CARMA OANT SEMINAR Speaker: Dr Francisco Aragón Artacho, CARMA, The University of Newcastle Title: Walking on real numbers Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle Access Grid Venue: UNewcastle [ENQUIRIES] Time and Date: 3:00 pm, Mon, 22nd Oct 2012 Abstract: Motivated by the desire to visualise large mathematical data sets, especially in number theory, we offer various tools for representing floating point numbers as planar walks and for quantitatively measuring their “randomness”. What to expect: some interesting ideas, many beautiful pictures (including a 108-gigapixel picture of π), and some easy-to-understand maths. What you won’t get: too many equations, difficult proofs, or any “real walking”. This is a joint work with David Bailey, Jon Borwein and Peter Borwein. [Permanent link] CARMA SEMINAR Speaker: Dr Francisco Aragón Artacho, CARMA, The University of Newcastle Title: Douglas-Rachford: an algorithm that mysteriously solves sudokus and other non-convex problems Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 4:00 pm, Thu, 21st Jun 2012 Abstract: The Douglas-Rachford algorithm is an iterative method for finding a point in the intersection of two (or more) closed sets. It is well-known that the iteration (weakly) converges when it is applied to convex subsets of a Hilbert space. Despite the absence of a theoretical justification, the algorithm has also been successfully applied to various non-convex practical problems, including finding solutions for the eight queens problem, or sudoku puzzles. In particular, we will show how these two problems can be easily modelled. With the aim providing some theoretical explanation of the convergence in the non-convex case, we have established a region of convergence for the prototypical non-convex Douglas-Rachford iteration which finds a point on the intersection of a line and a circle. Previous work was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given. PS: Bring your hardest sudoku puzzle :) Download: Talk slides (4.7 MB) [Permanent link] SIGMAOPT SEMINAR Speaker: Dr Francisco Aragón Artacho, CARMA, The University of Newcastle Title: Lipschitzian properties of a generalized proximal point algorithm Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle Access Grid Venue: UNewcastle [ENQUIRIES] Time and Date: 4:00 pm, Thu, 1st Sep 2011 Abstract: Basically, a function is Lipschitz continuous if it has a bounded slope. This notion can be extended to set-valued maps in different ways. We will mainly focus on one of them: the so-called Aubin (or Lipschitz-like) property. We will employ this property to analyze the iterates generated by an iterative method known as the proximal point algorithm. Specifically, we consider a generalized version of this algorithm for solving a perturbed inclusion $$y \in T(x),$$ where $y$ is a perturbation element near 0 and $T$ is a set-valued mapping. We will analyze the behavior of the convergent iterates generated by the algorithm and we will show that they inherit the regularity properties of $T$, and vice versa. We analyze the cases when the mapping $T$ is metrically regular (the inverse map has the Aubin property) and strongly regular (the inverse is locally a Lipschitz function). We will not assume any type of monotonicity. [Permanent link]