Gallery: Douglas-Rachford Algorithm

A visualisation of the Douglas-Rachford Algorithm created by Matthew P. Skerritt and Scott Lindstrom as part of research conducted with Jonathan M. Borwein, Brailey Sims, and Anna Schneider.

Here we see basins of periodicity for an iterative method applied to solving a perturbed version of the phase retrieval problem. We run the Douglas-Rachford algorithm for each individual pixel in the plane. We begin from the centre of that pixel, and color it according to the behaviour of the point under the dynamical system. Colours were inspired by Australian aboriginal artwork. The patterns which emerge in this image are more polyhedral than one might expect.

This picture is used in the main poster for Australian Mathematical Society special interest group Mathematics of Computation and Optimization (MoCaO). A variant of it was featured at the Bridges Math Art conference in Stockholm, Sweden.

Other visualisations by Scott Lindstrom can be found on his web page. Other visualisations by Matthew P. Skerritt will be available on his upcoming web page.