In high-dimensional linear models the problem of constructing adaptive confidence sets for the full parameter is known to be generally impossible. We propose re-weighted loss functions under which constructing fully adaptive confidence sets for the parameter is shown to be possible. We give necessary and sufficient conditions on the loss functions for which adaptive confidence sets exist, and exhibit a concrete rate-optimal procedure for construction of such confidence sets.

Recovering sparse generative models from limited and noisy measurements presents a significant and complex challenge. Given that the available data is frequently inadequate and affected by noise, it is crucial to assess the resulting uncertainty in the relevant parameters. Notably, this uncertainty in the parameters directly impacts the reliability of predictions and decision-making processes.

In this talk, we explore the Bayesian framework, which facilitates the quantification of uncertainty in parameter estimates by treating involved quantities as random variables and leveraging the posterior distribution. Within the Bayesian framework, sparsity promotion and computational efficiency can be attained with hierarchical models with conditionally Gaussian priors and gamma hyper-priors. However, most of the existing literature focuses on the numerical approximation of maximum a posteriori (MAP) estimates, and less attention has been given to sampling methods or other means for uncertainty quantification. To address this gap, our talk will delve into recent advancements and developments in uncertainty quantification and sampling techniques for sparsity-promoting hierarchical Bayesian inverse problems.

Parts of this talk are joint work with Anne Gelb (Dartmouth), Youssef Marzouk (MIT), and Jonathan Lindbloom (Dartmouth).

Conductivity Reconstruction serves as one of the most critical tasks of medical imaging, while approaches concerning interior current density information (ICDI) have drawn a lot of attention recently. However, they face challenges due to the nonlinearity between the conductivity and the interior current density and the high contrast of the conductivity. In this work, we propose a novel Bayesian framework to tackle these difficulties. We incorporate and iteratively update the model error introduced by linearization in the framework, and we also reform the linearization operator recursively to obtain better approximation. Numerical implementation shows that our method outperforms other approaches in terms of both relative errors of estimates and Kullback-Leibler divergence between distributions.

Image reconstruction problems, like image deblurring and computer tomography, are usually ill-posed and require regularization. A popular approach to regularization is to substitute the original problem with an optimization problem that minimizes the sum of two terms, an $\ell^2$ term and an $\ell^q$ term with $0<q\leq 1$. The first penalizes the distance between the measured data and the reconstructed one, the latter imposes sparsity on some features of the computed solution.

In this work, we propose to use the fractional Laplacian of a properly constructed graph in the $\ell^q$ term to compute extremely accurate reconstructions of the desired images. A simple model with a fully plug-and-play method is used to construct the graph and enhanced diffusion on the graph is achieved with the use of a fractional exponent in the Laplacian operator. Since this is a global operator, we propose to replace it with an approximation in an appropriate Krylov subspace. We show that the algorithm is a regularization method under some reasonable assumptions. Some selected numerical examples in image deblurring and computer tomography show the performances of our proposal.

[1] D. Bianchi, A. Buccini, M. Donatelli, E. Randazzo. Graph Laplacian for image deblurring. Electronic Transactions on Numerical Analysis, 55:169-186, 2021.

[2] A. Buccini, M. Donatelli. Graph Laplacian in $\ell^2-\ell^q$ regularization for image reconstruction. Proceedings - 2021 21st International Conference on Computational Science and Its Applications, ICCSA 2021:29-38, 2021.

[3] S. Aleotti, A. Buccini, M. Donatelli. Fractional Graph Laplacian for image reconstruction.In progress, Applied Numerical Mathematics, 2023