Data-Free Likelihood-Informed Dimension Reduction of Bayesian Inverse Problems

TL;DR

This paper presents a data-independent, gradient-based dimension reduction method for Bayesian inverse problems, enabling efficient sampling and reuse of the informed subspace across multiple problems, with controlled approximation error.

Contribution

The paper introduces a novel data-free, gradient-based dimension reduction technique that precomputes the informed subspace offline, facilitating efficient Bayesian inference and reuse across multiple inverse problems.

Findings

Successfully applied to PDE-based inverse problem with Gaussian prior

Effective in tomography with Poisson data and Besov prior

Allows controlled approximation error in posterior distribution

Abstract

Identifying a low-dimensional informed parameter subspace offers a viable path to alleviating the dimensionality challenge in the sampled-based solution to large-scale Bayesian inverse problems. This paper introduces a novel gradient-based dimension reduction method in which the informed subspace does not depend on the data. This permits an online-offline computational strategy where the expensive low-dimensional structure of the problem is detected in an offline phase, meaning before observing the data. This strategy is particularly relevant for multiple inversion problems as the same informed subspace can be reused. The proposed approach allows controlling the approximation error (in expectation over the data) of the posterior distribution. We also present sampling strategies that exploit the informed subspace to draw efficiently samples from the exact posterior distribution. The method is successfully illustrated on two numerical examples: a PDE-based inverse problem with a Gaussian process prior and a tomography problem with Poisson data and a Besov-$\mathcal{B}^2_{11}$ prior.