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Abstract This paper introduces new solvers for efficiently computing solutions to large-scale inverse problems with group sparsity regularization, including both non-overlapping and overlapping groups. Group sparsity regularization refers to a type of structured sparsity regularization, where the goal is to impose additional structure in the regularization process by assigning variables to predefined groups that may represent graph or network structures. Special cases of group sparsity regularization includeℓ₁and isotropic total variation regularization. In this work, we develop hybrid projection methods based on flexible Krylov subspaces, where we first recast the group sparsity regularization term as a sequence of 2-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion. Then we exploit flexible preconditioning techniques to efficiently incorporate the weight updates. The main advantages of these methods are that they are computationally efficient (leveraging the advantages of flexible methods), they are general (and therefore very easily adaptable to new regularization term choices), and they are able to select the regularization parameters automatically and adaptively (exploiting the advantages of hybrid methods). Extensions to multiple regularization terms and solution decomposition frameworks (e.g. for anomaly detection) are described, and a variety of numerical examples demonstrate both the efficiency and accuracy of the proposed approaches compared to existing solvers. 

Inverse models arise in various environmental applications, ranging from atmospheric modeling to geosciences. Inverse models can often incorporate predictor variables, similar to regression, to help estimate natural processes or parameters of interest from observed data. Although a large set of possible predictor variables may be included in these inverse or regression models, a core challenge is to identify a small number of predictor variables that are most informative of the model, given limited observations. This problem is typically referred to as model selection. A variety of criterion-based approaches are commonly used for model selection, but most follow a two-step process: first, select predictors using some statistical criteria, and second, solve the inverse or regression problem with these predictor variables. The first step typically requires comparing all possible combinations of candidate predictors, which quickly becomes computationally prohibitive, especially for large-scale problems. In this work, we develop a one-step approach for linear inverse modeling, where model selection and the inverse model are performed in tandem. We reformulate the problem so that the selection of a small number of relevant predictor variables is achieved via a sparsity-promoting prior. Then, we describe hybrid iterative projection methods based on flexible Krylov subspace methods for efficient optimization. These approaches are well-suited for large-scale problems with many candidate predictor variables. We evaluate our results against traditional, criteria-based approaches. We also demonstrate the applicability and potential benefits of our approach using examples from atmospheric inverse modeling based on NASA's Orbiting Carbon Observatory-2 (OCO-2) satellite. 

Abstract Computed Tomography (CT) has been widely adopted in medicine and it is increasingly being used in scientific and industrial applications. Parallelly, research in different mathematical areas concerning discrete inverse problems has led to the development of new sophisticated numerical solvers that can be applied in the context of CT. The Tomographic Iterative GPU-based Reconstruction (TIGRE) toolbox was born almost a decade ago precisely in the gap between mathematics and high performance computing for real CT data, providing user-friendly open-source software tools for image reconstruction. However, since its inception, the tools’ features and codebase have had over a twenty-fold increase, and are now including greater geometric flexibility, a variety of modern algorithms for image reconstruction, high-performance computing features and support for other CT modalities, like proton CT. The purpose of this work is two-fold: first, it provides a structured overview of the current version of the TIGRE toolbox, providing appropriate descriptions and references, and serving as a comprehensive and peer-reviewed guide for the user; second, it is an opportunity to illustrate the performance of several of the available solvers showcasing real CT acquisitions, which are typically not be openly available to algorithm developers. 

Abstract. Inverse models arise in various environmental applications, ranging from atmospheric modeling to geosciences. Inverse models can often incorporate predictor variables, similar to regression, to help estimate natural processes or parameters of interest from observed data. Although a large set of possible predictor variables may be included in these inverse or regression models, a core challenge is to identify a small number of predictor variables that are most informative of the model, given limited observations. This problem is typically referred to as model selection. A variety of criterion-based approaches are commonly used for model selection, but most follow a two-step process: first, select predictors using some statistical criteria, and second, solve the inverse or regression problem with these predictor variables. The first step typically requires comparing all possible combinations of candidate predictors, which quickly becomes computationally prohibitive, especially for large-scale problems. In this work, we develop a one-step approach, where model selection and the inverse model are performed in tandem. We reformulate the problem so that the selection of a small number of relevant predictor variables is achieved via a sparsity-promoting prior. Then, we describe hybrid iterative projection methods based on flexible Krylov subspace methods for efficient optimization.  These approaches are well-suited for large-scale problems with many candidate predictor variables. We evaluate our results against traditional, criteria-based approaches. We also demonstrate the applicability and potential benefits of our approach using examples from atmospheric inverse modeling based on NASA's Orbiting Carbon Observatory 2 (OCO-2) satellite.