Random Wavelet Series in Besov Spaces: Construction, Characterization, and Applications in Bayesian Inverse Problems

Inverse problems arise when we seek to infer unknown quantities from indirect and noisy observations. These problems are typically ill-posed, meaning small perturbations in the data leads to large variations in the solution. In this context, uncertainty quantification (UQ) plays a critical role: rather than seeking a single solution, we aim to characterize the range of plausible solutions and assess the confidence in inferred reconstructions. Bayesian methods provide a natural framework for UQ by combining prior knowledge with observed data to produce a posterior distribution over possible solutions.

This thesis focuses on developing probabilistic models on function spaces that effectively capture a wide range of regularity, emphasizing both smooth and non-smooth behavior. These models serve as prior distributions in Bayesian inverse problems, where accurately encoding regularity assumptions is crucial. At the core of this work is the construction and analysis of random wavelet series within Besov spaces, which provide a general mathematical framework for representing functions with varying degrees of regularity.

We investigated the design of random wavelet series that almost surely belong to the full range of Besov spaces, including quasi-Banach cases. By imposing mild assumptions on the random coefficients and controlling regularity through tunable decay rates, we obtained a full characterization of these series and their membership in Besov spaces. The approach supports a broad range of random coefficient distributions, from heavy- to light-tailed, enabling flexible modeling of both stochastic behavior and regularity.

We applied Besov priors to linear Bayesian inverse problems, illustrating how their tunable parameters facilitate the reconstruction of solutions with different regularity. To enable posterior inference, we developed a sampler based on the Randomize-Then-Optimize method tailored for linear inverse problems with Besov priors. Numerical experiments showed that the choice of wavelet basis strongly influences the recovery of discontinuous or smooth features, while the Besov parameters control the strength and sparsity of the inferred solutions. These insights provide valuable guidance for tuning Besov priors to the prior assumptions of specific inverse problem.

We consider Bayesian decomposition models using Besov priors to separate solution components by their regularity properties. In particular, we proposed a model combining two Besov priors with distinct wavelets, where one prior captures non-smooth components and the other smooth features. Additionally, we developed a hierarchical decomposition model that combines a hierarchical Gaussian prior on discrete derivatives, promoting piecewise-constant structure, with a Besov prior imposing smoothness. A Gibbs sampling scheme was designed for posterior inference and automatic estimation of prior strengths. Our experiments confirm that both models effectively decompose solutions into meaningful components with quantified uncertainty, with the hierarchical model demonstrating superior accuracy and flexibility through automatic parameter
estimation.

Overall, this work advances Bayesian inverse problem methodology by providing flexible, theoretically grounded prior constructions and practical inference tools. It offers a robust framework for modeling, decomposing, and quantifying uncertainty in solutions exhibiting complex regularity patterns, enabling more precise and interpretable inferences in applied settings.