High-dimensional Stochastic Inversion via Adjoint Models and Machine Learning

TL;DR

This paper introduces a new Bayesian stochastic inversion method combining Langevin MCMC and kernel PCA to efficiently handle high-dimensional, nonlinear, and non-Gaussian inverse problems, demonstrated in linear elasticity.

Contribution

It presents a novel coupling of gradient-based LMCMC with KPCA for dimension reduction and non-Gaussian posterior estimation in high-dimensional inverse problems.

Findings

Effective dimension reduction via KPCA in high-dimensional spaces.

Accurate non-Gaussian posterior estimation using LMCMC.

Successful application to inverse problems in linear elasticity.

Abstract

Performing stochastic inversion on a computationally expensive forward simulation model with a high-dimensional uncertain parameter space (e.g. a spatial random field) is computationally prohibitive even with gradient information provided. Moreover, the `nonlinear' mapping from parameters to observables generally gives rise to non-Gaussian posteriors even with Gaussian priors, thus hampering the use of efficient inversion algorithms designed for models with Gaussian assumptions. In this paper, we propose a novel Bayesian stochastic inversion methodology, characterized by a tight coupling between a gradient-based Langevin Markov Chain Monte Carlo (LMCMC) method and a kernel principal component analysis (KPCA). This approach addresses the `curse-of-dimensionality' via KPCA to identify a low-dimensional feature space within the high-dimensional and nonlinearly correlated spatial random field. Moreover, non-Gaussian full posterior probability distribution functions are estimated via an efficient LMCMC method on both the projected low-dimensional feature space and the recovered high-dimensional parameter space. We demonstrate this computational framework by integrating and adapting recent developments such as data-driven statistics-on-manifolds constructions and reduction-through-projection techniques to solve inverse problems in linear elasticity.