Breakdowns in the implementation of the Lánczos method for solving linear systems

AbstractThe Lánczos method for solving systems of linear equations is based on formal orthogonal polynomials. Its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials. A division by zero can occur in such recurrence relations, thus causing a breakdown in the algorithm which has to be stopped. In this paper, two types of breakdowns are discussed. The true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used. Among all the recurrence relationships which can be used and all the algorithms for implementing the Lánczos method which came out from them, the only reliable algorithm is Lánczos/Orthodir which can only suffer from true breakdowns. It is shown how to avoid true breakdowns in this algorithm. Other algorithms are also discussed and the case of near-breakdown is treated. The same treatment applies to other methods related to Lánczos'

Closer to the solutions: iterative linear solvers

The solution of dense linear systems received much attention after the second world war, and by the end of the sixties, most of the problems associated with it had been solved. For a long time, Wilkinson's \The Algebraic Eigenvalue Problem" [107], other than the title suggests, became also the standard textbook for the solution of linear systems. When it became clear that partial dierential equations could be solved numerically, to a level of accuracy that was of interest for application areas (such as reservoir engineering, and reactor diusion modeling), there was a strong need for the fast solution of the discretized systems, and iterative methods became popular for these problems

A Regularized Boundary Element Formulation for Contactless SAR Evaluations within Homogeneous and Inhomogeneous Head Phantoms

This work presents a Boundary Element Method (BEM) formulation for contactless electromagnetic field assessments. The new scheme is based on a regularized BEM approach that requires the use of electric measurements only. The regularization is obtained by leveraging on an extension of Calderon techniques to rectangular systems leading to well-conditioned problems independent of the discretization density. This enables the use of highly discretized Huygens surfaces that can be consequently placed very near to the radiating source. In addition, the new regularized scheme is hybridized with both surfacic homogeneous and volumetric inhomogeneous forward BEM solvers accelerated with fast matrix-vector multiplication schemes. This allows for rapid and effective dosimetric assessments and permits the use of inhomogeneous and realistic head phantoms. Numerical results corroborate the theory and confirms the practical effectiveness of all newly proposed formulations

The Ultraviolet and Infrared Star Formation Rates of Compact Group Galaxies: An Expanded Sample

Compact groups of galaxies provide insight into the role of low-mass, dense environments in galaxy evolution because the low velocity dispersions and close proximity of galaxy members result in frequent interactions that take place over extended timescales. We expand the census of star formation in compact group galaxies by \citet{tzanavaris10} and collaborators with Swift UVOT, Spitzer IRAC and MIPS 24 \micron\ photometry of a sample of 183 galaxies in 46 compact groups. After correcting luminosities for the contribution from old stellar populations, we estimate the dust-unobscured star formation rate (SFR$_{\mathrm{UV}}$) using the UVOT uvw2photometry. Similarly, we use the MIPS 24 \micron\ photometry to estimate the component of the SFR that is obscured by dust (SFR$_{\mathrm{IR}}$). We find that galaxies which are MIR-active (MIR-"red"), also have bluer UV colours, higher specific star formation rates, and tend to lie in H~{\sc i}-rich groups, while galaxies that are MIR-inactive (MIR-"blue") have redder UV colours, lower specific star formation rates, and tend to lie in H~{\sc i}-poor groups. We find the SFRs to be continuously distributed with a peak at about 1 M$_{\odot}$ yr$^{-1}$, indicating this might be the most common value in compact groups. In contrast, the specific star formation rate distribution is bimodal, and there is a clear distinction between star-forming and quiescent galaxies. Overall, our results suggest that the specific star formation rate is the best tracer of gas depletion and galaxy evolution in compact groups.Comment: 19 pages, 17 figure

A framework for deflated and augmented Krylov subspace methods

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately. We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MINRES method. Numerical experiments illustrate properties of different variants of deflated MINRES analyzed in this paper.Comment: 24 pages, 3 figure

Measurements of the Flow and Pressure within an Axial Waterjet Pump during Cavitation Breakdown

International audienceThe flow mechanisms causing cavitation breakdown in axial turbomachines have been a long-standing puzzle. Recently, observations in our laboratory have associated this phenomenon with blockage caused by perpendicular cavitating vortices (PCVs) in the blade tip region. In follow-up experiments, circumferential casing grooves (CGs) have been used to manipulate the tip leakage flow, intending to suppress the formation of PVCs. While the PCV formation is affected by a CG located near the trailing edge, it shows very little effect in alleviating the rapid performance deterioration during breakdown. High-speed movies show that breakdown begins when a slight decrease in inlet pressure extends the attached/sheet cavitation on the blade suction side (SS) into the blade overlap region. At this phase, the area covered by the sheet cavitation initially fluctuates and then expands rapidly to cover a large portion of the surface. Stereoscopic particle image velocimetry (SPIV) measurements performed in non-cavitating fractions of the passage show that this process involves a significant increase in liquid velocity and decrease in pressure along the blade pressure side (PS) in the overlap region. It appears that blockage caused by the cavitation on the SS accelerates the flow along the PS, which results in further reduction in pressure over the entire passage and further expansion of the cavitation on the SS. However, the pressure difference across the blade tip measured at mid span increases, compensating for the reduction in PS pressure. Hence, in some cases, the pump head increases slightly at the onset of breakdown. Further reduction in inlet pressure causes an increase in blockage in the tip region, but still a slight increase in velocity deeper in the passage. The reduced flowrate combined with a decrease in mid passage pressure results in a rapid drop in head as well. The CG is only partially effective in reducing the tip region blockage, presumed due to the inability of the shallow grooves to affect the cloud cavitation that fills the trailing edge tip region at this stage

Additional application of the NASCAP code. Volume 1: NASCAP extension

The NASCAP computer program comprehensively analyzes problems of spacecraft charging. Using a fully three dimensional approach, it can accurately predict spacecraft potentials under a variety of conditions. Several changes were made to NASCAP, and a new code, NASCAP/LEO, was developed. In addition, detailed studies of several spacecraft-environmental interactions and of the SCATHA spacecraft were performed. The NASCAP/LEO program handles situations of relatively short Debye length encountered by large space structures or by any satellite in low earth orbit (LEO)

A platform for numerical computations with special application to preconditioning

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