Inhomogeneous Priors for Bayesian Inverse Problems

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.

💡 Research Summary

The paper addresses a fundamental limitation of widely used Gaussian priors in Bayesian inverse problems: the assumption of spatial homogeneity. Classical Whittle‑Matern priors are defined by a constant correlation length, smoothness exponent, and variance, which forces the random field to exhibit the same statistical properties everywhere. In many engineering and scientific applications—such as defect detection in composites, heterogeneous biological tissues, or layered geological media—the unknown quantity displays pronounced local variations that cannot be captured by a globally stationary model.

To overcome this, the authors introduce a new class of inhomogeneous priors constructed by convolving spatial white noise with a location‑dependent kernel. Mathematically, the prior is defined as the solution of a stochastic pseudo‑differential equation (SPDE) of the form

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