A Variational Inequality Model for the Construction of Signals from Inconsistent Nonlinear Equations

TL;DR

This paper introduces a variational inequality approach for constructing signals from inconsistent nonlinear equations, providing a convergent algorithm and demonstrating robustness in signal and image processing tasks.

Contribution

It proposes a novel variational inequality framework for signal synthesis from inconsistent nonlinear equations and develops a block-iterative algorithm with proven convergence.

Findings

Robust signal recovery demonstrated in numerical experiments.

Convergent block-iterative algorithm for the variational inequality model.

Effective handling of inconsistent nonlinear equations in signal processing.

Abstract

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.