vector that capture as much as possible of the associated posterior density, and secondly to enable the computation of the approximate posterior density with as few forward calls as possible. We discuss how these objectives can be achieved by using a fully Bayesian argumentation and employing the marginal likelihood or evidence as the ultimate model validation metric for any proposed dimensionality reduction. We demonstrate the performance of the proposed methodology for problems in nonlinear elastography where the identification of the mechanical properties of biological materials can inform non-invasive, medical diagnosis. An Importance Sampling scheme is finally employed in order to validate the results and assess the efficacy of the approximations provided.