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Inhomogeneous Priors for Bayesian Inverse Problems

Babak Maboudi Afkham, Tomas Soto, Mirza Karamehmedovic, Lassi Roininen

TL;DR

This work addresses uncertainty quantification in inverse problems with pronounced spatial inhomogeneity by introducing inhomogeneous priors constructed through convolution with white noise, yielding nonstationary Whittle–Matérn-type fields. These priors are realized as solutions to spatially inhomogeneous SPDEs with pseudo-differential symbols p_{ ext{α}}(x,η) = c(σ,α)(σ(x)+|η|^2)^{α/2}, and fit within the infinite-dimensional Bayesian framework with a tractable parametrix-based sampling scheme. The authors develop a rigorous discretization and sampling procedure, provide error-quantified approximations, and demonstrate improved reconstruction quality and uncertainty quantification in 1D denoising and 2D limited-angle X-ray CT, including hierarchical priors for unknown inhomogeneity. The approach integrates with modern inference algorithms (NUTS/HMC, L-BFGS for MAP) and is backed by white-noise analysis, offering practical, discretization-stable tools for large-scale inverse problems with localized structure.

Abstract

Many inverse problems arising in engineering and applied sciences involve unknown quantities with pronounced spatial inhomogeneity, such as localized defects or spatially varying material properties, making reliable uncertainty quantification particularly challenging. While Bayesian inverse problem methodologies provide a principled framework for assessing reconstruction reliability, commonly used Gaussian priors, such as Whittle-Matern models, impose globally homogeneous assumptions that limit their ability to capture such structure in large-scale settings. We introduce a new class of inhomogeneous priors defined via convolution with white noise, yielding nonstationary Whittle-Matern-type random fields with a rigorous mathematical construction. These priors fit naturally within existing Bayesian well-posedness theory and enable efficient sampling by reducing prior realizations to the solution of a pseudo-differential equation, for which we develop numerical schemes with quantified approximation error. Numerical experiments in one-dimensional denoising and two-dimensional limited-angle X-ray tomography demonstrate significant improvements in reconstruction quality and uncertainty quantification, particularly in data-limited scenarios.

Inhomogeneous Priors for Bayesian Inverse Problems

TL;DR

This work addresses uncertainty quantification in inverse problems with pronounced spatial inhomogeneity by introducing inhomogeneous priors constructed through convolution with white noise, yielding nonstationary Whittle–Matérn-type fields. These priors are realized as solutions to spatially inhomogeneous SPDEs with pseudo-differential symbols p_{ ext{α}}(x,η) = c(σ,α)(σ(x)+|η|^2)^{α/2}, and fit within the infinite-dimensional Bayesian framework with a tractable parametrix-based sampling scheme. The authors develop a rigorous discretization and sampling procedure, provide error-quantified approximations, and demonstrate improved reconstruction quality and uncertainty quantification in 1D denoising and 2D limited-angle X-ray CT, including hierarchical priors for unknown inhomogeneity. The approach integrates with modern inference algorithms (NUTS/HMC, L-BFGS for MAP) and is backed by white-noise analysis, offering practical, discretization-stable tools for large-scale inverse problems with localized structure.

Abstract

Many inverse problems arising in engineering and applied sciences involve unknown quantities with pronounced spatial inhomogeneity, such as localized defects or spatially varying material properties, making reliable uncertainty quantification particularly challenging. While Bayesian inverse problem methodologies provide a principled framework for assessing reconstruction reliability, commonly used Gaussian priors, such as Whittle-Matern models, impose globally homogeneous assumptions that limit their ability to capture such structure in large-scale settings. We introduce a new class of inhomogeneous priors defined via convolution with white noise, yielding nonstationary Whittle-Matern-type random fields with a rigorous mathematical construction. These priors fit naturally within existing Bayesian well-posedness theory and enable efficient sampling by reducing prior realizations to the solution of a pseudo-differential equation, for which we develop numerical schemes with quantified approximation error. Numerical experiments in one-dimensional denoising and two-dimensional limited-angle X-ray tomography demonstrate significant improvements in reconstruction quality and uncertainty quantification, particularly in data-limited scenarios.
Paper Structure (16 sections, 2 theorems, 72 equations, 12 figures, 2 algorithms)

This paper contains 16 sections, 2 theorems, 72 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Bayesian Inverse Problems on a Continuum
  3. Gaussian Random Functions
  4. Spatially Inhomogeneous SPDEs
  5. Setup of the Functional Bayesian Inverse Problem
  6. Computational Algorithm for Inhomogeneous Priors
  7. Numerical Demonstrations
  8. Prior Discretization
  9. 1D De-noising
  10. 2D Inverse Problems
  11. 2D Hierarchical Inhomogeneous Priors
  12. Limited Angle X-ray Computed Tomography
  13. Conclusion
  14. White Noise Analysis
  15. White Noise in Sobolev Spaces
  16. ...and 1 more sections

Key Result

Proposition 2.1

kuo2006gaussianibragimov2012gaussian Let $m\in H$ and $\mathcal{C}:H\to H$ be a mean function and a covariance operator as stated above. Let $\{e_j\}_{j=1}^{\infty}$ be the eigenfunctions and $\{ \lambda_j \}_{j=1}^{\infty}$ be the corresponding eigenvalues of $\mathcal{C}$, sorted in decreasing ord where $X_j\sim \mathcal{N}(0,1)$ are independent real-valued random variables for all $j \geq 1$. T

Figures (12)

  • Figure 1: Coefficients of the SPDE \ref{['eq:SPDE_inhomogeneous']} with principal symbol \ref{['eq:psido-symbol']}
  • Figure 2: $\ell^2$-norm of the expansion coefficients of the SPDE \ref{['eq:SPDE_inhomogeneous']} with principal symbol \ref{['eq:psido-symbol']}, up to integration constant.
  • Figure 3: Comparison between the truncated parametrix solution of the SPDE \ref{['eq:SPDE_inhomogeneous']} and the finite-difference method.
  • Figure 4: MAP estimation and HPD interval in the denoising problem with various noise levels.
  • Figure 5: MAP estimation and HPD interval for Fourier coefficients in the denoising problem with various noise levels.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 3.1