# Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems

## TL;DR

This paper introduces a Hessian-based adaptive sparse quadrature method for efficiently computing infinite-dimensional integrals in Bayesian inverse problems, improving accuracy by better capturing the posterior distribution.

## Contribution

The paper develops a novel Hessian-based parametrization and adaptive sparse quadrature that achieve dimension-independent convergence rates for Bayesian inverse problems.

## Key findings

- Achieves dimension-independent convergence rates.
- Demonstrates faster convergence than O(N^{-1/2}) in experiments.
- Effectively approximates posterior distributions in linear and nonlinear problems.

## Abstract

In this work we propose and analyze a Hessian-based adaptive sparse quadrature to compute infinite-dimensional integrals with respect to the posterior distribution in the context of Bayesian inverse problems with Gaussian prior. Due to the concentration of the posterior distribution in the domain of the prior distribution, a prior-based parametrization and sparse quadrature may fail to capture the posterior distribution and lead to erroneous evaluation results. By using a parametrization based on the Hessian of the negative log-posterior, the adaptive sparse quadrature can effectively allocate the quadrature points according to the posterior distribution. A dimension-independent convergence rate of the proposed method is established under certain assumptions on the Gaussian prior and the integrands. Dimension-independent and faster convergence than $O(N^{-1/2})$ is demonstrated for a linear as well as a nonlinear inverse problem whose posterior distribution can be effectively approximated by a Gaussian distribution at the MAP point.