Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres

TL;DR

This paper investigates the behavior of the Douglas-Rachford method when applied to ellipses and p-spheres, revealing geometric phenomena that influence convergence and providing new theoretical insights.

Contribution

It extends previous analyses to ellipses and p-spheres, combining computational and theoretical methods to understand convergence behavior in these generalized settings.

Findings

Geometric phenomena can slow or inhibit convergence.

Local convergence is proven near feasible points.

Infeasible case behavior is characterized theoretically.

Abstract

We expand upon previous work that examined behavior of the iterated Douglas-Rachford method for a line and a circle by considering two generalizations: that of a line and an ellipse and that of a line together with a $p$-sphere. With computer assistance we discover a beautiful geometry that illustrates phenomena which may affect the behavior of the iterates by slowing or inhibiting convergence for feasible cases. We prove local convergence near feasible points, and---seeking a better understanding of the behavior---we employ parallelization in order to study behavior graphically. Motivated by the computer-assisted discoveries, we prove a result about behavior of the method in infeasible cases.