A preconditioned deepest descent algorithm for a class of optimization problems involving the $p(x)$-Laplacian operator

TL;DR

This paper introduces a preconditioned descent algorithm for optimization problems involving the variable exponent p(x)-Laplacian, demonstrating its effectiveness through numerical experiments in imaging applications.

Contribution

It proposes a novel preconditioned descent method using a frozen exponent approach for p(x)-Laplacian problems, with analysis of well-posedness and practical numerical validation.

Findings

The algorithm effectively solves p(x)-Laplacian optimization problems.

Numerical experiments show advantages in image processing tasks.

The method is robust under log-Hölder continuous p(x) functions.

Abstract

In this paper we are concerned with a class of optimization problems involving the $p(x)$-Laplacian operator, which arise in imaging and signal analysis. We study the well-posedness of this kind of problems in an amalgam space considering that the variable exponent $p(x)$ is a log-H\"older continuous function. Further, we propose a preconditioned descent algorithm for the numerical solution of the problem, considering a "frozen exponent" approach in a finite dimension space. Finally, we carry on several numerical experiments to show the advantages of our method. Specifically, we study two detailed example whose motivation lies in a possible extension of the proposed technique to image processing.