Distance-like based accelerated regularization methods

Inverse problems constitute an essential framework for approaching a large variety of issues in technical and medical domains. However, most inverse problems are ill-posed, meaning that small perturbations in the data can trigger high oscillations in the solution. Thus, rapid developments in the above-mentioned domains make designing efficient and stable inverse problems algorithms a continuous challenge. A promising direction in this respect is to investigate methods using more sophisticated measures for distances between points, rather than the Euclidean-type distances. In this project, we aim at efficiently recovering stable approximations of inverse problems solutions with certain features, such as nonnegativity, sparsity, and piecewise constant structure. The main novelty here is solving such problems by iterative methods that exploit the versatile role of distance-like functions in promoting the solution features and accelerating the convergence of the iterates. The infinite dimensional setting that naturally anchors these problems brings even more complexity to the envisaged framework. We would like to propose and analyze accelerated versions of several iterative methods based on such distance-like functions, both theoretically and computationally, and to compare them with well-established methods. Interesting links to machine learning and image processing can also be investigated.

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