Iterative Methods for Sparse Linear Systems

Second Edition

Author: Yousef Saad

Publisher: SIAM

ISBN: 9780898718003

Category: Differential equations, Partial

Page: 528

View: 3050

Since the first edition of this book was published in 1996, tremendous progress has been made in the scientific and engineering disciplines regarding the use of iterative methods for linear systems. The size and complexity of the new generation of linear and nonlinear systems arising in typical applications has grown. Solving the three-dimensional models of these problems using direct solvers is no longer effective. At the same time, parallel computing has penetrated these application areas as it became less expensive and standardized. Iterative methods are easier than direct solvers to implement on parallel computers but require approaches and solution algorithms that are different from classical methods. Iterative Methods for Sparse Linear Systems, Second Edition gives an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations. These equations can number in the millions and are sparse in the sense that each involves only a small number of unknowns. The methods described are iterative, i.e., they provide sequences of approximations that will converge to the solution.

Computer Solution of Large Linear Systems

Author: Gerard Meurant

Publisher: Elsevier

ISBN: 9780080529516

Category: Mathematics

Page: 776

View: 6610

This book deals with numerical methods for solving large sparse linear systems of equations, particularly those arising from the discretization of partial differential equations. It covers both direct and iterative methods. Direct methods which are considered are variants of Gaussian elimination and fast solvers for separable partial differential equations in rectangular domains. The book reviews the classical iterative methods like Jacobi, Gauss-Seidel and alternating directions algorithms. A particular emphasis is put on the conjugate gradient as well as conjugate gradient -like methods for non symmetric problems. Most efficient preconditioners used to speed up convergence are studied. A chapter is devoted to the multigrid method and the book ends with domain decomposition algorithms that are well suited for solving linear systems on parallel computers.

Iterative Methods for Large Linear Systems

Author: David R. Kincaid,Linda J. Hayes

Publisher: Academic Press

ISBN: 1483260208

Category: Mathematics

Page: 350

View: 3072

Iterative Methods for Large Linear Systems contains a wide spectrum of research topics related to iterative methods, such as searching for optimum parameters, using hierarchical basis preconditioners, utilizing software as a research tool, and developing algorithms for vector and parallel computers. This book provides an overview of the use of iterative methods for solving sparse linear systems, identifying future research directions in the mainstream of modern scientific computing with an eye to contributions of the past, present, and future. Different iterative algorithms that include the successive overrelaxation (SOR) method, symmetric and unsymmetric SOR methods, local (ad-hoc) SOR scheme, and alternating direction implicit (ADI) method are also discussed. This text likewise covers the block iterative methods, asynchronous iterative procedures, multilevel methods, adaptive algorithms, and domain decomposition algorithms. This publication is a good source for mathematicians and computer scientists interested in iterative methods for large linear systems.

Iterative Methods for Solving Linear Systems

Author: Anne Greenbaum

Publisher: SIAM

ISBN: 9781611970937

Category: Equations, Simultaneous

Page: 220

View: 6544

Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods. A language loaded with acronyms for a thousand different algorithms has developed, and it is often difficult even for specialists to identify the basic principles involved. Here is a book that focuses on the analysis of iterative methods. The author includes the most useful algorithms from a practical point of view and discusses the mathematical principles behind their derivation and analysis. Several questions are emphasized throughout: Does the method converge? If so, how fast? Is it optimal, among a certain class? If not, can it be shown to be near-optimal? The answers are presented clearly, when they are known, and remaining important open questions are laid out for further study. Greenbaum includes important material on the effect of rounding errors on iterative methods that has not appeared in other books on this subject. Additional important topics include a discussion of the open problem of finding a provably near-optimal short recurrence for non-Hermitian linear systems; the relation of matrix properties such as the field of values and the pseudospectrum to the convergence rate of iterative methods; comparison theorems for preconditioners and discussion of optimal preconditioners of specified forms; introductory material on the analysis of incomplete Cholesky, multigrid, and domain decomposition preconditioners, using the diffusion equation and the neutron transport equation as example problems. A small set of recommended algorithms and implementations is included.

Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications

Author: Daniele Bertaccini,Fabio Durastante

Publisher: CRC Press

ISBN: 1498764177

Category: Mathematics

Page: 354

View: 9140

This book describes, in a basic way, the most useful and effective iterative solvers and appropriate preconditioning techniques for some of the most important classes of large and sparse linear systems. The solution of large and sparse linear systems is the most time-consuming part for most of the scientific computing simulations. Indeed, mathematical models become more and more accurate by including a greater volume of data, but this requires the solution of larger and harder algebraic systems. In recent years, research has focused on the efficient solution of large sparse and/or structured systems generated by the discretization of numerical models by using iterative solvers.

Templates for the Solution of Linear Systems

Building Blocks for Iterative Methods

Author: Richard Barrett,Michael W. Berry,Tony F. Chan,James Demmel,June Donato,Jack Dongarra,Victor Eijkhout,Roldan Pozo,Charles Romine,Henk van der Vorst

Publisher: SIAM

ISBN: 0898713285

Category: Mathematics

Page: 112

View: 1440

Mathematics of Computing -- Numerical Analysis.

A Survey of Preconditioned Iterative Methods

Author: Are Magnus Bruaset

Publisher: CRC Press

ISBN: 9780582276543

Category: Mathematics

Page: 176

View: 9252

The problem of solving large, sparse, linear systems of algebraic equations is vital in scientific computing, even for applications originating from quite different fields. A Survey of Preconditioned Iterative Methods presents an up to date overview of iterative methods for numerical solution of such systems. Typically, the methods considered are well suited for the kind of systems arising from the discretization of partial differential equations. The focus of this presentation is on the family of Krylov subspace solvers, of which the Conjugate Gradient algorithm is a typical example. In addition to an introduction to the basic principles of such methods, a large number of specific algorithms for symmetric and nonsymmetric problems are discussed. When solving linear systems by iteration, a preconditioner is usually introduced in order to speed up convergence. In many cases, the selection of a proper preconditioner is crucial to the resulting computational performance. For this reason, this book pays special attention to different preconditioning strategies. Although aimed at a wide audience, the presentation assumes that the reader has basic knowledge of linear algebra, and to some extent, of partial differential equations. The comprehensive bibliography in this survey is provides an entry point to the enormous amount of published research in the field of iterative methods.

A Family of Preconditioned Iterative Solvers for Sparse Linear Systems

Author: Ulrike Meier Yang,University of Illinois at Urbana-Champaign. Department of Computer Science

Publisher: N.A


Category: Cellular automata

Page: 134

View: 1490

Based on the relationship between the family of Broyden methods and the EN method, a new family of iterative methods, the family of EN-like methods, is developed and analyzed. These methods are shown to be related to a variety of other known methods, which comprise the Broyden methods, GCR, GMRES, Newton's method for approximating the inverse, and a combination of a Galerkin step followed by a step of Richardson's method. Scaling-invariant versions and implementations of higher efficiency are developed, and their complexity is examined. The convergence of the new methods, as well as their restarted and truncated versions, are examined. Various convergence results are derived, which include termination within a finite number of steps and estimates for residuals and errors. The methods are also shown to be suitable in the context of inner/outer iteration schemes, and, for two of the methods, orthogonality preserving inner/outer iteration schemes are developed. Adaptive versions, which are a combination of truncated and restarted versions, and which automatically increase the size of the Krylov subspace, are included in the software package PARASPAR, which achieves robustness by reevaluating a parametrized preconditioner whenever poor convergence or instability is encountered. We present numerical experiments that demonstrate the efficiency of several members of this new family in comparison with other known methods, in the context of PARASPAR, and in the context of inner/outer iteration schemes. The experiments show that even though methods like CGS and BiCGSTAB may converge faster for many problems, EN-like methods are in general more robust, since, like GMRES, they have the option of increasing the size of the Krylov subspace. Additionally, they often require less memory than GMRES or ORTHOMIN. They are also very suitable as iterative solvers inside PARASPAR, since they evaluate, without additional cost, parameters that estimate the quality of the preconditioner. Some EN-like methods also show drastic divergence when applied to an ill-conditioned problem. Consequently, it is possible to quickly recognize when to reevaluate the preconditioner. Finally, nonlinear EN-like methods are developed, and their convergence behavior is investigated.

Numerische Methoden der Technischen Akustik

Author: Gerhard Müller,Michael Möser

Publisher: Springer-Verlag

ISBN: 3662554097

Category: Technology & Engineering

Page: 36

View: 4813

Dieser Band der Reihe Fachwissen Technische Akustik behandelt die am weitesten verbreiteten, wellentheoretischen Verfahren der numerischen Akustik. Die Randelementemethode, die Finite-Elemente-Methode und die Ersatzstrahlermethode werden in den Kapiteln ausführlich behandelt. Weitere Methoden wie z. B. Approximationen für hohe Frequenzen, Verfahren der geometrischen Akustik oder die statistische Energieanalyse werden ebenfalls kurz angesprochen.

Computational Methods for Inverse Problems

Author: Curtis R. Vogel

Publisher: SIAM

ISBN: 0898717574

Category: Mathematics

Page: 183

View: 3628

Provides a basic understanding of both the underlying mathematics and the computational methods used to solve inverse problems.

Iterative Methods for Linear Systems

Theory and Applications

Author: Maxim A. Olshanskii,Eugene E. Tyrtshnikov

Publisher: SIAM

ISBN: 1611973457

Category: Mathematics

Page: 247

View: 5923

Iterative Methods for Linear Systems offers a mathematically rigorous introduction to fundamental iterative methods for systems of linear algebraic equations. The book distinguishes itself from other texts on the topic by providing a straightforward yet comprehensive analysis of the Krylov subspace methods, approaching the development and analysis of algorithms from various algorithmic and mathematical perspectives, and going beyond the standard description of iterative methods by connecting them in a natural way to the idea of preconditioning.

Matrix Computations

Author: Gene H. Golub,Charles F. Van Loan

Publisher: JHU Press

ISBN: 1421407949

Category: Mathematics

Page: 756

View: 2642

The fourth edition of Gene H. Golub and Charles F. Van Loan's classic is an essential reference for computational scientists and engineers in addition to researchers in the numerical linear algebra community. Anyone whose work requires the solution to a matrix problem and an appreciation of its mathematical properties will find this book to be an indispensible tool. This revision is a cover-to-cover expansion and renovation of the third edition. It now includes an introduction to tensor computations and brand new sections on • fast transforms• parallel LU• discrete Poisson solvers• pseudospectra• structured linear equation problems• structured eigenvalue problems• large-scale SVD methods• polynomial eigenvalue problems Matrix Computations is packed with challenging problems, insightful derivations, and pointers to the literature—everything needed to become a matrix-savvy developer of numerical methods and software.

Numerical Methods for Elliptic and Parabolic Partial Differential Equations

Author: Peter Knabner,Lutz Angerman

Publisher: Springer Science & Business Media

ISBN: 038795449X

Category: Mathematics

Page: 426

View: 3305

This text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises.

Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations

Author: David E. Keyes

Publisher: SIAM

ISBN: 9780898712889

Category: Mathematics

Page: 623

View: 3075

Papers presented at the May 1991 symposium reflect continuing interest in the role of domain decomposition in the effective utilization of parallel systems; applications in fluid mechanics, structures, biology, and design optimization; and maturation of analysis of elliptic equations, with theoretic

Inherently Parallel Algorithms in Feasibility and Optimization and their Applications

Author: D. Butnariu,S. Reich,Y. Censor

Publisher: Elsevier

ISBN: 9780080508764

Category: Mathematics

Page: 516

View: 6398

The Haifa 2000 Workshop on "Inherently Parallel Algorithms for Feasibility and Optimization and their Applications" brought together top scientists in this area. The objective of the Workshop was to discuss, analyze and compare the latest developments in this fast growing field of applied mathematics and to identify topics of research which are of special interest for industrial applications and for further theoretical study. Inherently parallel algorithms, that is, computational methods which are, by their mathematical nature, parallel, have been studied in various contexts for more than fifty years. However, it was only during the last decade that they have mostly proved their practical usefulness because new generations of computers made their implementation possible in order to solve complex feasibility and optimization problems involving huge amounts of data via parallel processing. These led to an accumulation of computational experience and theoretical information and opened new and challenging questions concerning the behavior of inherently parallel algorithms for feasibility and optimization, their convergence in new environments and in circumstances in which they were not considered before their stability and reliability. Several research groups all over the world focused on these questions and it was the general feeling among scientists involved in this effort that the time has come to survey the latest progress and convey a perspective for further development and concerted scientific investigations. Thus, the editors of this volume, with the support of the Israeli Academy for Sciences and Humanities, took the initiative of organizing a Workshop intended to bring together the leading scientists in the field. The current volume is the Proceedings of the Workshop representing the discussions, debates and communications that took place. Having all that information collected in a single book will provide mathematicians and engineers interested in the theoretical and practical aspects of the inherently parallel algorithms for feasibility and optimization with a tool for determining when, where and which algorithms in this class are fit for solving specific problems, how reliable they are, how they behave and how efficient they were in previous applications. Such a tool will allow software creators to choose ways of better implementing these methods by learning from existing experience.

Parallel Processing for Scientific Computing

Author: Michael A. Heroux,Padma Raghavan,Horst D. Simon

Publisher: SIAM

ISBN: 9780898718133

Category: Parallel processing (Electronic computers)

Page: 397

View: 8747

Parallel processing has been an enabling technology in scientific computing for more than 20 years. This book is the first in-depth discussion of parallel computing in 10 years; it reflects the mix of topics that mathematicians, computer scientists, and computational scientists focus on to make parallel processing effective for scientific problems. Presently, the impact of parallel processing on scientific computing varies greatly across disciplines, but it plays a vital role in most problem domains and is absolutely essential in many of them. Parallel Processing for Scientific Computing is divided into four parts: The first concerns performance modeling, analysis, and optimization; the second focuses on parallel algorithms and software for an array of problems common to many modeling and simulation applications; the third emphasizes tools and environments that can ease and enhance the process of application development; and the fourth provides a sampling of applications that require parallel computing for scaling to solve larger and realistic models that can advance science and engineering.

Computational Partial Differential Equations

Numerical Methods and Diffpack Programming

Author: Hans Petter Langtangen

Publisher: Springer Science & Business Media

ISBN: 3662011700

Category: Mathematics

Page: 685

View: 8112

Targeted at students and researchers in computational sciences who need to develop computer codes for solving PDEs, the exposition here is focused on numerics and software related to mathematical models in solid and fluid mechanics. The book teaches finite element methods, and basic finite difference methods from a computational point of view, with the main emphasis on developing flexible computer programs, using the numerical library Diffpack. Diffpack is explained in detail for problems including model equations in applied mathematics, heat transfer, elasticity, and viscous fluid flow. All the program examples, as well as Diffpack for use with this book, are available on the Internet. XXXXXXX NEUER TEXT This book is for researchers who need to develop computer code for solving PDEs. Numerical methods and the application of Diffpack are explained in detail. Diffpack is a modern C++ development environment that is widely used by industrial scientists and engineers working in areas such as oil exploration, groundwater modeling, and materials testing. All the program examples, as well as a test version of Diffpack, are available for free over the Internet.

Verification and Validation in Scientific Computing

Author: William L. Oberkampf,Christopher J. Roy

Publisher: Cambridge University Press

ISBN: 1139491768

Category: Computers

Page: N.A

View: 5617

Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.