Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem

TL;DR

This paper proves that Bayesian methods using Gaussian process priors reliably recover the true conductivity in a nonlinear elliptic inverse problem as the number of measurements increases, with quantifiable convergence rates.

Contribution

It establishes the consistency and convergence rates of Gaussian process-based Bayesian inference for a nonlinear elliptic inverse problem, demonstrating practical feasibility with MCMC methods.

Findings

Posterior distributions concentrate around the true parameter as measurements increase.

Convergence rate of the reconstruction error is quantified as N^{-mbda}.

Bayesian procedures are effective for nonlinear inverse problems with Gaussian process priors.

Abstract

For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth function $g:\mathcal{O}\to\mathbb{R}$, we consider the statistical nonlinear inverse problem of recovering the conductivity $f>0$ in the divergence form equation $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from $N$ discrete noisy point evaluations of the solution $u=u_f$ on $\mathcal O$. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number $N$ of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate $N^{-\lambda}, \lambda>0,$ for the reconstruction error of the associated posterior means, in $L^2(\mathcal{O})$-distance.