Catching-up Algorithm with Approximate Projections for Moreau's Sweeping Processes

TL;DR

This paper introduces an improved catching-up algorithm for Moreau's sweeping processes using approximate projections, providing convergence analysis and efficient numerical methods, advancing both theoretical understanding and practical simulation capabilities.

Contribution

It presents a novel enhanced catching-up algorithm with approximate projections, offering convergence proofs and practical numerical strategies for sweeping processes.

Findings

Convergence of the enhanced algorithm under certain set conditions

Development of efficient numerical methods for approximate projections

Recovery of classical existence results with new insights

Abstract

In this paper, we develop an enhanced version of the catching-up algorithm for sweeping processes through an appropriate concept of approximate projections. We establish some properties of this notion of approximate projection. Then, under suitable assumptions, we show the convergence of the enhanced catching-up algorithm for prox-regular, subsmooth, and merely closed sets. Finally, we discuss efficient numerical methods for obtaining approximate projections. Our results recover classical existence results in the literature and provide new insight into the numerical simulation of sweeping processes.