Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems

TL;DR

This paper critically examines the limitations of polynomial chaos expansions in Bayesian inverse problems, showing that they can lead to inaccurate results when data contain more information than the prior assumes, and discusses the trade-offs of adaptive methods.

Contribution

It provides analysis and examples demonstrating the failure modes of polynomial chaos surrogates in Bayesian inverse problems with informative data.

Findings

Surrogate posteriors can significantly deviate from true posteriors when data are highly informative.

Increasing polynomial chaos order improves accuracy but can become computationally prohibitive.

Monte Carlo sampling without surrogates may be more cost-effective in certain scenarios.

Abstract

Polynomial chaos expansions are used to reduce the computational cost in the Bayesian solutions of inverse problems by creating a surrogate posterior that can be evaluated inexpensively. We show, by analysis and example, that when the data contain significant information beyond what is assumed in the prior, the surrogate posterior can be very different from the posterior, and the resulting estimates become inaccurate. One can improve the accuracy by adaptively increasing the order of the polynomial chaos, but the cost may increase too fast for this to be cost effective compared to Monte Carlo sampling without a surrogate posterior.