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Phase plotting is a useful way of visualising functions on complex space. We reinvent the method in the context of hyperbolic geometry, and we use it to plot functions on various representative surfaces for hyperbolic space, illustrating with direct motions in particular. The reinvention is nontrivial, and we discuss the essential features for ensuring that such visualisations convey useful information. Our approach is to exploit conformal maps between representative surfaces, in order to uniquely represent the preimages of geodesics. Our considerations and methods are prototypical of what one might consider for adapting similar methods of visualisation in other contexts.

The Douglas--Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which --- somewhat surprisingly --- depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails. doi:10.1017/S1446181118000391

Moreau's seminal paper, introducing what is now called the Moreau envelope and the proximity operator (also known as the proximal mapping), appeared in 1965. The Moreau envelope of a given convex function provides a regularized version which has additional desirable properties such as differentiability and full domain. Fifty years ago, Attouch proposed using the Moreau envelope for regularization. Since then, this branch of convex analysis has developed in many fruitful directions. In 1967, Bregman introduced what is nowadays known as the Bregman distance as a measure of discrepancy between two points generalizing the square of the Euclidean distance. Proximity operators based on the Bregman distance have become a topic of significant research as they are useful in the algorithmic solution of optimization problems. More recently, in 2012, Kan and Song studied regularization aspects of the left Bregman-Moreau envelope even for nonconvex functions. In this paper, we complement previous works by analyzing the left and right Bregman-Moreau envelopes and by providing additional asymptotic results. Several examples are provided.