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Jonathan M. Borwein

Publications and source records attributed to Jonathan M. Borwein.

At least 19 recordsLinked to original sources
arXiv2018

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

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.

math.FA↗
arXiv2017

Ergodic behaviour of a Douglas-Rachford operator away from the origin

It is shown that away from the origin, the Douglas-Rachford operator with respect to a sphere and a convex set in a Hilbert space can be approximated by a another operator which satisfies a weak ergodic theorem. Similar results for other projection and reflection operators are also discussed.

math.OC↗
arXiv2017

Gamma and Factorial in the Monthly

The Monthly has published roughly fifty papers on the $Γ$ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the context of the larger mathematical literature on $Γ$.

math.HO↗
arXiv2016

Convergence rate analysis for averaged fixed point iterations in the presence of Hölder regularity

In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded Hölder regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for Krasnoselskii-Mann iterations, the cyclic projection algorithm, and the Douglas-Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a finite dimensional space, we show that the Hölder regularity properties are automatically satisfied, from which sublinear convergence follows.

math.OC↗
arXiv2016

Generalized Continued Logarithms and Related Continued Fractions

We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchine's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchine constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize past continued logarithms.

math.NT↗
arXiv2015

Convex analysis in groups and semigroups: a sampler

We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only on vector spaces. Some examples and counter-examples are also discussed.

math.OC↗
arXiv2015

Nearest points and delta convex functions in Banach spaces

Given a closed set $C$ in a Banach space $(X, \|\cdot\|)$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_C(x) =\|x-z\|$, where $d_C$ is the distance of $x$ from $C$. We shortly survey the problem of studying how large is the set of points in $X$ which have nearest points in $C$. We then discuss the topic of delta-convex functions and how it is related to finding nearest points.

math.FA↗
arXiv2015

Densities of short uniform random walks in higher dimensions

We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of up to five steps. Somewhat to our surprise, we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two-dimensional case.

math.CA↗
arXiv2015

Global Behavior of the Douglas-Rachford Method for a Nonconvex Feasibility Problem

In recent times the Douglas-Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a well-quasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems.

math.OC↗
arXiv2014

Global convergence of a non-convex Douglas-Rachford iteration

We establish a region of convergence for the proto-typical non-convex Douglas-Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration [2] was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given.

math.OC↗
arXiv2014

Reflection methods for inverse problems with application to protein conformation determination

The Douglas-Rachford reflection method is a general purpose algorithm useful for solving the feasibility problem of finding a point in the intersection of finitely many sets. In this chapter we demonstrate that applied to a specific problem, the method can benefit from heuristics specific to said problem which exploit its special structure. In particular, we focus on the problem of protein conformation determination formulated within the framework of matrix completion, as was considered in a recent paper of the present authors.

math.OC↗
arXiv2013

Analysis of the convergence rate for the cyclic projection algorithm applied to basic semi-algebraic convex sets

In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semi-algebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semi-algebraic convex sets and the dimension of the underlying space. We achieve our results by exploiting the algebraic structure of the basic semi-algebraic convex sets.

math.FA↗
arXiv2013

On lattice sums and Wigner limits

Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static electron lattices, in a mathematically rigorous way one commonly truncates the lattice sum and the corresponding integral and takes the limit along expanding hypercubes or other regular geometric shapes. We generalize the known mathematically rigorous two and three dimensional results regarding Wigner limits, as laid down in [Analysis of certain lattice sums, D. Borwein, J. M. Borwein, and R. Shail, 1989], to integer lattices of arbitrary dimension. In doing so, we also resolve a problem posed in Chapter 7 of [Lattice Sums: Then and Now, J. M. Borwein, L. Glasser, R. McPhedran, J. G. Wan, and I. J. Zucker, 2013]. For the sake of clarity, we begin by considering the simpler case of cubic lattice sums first, before treating the case of arbitrary quadratic forms. We also consider limits taken along expanding hyperballs with respect to general norms, and connect with classical topics such as Gauss's circle problem. An appendix is included to recall certain properties of Epstein zeta functions that are either used in the paper or serve to provide perspective.

math-ph↗
arXiv2013

Recent progress on Monotone Operator Theory

In this paper, we survey recent progress on the theory of maximally monotone operators in general Banach space. We also extend various of the results and leave some open questions.

math.FA↗