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

A continuous low star formation rate in IZw 18 ?

Deep long-slit spectroscopic observations of the blue compact galaxy IZw 18 obtained with the CFH 3.6 m Telescope are presented. The very low value of oxygen abundance previously reported is confirmed and a very homogeneous abundance distribution is found (no variation larger than 0.05 dex) over the whole ionized region. We concur with Tenorio-Tagle (1996) and Devost et al. (1997) that the observed abundance level cannot result from the material ejected by the stars formed in the current burst, and propose that the observed metals were formed in a previous star formation episode. Metals ejected in the current burst of star formation remain most probably hidden in a hot phase and are undetectable using optical spectroscopy. We discuss different scenarios of star formation in IZw 18. Combining various observational facts, for instance the faint star formation rate observed in low surface brightness galaxies van Zee et al. (1997), it is proposed that a low and continuous rate of star formation occurring during quiescent phases between bursts could be a significant source of metal enrichment of the interstellar medium.Comment: 10 pages, 4 Postscript figures, to be published in Astronomy and Astrophysics main journa

Physical properties and evolutionary state of the Lyman alpha emitting starburst galaxy IRAS 08339+6517

Though Lyman alpha emission (Lya) is one of the most used tracers of massive star formation at high redshift, a correct understanding of radiation transfer effects by neutral gas is required to properly quantify the star formation rate along the history of the Universe. We are embarked in a program to study the properties of the Lya emission (spectral profile, spatial distribution, relation to Balmer lines intensity,...) in several local starburst galaxies. We present here the results obtained for IRAS 08339+6517. Using evolutionary population synthesis models, we have characterized the properties of the starburst (UV continuum, Halpha, total infrared and X-ray emissions, etc.), which transformed 1.4e+8 Mo of gas into stars around 5-6 Myr ago. In addition to the central compact emission blob, we have identified a diffuse Lya emission component smoothly distributed over the whole central area of IRAS 08339+6517. This diffuse emission is spatially decoupled from the UV continuum, the Halpha emission or the Halpha/Hbeta ratio. Both locally and globally, the Lya/Halpha ratio is lower than the Case B predictions, even after reddening correction, with an overall Lya escape fraction of only 4%. We conclude that in IRAS 08339+6517 the resonant scattering of Lya photons by an outflowing shell of neutral gas causes their highly-efficient destruction by dust, which explains the low Lya escape fraction measured. These results stress again the importance of a proper correction of scattering and transfer effects when using Lya to derive the star formation rate in high-redshift galaxies.Comment: Accepted for publication in A&A, 17 pages, 13 figures, 8 tables. If problems with quality of images, see https://cloud.cab.inta-csic.es/public.php?service=files&file=%2Fotih%2Ffiles%2Foti_mas%2Firas%2Firas-v53.ack_referee.pd