# A variational inequality model for the construction of signals from inconsistent nonlinear equations

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.

Cite this Paper (BibTeX)
@article{woodstock:20220115,
author={Patrick L. Combettes and Zev C. Woodstock},
title={A variational inequality model for the construction of signals from inconsistent nonlinear equations},
journal={SIAM Journal on Imaging Sciences},
year={2022},
volume={15},
number={1},
pages={84--109},
DOI={10.1137/21M1420368}}