A multi-resolution, non-parametric, Bayesian framework for identification of spatially-varying model parameters

TL;DR

This paper introduces a hierarchical, multi-resolution Bayesian framework for efficiently identifying spatially-varying parameters in PDE models from noisy data, combining non-parametric sparsity with adaptive sampling for improved accuracy and computational savings.

Contribution

It presents a non-intrusive, multi-resolution Bayesian approach that uses adaptive Sequential Monte Carlo sampling to efficiently infer sparse, spatially-varying parameters with uncertainty quantification.

Findings

Significant computational savings with coarse-to-fine solver hierarchy

Enhanced accuracy in parameter identification

Effective uncertainty quantification for model predictions

Abstract

This paper proposes a hierarchical, multi-resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output. Such parameters are frequently encountered in PDE-based models and correspond to quantities such as density or pressure fields, elasto-plastic moduli and internal variables in solid mechanics, conductivity fields in heat diffusion problems, permeability fields in fluid flow through porous media etc. The proposed model has all the advantages of traditional Bayesian formulations such as the ability to produce measures of confidence for the inferences made and providing not only predictive estimates but also quantitative measures of the predictive uncertainty. In contrast to existing approaches it utilizes a parsimonious, non-parametric formulation that favors sparse representations and whose complexity can be determined from the data. The proposed framework in non-intrusive and makes use of a sequence of forward solvers operating at various resolutions. As a result, inexpensive, coarse solvers are used to identify the most salient features of the unknown field(s) which are subsequently enriched by invoking solvers operating at finer resolutions. This leads to significant computational savings particularly in problems involving computationally demanding forward models but also improvements in accuracy. It is based on a novel, adaptive scheme based on Sequential Monte Carlo sampling which is embarrassingly parallelizable and circumvents issues with slow mixing encountered in Markov Chain Monte Carlo schemes.