Efficient hyperparameter estimation in Bayesian inverse problems using sample average approximation

TL;DR

This paper introduces an efficient method for hyperparameter estimation in Bayesian inverse problems, leveraging sample average approximation and preconditioned Lanczos techniques to reduce computational costs in seismic tomography applications.

Contribution

The authors develop a novel, computationally efficient approach for hyperparameter estimation in Bayesian inverse problems using SAA and a new preconditioner, improving scalability.

Findings

Significant reduction in computation time for hyperparameter estimation.

Effective application to seismic tomography problems.

Robustness of the method across static and dynamic scenarios.

Abstract

In Bayesian inverse problems, it is common to consider several hyperparameters that define the prior and the noise model that must be estimated from the data. In particular, we are interested in linear inverse problems with additive Gaussian noise and Gaussian priors defined using Mat\'{e}rn covariance models. In this case, we estimate the hyperparameters using the maximum a posteriori (MAP) estimate of the marginalized posterior distribution. However, this is a computationally intensive task since it involves computing log determinants. To address this challenge, we consider a stochastic average approximation (SAA) of the objective function and use the preconditioned Lanczos method to compute efficient approximations of the function and gradient evaluations. We propose a new preconditioner that can be updated cheaply for new values of the hyperparameters and an approach to compute approximations of the gradient evaluations, by reutilizing information from the function evaluations. We demonstrate the performance of our approach on static and dynamic seismic tomography problems.