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arXiv · 1610.03975

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

Jonathan M. Borwein·Scott B. Lindstrom·Brailey Sims·Anna Schneider·Matthew P. Skerritt
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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.

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Jonathan M. Borwein, Scott B. Lindstrom, Brailey Sims, Anna Schneider, Matthew P. Skerritt. 2016-10-13. Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres. https://doi.org/10.1007/s11228-017-0457-0

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