2021/05/16 by Patrick L. Combettes, Combettes, Patrick L., Zev C. Woodstock +1 · 1 citation
Engineering · Mathematics · #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Photoacoustic and Ultrasonic Imaging #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.2105.07380
openalex publication_date 2021/05/16 · openalex created_date 2022/07/25 · openalex updated_date 2026/08/01
Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies a number of nonlinear equations involving firmly nonexpansive operators. We investigate this formalism in the case when, due to inaccurate modeling or perturbations, the nonlinear equations are inconsistent. A relaxed formulation of the original problem is proposed in the form of a variational inequality. The properties of the relaxed problem are investigated and a provenly convergent block-iterative algorithm, whereby only blocks of the underlying firmly nonexpansive operators are activated at a given iteration, is devised to solve it. Numerical experiments illustrate robust recoveries in several signal and image processing applications.