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Discretization-free Bayesian inverse problems in distribution spaces

Daniela Calvetti, Erkki Somersalo

TL;DR

The paper addresses solving Bayesian inverse problems directly in distribution spaces to bypass discretization of the unknown. It develops a discretization-free framework where measurements are treated as dual evaluations and posterior covariances are obtained via finite-dimensional projections computed by quadrature, extending Gaussian linear theory to infinite dimensions. A key contribution is expressing the posterior of dual projections as Gaussian with mean and covariance given by blocks of a cross-covariance matrix, without discretizing the unknown, and demonstrating this with fan-beam tomography. This approach reduces discretization-induced errors and offers flexibility when changing measurement sets, bridging rigorous infinite-dimensional theory with practical computation in inverse problems.

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.

Discretization-free Bayesian inverse problems in distribution spaces

TL;DR

The paper addresses solving Bayesian inverse problems directly in distribution spaces to bypass discretization of the unknown. It develops a discretization-free framework where measurements are treated as dual evaluations and posterior covariances are obtained via finite-dimensional projections computed by quadrature, extending Gaussian linear theory to infinite dimensions. A key contribution is expressing the posterior of dual projections as Gaussian with mean and covariance given by blocks of a cross-covariance matrix, without discretizing the unknown, and demonstrating this with fan-beam tomography. This approach reduces discretization-induced errors and offers flexibility when changing measurement sets, bridging rigorous infinite-dimensional theory with practical computation in inverse problems.

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.
Paper Structure (9 sections, 1 theorem, 66 equations, 2 figures)

This paper contains 9 sections, 1 theorem, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Measurements as dual evaluations
  3. Inverse problems in the Bayesian framework
  4. Gaussian models in the traditional setting: a review
  5. Posterior density through dual evaluations
  6. Discretization-free matrix evaluations
  7. Example: Fan beam tomography
  8. Comparison with discretized model
  9. Discussion

Key Result

Theorem 3.1

Let $Z:\Omega\to{\mathbb R}^n$ and $B:\Omega\to{\mathbb R}^m$ be two multivariate zero mean Gaussian random variables, and denote by $Y$ the combined random variable, and partition joint covariance matrix as where Then the posterior distribution of $Z$ conditioned on $B=b$ is Gaussian with mean and covariance i.e., the conditional covariance is the Schur complement of ${\mathsf C}^{22}$.

Figures (2)

  • Figure 1: The geometric arrangement of a single projection in the fanbeam tomography measurement (left). The imaging window $\Omega$ is a square, and the unknown density is supported on the disc in $\Omega$. The red dot indicates the X-ray source and $S$ is the detector screen. On the right, the screen $S$ is represented as a linear array of detectors $s_k$ collecting photons and integrating the result in a single pixel value, the sensitivity of the detector being represented by some device function $\psi_k$. The full data consists of several projections obtained by rotating the object $\Omega$ over a discrete set of projection angles.
  • Figure 2: The linear mapping ${\mathsf A}:V\to U$ corresponding to a single fanbeam projection. The test function $\psi$ is a one--dimensional profile defined over the screen $S$, and ${\mathsf A}\psi$ is a test function defined over the imaging area $\Omega$. Observe that in this visualization of the mapping ${\mathsf A}$, we do not assume that the test function $\psi$ is necessarily associated to any physical measurement device.

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Definition 3.2