Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems

We consider projection algorithms for solving (nonconvex) feasibility problems in Euclidean spaces. Of special interest are the Method of Alternating Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection Algorithm (AAR). In the case of convex feasibility, firm nonexpansiveness of projection mappings is a global property that yields global convergence of MAP and for consistent problems AAR. Based on (\epsilon, \delta)-regularity of sets developed by Bauschke, Luke, Phan and Wang in 2012, a relaxed local version of firm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems. Together with a coercivity condition that relates to the regularity of the intersection, this yields local linear convergence of MAP for a wide class of nonconvex problems,Comment: 22 pages, no figures, 30 reference

Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved nonconvex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. In a recent paper Bauschke, Luke, Phan and Wang (2014) showed that, locally, the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint. In this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications. These analytical tools can also be applied to other algorithms. This is demonstrated with the prominent Douglas-Rachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem.Comment: 29 pages, 2 figures, 37 references. Much expanded version from last submission. Title changed to reflect new development

Magnetic Properties of Pr0.7Ca0.3MnO3/SrRuO3 Superlattices

High-quality Pr0.7Ca0.3MnO3/SrRuO3 superlattices were fabricated by pulsed laser deposition and were investigated by high-resolution transmission electron microscopy and SQUID magnetometry. Superlattices with orthorhombic and tetragonal SrRuO3 layers were investigated. The superlattices grew coherently; in the growth direction Pr0.7Ca0.3MnO3 layers were terminated by MnO2- and SrRuO3 layers by RuO2-planes. All superlattices showed antiferromagnetic interlayer coupling in low magnetic fields. The coupling strength was significantly higher for orthorhombic than for tetragonal symmetry of the SrRuO3 layers. The strong interlayer exchange coupling in the superlattice with orthorhombic SrRuO3 layers led to a magnetization reversal mechanism with a partially inverted hysteresis loop.Comment: 12 pages, 4 figure

Activity Identification and Local Linear Convergence of Douglas--Rachford/ADMM under Partial Smoothness

Convex optimization has become ubiquitous in most quantitative disciplines of science, including variational image processing. Proximal splitting algorithms are becoming popular to solve such structured convex optimization problems. Within this class of algorithms, Douglas--Rachford (DR) and alternating direction method of multipliers (ADMM) are designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local convergence behaviour of DR (resp. ADMM) when the involved functions (resp. their Legendre-Fenchel conjugates) are moreover partly smooth. More precisely, when both of the two functions (resp. their conjugates) are partly smooth relative to their respective manifolds, we show that DR (resp. ADMM) identifies these manifolds in finite time. Moreover, when these manifolds are affine or linear, we prove that DR/ADMM is locally linearly convergent. When $J$ and $G$ are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified manifolds. This is illustrated by several concrete examples and supported by numerical experiments.Comment: 17 pages, 1 figure, published in the proceedings of the Fifth International Conference on Scale Space and Variational Methods in Computer Visio

The INTEGRAL-OMC Scientific Archive

The Optical Monitoring Camera (OMC) on-board the INTEGRAL satellite has, as one of its scientific goals, the observation of a large number of variable sources previously selected. After almost 6 years of operations, OMC has monitored more than 100 000 sources of scientific interest. In this contribution we present the OMC Scientific Archive (http://sdc.laeff.inta.es/omc/) which has been developed to provide the astronomical community with a quick access to the light curves generated by this instrument.We describe the main characteristics of this archive, as well as important aspects for the users: object types, temporal sampling of light curves and photometric accuracy.Comment: 4 pages, 5 figures. "Highlights of Spanish Astrophysics V" Proceedings of the VIII Scientific Meeting of the Spanish Astronomical Society (SEA) held in Santander, July 7-11, 200