Discretization-free Bayesian inverse problems in distribution spaces

TL;DR

This paper introduces a discretization-free Bayesian method for inverse problems in distribution spaces, enabling direct computation without discretizing the unknown, relying only on numerical quadratures.

Contribution

It develops a theoretical framework for solving linear inverse problems in distribution spaces without discretizing the unknown, bridging infinite-dimensional theory and practical computation.

Findings

Discretization of the unknown is unnecessary under certain conditions.

Numerical quadratures suffice for computational treatment.

The approach connects with and extends discretization-based methods.

Abstract

The Bayesian approach to inverse problems provides a practical way to solve ill-posed problems by augmenting the observation model with prior information. Due to the measure-theoretic underpinnings, the approach has raised theoretical interest, leading to a rather comprehensive description in infinite-dimensional function spaces. The goal of this article is to bridge the infinite-dimensional theory for linear inverse problems in distribution spaces and associated computational inverse problems without resorting to a discrete approximation of the forward model. We will shown that under certain assumptions, discretization of the unknown of interest is not necessary for the numerical treatment of the problem, the only approximations required being numerical quadratures that are independent of any discrete representation of the unknown. An analysis of the connection between the proposed approach and discretization-based ones is also provided.