Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

The biconjugate gradient (BCG) method is the "natural" generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.

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QMR: a quasi-minimal residual method for non-Hermitian linear systems

http://scgroup.hpclab.ceid.upatras.gr/class/SCII/Various/SaadvdVorst2000.pdf

Iterative solution of linear systems in the 20th century

Recent advances in the field of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for non-Hermitian matrices.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0962492900002245

Iterative solution of linear systems

Data of vertically pointing microrain radars (MRRs), located at various sites around the Baltic Sea, were analyzed for a period of several years. From the Doppler spectra profiles of drop size distributions (DSDs) are obtained. A significant height dependence of the shape of the DSDs—and thus of the Z–R relations—is observed at high rain rates. This implies, for the considered sites, that ground-based Z–R relations lead to underestimation of high rain rates by weather radars.

Profiles of Raindrop Size Distributions as Retrieved by Microrain Radars

Measurements of rain were obtained with a vertically pointing micro radar (MRR) with 1 min time resolution and 50(100) m height resolution at the German Baltic coast on the Zingst peninsula (54.43°N, 12.67°E). The comparison with a conventional rain gauge (30 min integration time) for a five months summer period show a correlation coefficient of p = 0.87 for the rainrate and agreement within 5% for the total rainfall integrated over the whole period. Single measurements with 30 min integration time showed deviations up to a factor of 2 between MRR and rain gauge. Classification of the measurements into rainrate intervals shows that rainrates around 0.2 mm h -1 provide the highest contribution per rainrate interval to the total rainfall. Typical distributions of number-concentration, liquid-water- concentration and rainrate versus drop size, retrieved with the MRR, are presented. Simultaneous estimates of rainrate and reflectivity factor with data of a C-band (frequency 6 GHz) weather radar suggest that the MRR may be used to support quantitative rainrate estimates with weather radars. The weather radar used for comparison is operated by the German Weather Service and is situated 51 km from the MRR.

Rain observations with a vertically looking Micro Rain Radar (MRR)

PURPOSE: This study investigates the normal ultrasono graphic morphology of pelvic floor striated muscle and evaluates whether there are differences between males and females and between healthy subjects and patients with idiopathic fecal incontinence. METHODS: Manometric and ultrasonographic studies of the anal canal were performed in 30 healthy volunteers and in 26 patients with idiopathic fecal incontinence. Verification of ultrasonographic findings was obtained by studying anatomic preparations. RESULTS: Ultrasonography reliably identified the internal and external anal sphincter and frequently visualized neighboring structures. In addition, it detected muscular structures that have not clearly been described previously. However, differences in external sphincter function between males and females and between healthy subjects and incontinent patients were not reflected by alterations in muscle thickness. CONCLUSIONS: Endosonography of the anal canal clearly delineates muscular structures and may even visualize structures that have been previously incompletely defined. However, it remains to be shown that differences in certain muscular functions have a morphologic correlate that can be detected by endosonography.

Anal endosonography in healthy subjects and patients with idiopathic fecal incontinence

http://i.stanford.edu/pub/cstr/reports/cs/tr/89/1264/CS-TR-89-1264.pdf

Chebyshev polynomials are not always optimal

Discretisation of the classical Stokes problem gives rise to symmetric indefinite matrices with eigenvalues which, in a precise way, are not symmetric about the origin, but which do depend on a mesh size parameter. Convergence estimates for the Conjugate Residual or Minimum Residual iterative solution of such systems are given by best minimax polynomial approximations on an inclusion set for the eigenvalues. In this paper, an analytic convergence estimate for such problems is given in terms of an asymptotically small mesh size parameter.

Bernd Fischer

Papers

Profiles of Raindrop Size Distributions as Retrieved by Microrain Radars

Rain observations with a vertically looking Micro Rain Radar (MRR)

Anal endosonography in healthy subjects and patients with idiopathic fecal incontinence

Chebyshev polynomials are not always optimal

The convergence rate of the minimal residual method for the Stokes problem