Likelihood-informed dimension reduction for nonlinear inverse problems

Abstract

The intrinsic dimensionality of an inverse problem is affected by prior information, the accuracy and number of observations, and the smoothing properties of the forward operator. From a Bayesian perspective, changes from the prior to the posterior may, in many problems, be confined to a relatively lowdimensional subspace of the parameter space. We present a dimension reduction approach that defines and identifies such a subspace, called the "likelihoodinformed subspace" (LIS), by characterizing the relative influences of the prior and the likelihood over the support of the posterior distribution. This identification enables new and more efficient computational methods for Bayesian inference with nonlinear forward models and Gaussian priors. In particular, we approximate the posterior distribution as the product of a lower-dimensional posterior defined on the LIS and the prior distribution marginalized onto the complementary subspace. Markov chain Monte Carlo sampling can then proceed in lower dimensions, with significant gains in computational efficiency. We also introduce a Rao-Blackwellization strategy that de-randomizes Monte Carlo estimates of posterior expectations for additional variance reduction. We demonstrate the efficiency of our methods using two numerical examples: inference of permeability in a groundwater system governed by an elliptic PDE, and an atmospheric remote sensing problem based on Global Ozone Monitoring System (GOMOS) observations.

Top-down constraints on global N<sub>2</sub>O emissions at optimal resolution: application of a new dimension reduction technique

“…The eigenvectors v i can be interpreted as the most constrained modes in flux space, i.e., flux patterns that are independently constrained by the observations (Cui et al, 2014;Bousserez and Henze, 2017). These eigenvectors of the prior-preconditioned Hessian are efficiently calculated using a fully parallelized randomized algorithm (Halko et al, 2011), as in Bui-Thanh et al (2012) and Bousserez and Henze (2017).…”

Model reduction from partial observations

“…In the light of (14), we see that in the partially-informed setup considered in this paper, the optimal reduction performance is lower bounded by κ i (M) and upper bounded by κ i (M post ). The gap between κ i (M) and κ i (M post ) "materializes" the loss of reducibility which can occur by working in a partially-informed setting rather than a perfectly-informed one.…”

“…induced by the optimal approximation subspace S post i is bounded by the Kolmogorov i-width κ i (M post ), see (14). The upper bound in Theorem 1 thus also defines an upper limit on the projection error made by reducing the true, unknown, manifold M in the worst-case optimal subspace S post i .…”

Sampling via Measure Transport: An Introduction

“…A further source of low dimensionality in transports is low-rank structure, i.e., situations where a map departs from the identity only on a low-dimensional subspace of the input space [78]. This situation is fairly common in large-scale Bayesian inverse problems where the data are informative, relative to the prior, only about a handful of directions in the parameter space [25,79].…”

“…Thus, if we want to evaluate the direct transport at a particular x * ∈ R n , i.e., z * := T (x * ), then by (25) we can simply invert S at x * to obtain z * . In particular, if x * = (x * 1 , .…”