Complexity bounds on supermesh construction for quasi-uniform meshes

Projecting fields between different meshes commonly arises in computational physics. This operation requires a supermesh construction and its computational cost is proportional to the number of cells of the supermesh $n$. Given any two quasi-uniform meshes of $n_A$ and $n_B$ cells respectively, we show under standard assumptions that n is proportional to $n_A + n_B$. This result substantially improves on the best currently available upper bound on $n$ and is fundamental for the analysis of algorithms that use supermeshes

Multiple local minima of PDE-constrained optimisation problems via deflation

Nonconvex optimisation problems constrained by partial differential equations (PDEs) may permit distinct local minima. In this paper we present a numerical technique, called deflation, for computing multiple local solutions of such optimisation problems. The basic approach is to apply a nonlinear transformation to the Karush-Kuhn-Tucker optimality conditions that eliminates previously found solutions from consideration. Starting from some initial guess, Newton's method is used to find a stationary point of the Lagrangian; this solution is then deflated away, and Newton's method is initialised from the same initial guess to find other solutions. In this paper, we investigate how the Schur complement preconditioners widely used in PDE-constrained optimisation perform after deflation. We prove an upper bound on the number of new distinct eigenvalues of a matrix after an arbitrary additive perturbation; from this it follows that for diagonalisable operators the number of Krylov iterations required for exact convergence of the Newton step at most doubles compared to the undeflated problem. While deflation is not guaranteed to converge to all minima, these results indicate the approach scales to arbitrary-dimensional problems if a scalable Schur complement pre-conditioner is available. The technique is demonstrated on a discretised nonconvex PDE-constrained optimisation problem with approximately ten million degrees of freedom

BlogForever D5.1: Design and Specification of Case Studies

This document presents the specification and design of six case studies for testing the BlogForever platform implementation process. The report explains the data collection plan where users of the repository will provide usability feedback through questionnaires as well as details of scalability analysis through the creation of specific log files analytics. The case studies will investigate the sustainability of the platform, that it meets potential users’ needs and that is has an important long term impact

Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

We study the problem of a cholesteric liquid crystal confined to an elliptical channel. The system is geometrically frustrated because the cholesteric prefers to adopt a uniform rate of twist deformation, but the elliptical domain precludes this. The frustration is resolved by deformation of the layers or introduction of defects, leading to a particularly rich family of equilibrium configurations. To identify the solution set, we adapt and apply a new family of algorithms, known as deflation methods, that iteratively modify the free energy extremisation problem by removing previously known solutions. A second algorithm, deflated continuation, is used to track solution branches as a function of the aspect ratio of the ellipse and preferred pitch of the cholesteric.Comment: 9 pages, 6 figure

Searching for X-ray sources in nearby late-type galaxies with low star formation rates

Late type non-starburst galaxies have been shown to contain X-ray emitting objects, some being ultraluminous X-ray sources. We report on XMM-Newton observations of 11 nearby, late-type galaxies previously observed with the Hubble Space Telescope (HST) in order to find such objects. We found 18 X-ray sources in or near the optical extent of the galaxies, most being point-like. If associated with the corresponding galaxies, the source luminosities range from $2 \times 10^{37}$ erg s$^{-1}$ to $6 \times 10^{39}$ erg s$^{-1}$. We found one ultraluminous X-ray source, which is in the galaxy IC 5052, and one source coincident with the galaxy IC 4662 with a blackbody temperature of $0.166 \pm 0.015$ keV that could be a quasi-soft source or a quiescent neutron star X-ray binary in the Milky Way. One X-ray source, XMMU J205206.0$-$691316, is extended and coincident with a galaxy cluster visible on an HST image. The X-ray spectrum of the cluster reveals a redshift of $z = 0.25 \pm 0.02$ and a temperature of 3.6$\pm$0.4 keV. The redshift was mainly determined by a cluster of Fe XXIV lines between the observed energy range $0.8-1.0$ keV.Comment: 8 pages, to appear in MNRA