Change of Measure for Bayesian Field Inversion with Hierarchical Hyperparameters Sampling

TL;DR

This paper introduces a hierarchical Bayesian approach with a change of measure for efficient joint sampling of a scalar field and its hyperparameters in inverse problems, utilizing KL decomposition, surrogate models, and MCMC methods.

Contribution

It presents a novel change of measure technique to reformulate the joint posterior, enabling efficient MCMC sampling of hyperparameters and field coordinates in Bayesian inversion.

Findings

Method is consistent with existing approaches in diffusion problems.

Hyperparameter inference significantly improves inversion accuracy.

Application to complex geometries demonstrates method's versatility.

Abstract

This paper proposes an effective treatment of hyperparameters in the Bayesian inference of a scalar field from indirect observations. Obtaining the joint posterior distribution of the field and its hyperparameters is challenging. The infinite dimensionality of the field requires a finite parametrization that usually involves hyperparameters to reflect the limited prior knowledge. In the present work, we consider a Karhunen-Lo{\`e}ve(KL) decomposition for the random field and hyperparameters to account for the lack of prior knowledge of its autocovariance function. The hyperparameters must be inferred. To efficiently sample jointly the KL coordinates of the field and the autocovariance hyperparameters, we introduce a change of measure to reformulate the joint posterior distribution into a hierarchical Bayesian form. The likelihood depends only onthe field's coordinates in a fixed KL basis, with a prior conditioned on the hyperparameters. We exploit this structure to derive an efficient Markov Chain Monte Carlo (MCMC) sampling scheme based on an adapted Metropolis-Hasting algorithm. We rely on surrogate models (Polynomial Chaos expansions) of the forward model predictions to further accelerate the MCMC sampling. A first application to a transient diffusionproblem shows that our method is consistent with other approaches based on a change of coordinates (Sraj et al., 2016). A second application to a seismic traveltime tomography highlights the importance of inferring the hyperparameters. A third application to a 2D anisotropic groundwater flow problem illustrates the method on a more complex geometry.