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Author : Bertus Johann Steyn
Publisher :
ISBN 13 :
Total Pages : 120 pages
Book Rating : 4.:/5 (794 download)
Download or read book A Multigrid Algorithm for the Solution of the Poisson Equation written by Bertus Johann Steyn and published by . This book was released on 1999 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Yair Shapira
Publisher : Springer Science & Business Media
ISBN 13 : 1475737262
Total Pages : 225 pages
Book Rating : 4.4/5 (757 download)
Download or read book Matrix-Based Multigrid written by Yair Shapira and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many important problems in applied science and engineering, such as the Navier Stokes equations in fluid dynamics, the primitive equations in global climate mod eling, the strain-stress equations in mechanics, the neutron diffusion equations in nuclear engineering, and MRIICT medical simulations, involve complicated sys tems of nonlinear partial differential equations. When discretized, such problems produce extremely large, nonlinear systems of equations, whose numerical solution is prohibitively costly in terms of time and storage. High-performance (parallel) computers and efficient (parallelizable) algorithms are clearly necessary. Three classical approaches to the solution of such systems are: Newton's method, Preconditioned Conjugate Gradients (and related Krylov-space acceleration tech niques), and multigrid methods. The first two approaches require the solution of large sparse linear systems at every iteration, which are themselves often solved by multigrid methods. Developing robust and efficient multigrid algorithms is thus of great importance. The original multigrid algorithm was developed for the Poisson equation in a square, discretized by finite differences on a uniform grid. For this model problem, multigrid exhibits extremely rapid convergence, and actually solves the problem in the minimal possible time. The original algorithm uses rediscretization of the partial differential equation (POE) on each grid in the hierarchy of coarse grids that are used. However, this approach would not work for more complicated problems, such as problems on complicated domains and nonuniform grids, problems with variable coefficients, and non symmetric and indefinite equations. In these cases, matrix-based multi grid methods are in order.
Author : Taras Gerya
Publisher : Cambridge University Press
ISBN 13 : 0521887542
Total Pages : 359 pages
Book Rating : 4.5/5 (218 download)
Download or read book Introduction to Numerical Geodynamic Modelling written by Taras Gerya and published by Cambridge University Press. This book was released on 2010 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: This user-friendly reference for students and researchers presents the basic mathematical theory, before introducing modelling of key geodynamic processes.
Author : Stephen F. Mccormick
Publisher : CRC Press
ISBN 13 : 1000147223
Total Pages : 668 pages
Book Rating : 4.0/5 (1 download)
Download or read book multigrid methods written by Stephen F. Mccormick and published by CRC Press. This book was released on 2020-08-12 with total page 668 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of research papers on a wide variety of multigrid topics, including applications, computation and theory. It represents proceedings of the Third Copper Mountain Conference on Multigrid Methods, which was held at Copper Mountain, Colorado.
Author : W. Hackbusch
Publisher : Springer
ISBN 13 : 354039544X
Total Pages : 664 pages
Book Rating : 4.5/5 (43 download)
Download or read book Multigrid Methods written by W. Hackbusch and published by Springer. This book was released on 2006-11-15 with total page 664 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Ulrich Trottenberg
Publisher : Academic Press
ISBN 13 : 9780127010700
Total Pages : 652 pages
Book Rating : 4.0/5 (17 download)
Download or read book Multigrid Methods written by Ulrich Trottenberg and published by Academic Press. This book was released on 2001 with total page 652 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Author : William L. Briggs
Publisher : SIAM
ISBN 13 : 9780898714623
Total Pages : 318 pages
Book Rating : 4.7/5 (146 download)
Download or read book A Multigrid Tutorial written by William L. Briggs and published by SIAM. This book was released on 2000-07-01 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Author : Marián Vajtersic
Publisher : Springer Science & Business Media
ISBN 13 : 9401707014
Total Pages : 310 pages
Book Rating : 4.4/5 (17 download)
Download or read book Algorithms for Elliptic Problems written by Marián Vajtersic and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume deals with problems of modern effective algorithms for the numerical solution of the most frequently occurring elliptic partial differential equations. From the point of view of implementation, attention is paid to algorithms for both classical sequential and parallel computer systems. The first two chapters are devoted to fast algorithms for solving the Poisson and biharmonic equation. In the third chapter, parallel algorithms for model parallel computer systems of the SIMD and MIMD types are described. The implementation aspects of parallel algorithms for solving model elliptic boundary value problems are outlined for systems with matrix, pipeline and multiprocessor parallel computer architectures. A modern and popular multigrid computational principle which offers a good opportunity for a parallel realization is described in the next chapter. More parallel variants based in this idea are presented, whereby methods and assignments strategies for hypercube systems are treated in more detail. The last chapter presents VLSI designs for solving special tridiagonal linear systems of equations arising from finite-difference approximations of elliptic problems. For researchers interested in the development and application of fast algorithms for solving elliptic partial differential equations using advanced computer systems.
Author : William L. Briggs
Publisher : SIAM
ISBN 13 : 0898714621
Total Pages : 196 pages
Book Rating : 4.8/5 (987 download)
Download or read book A Multigrid Tutorial written by William L. Briggs and published by SIAM. This book was released on 2000-07-01 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Author : David L. Kamowitz
Publisher :
ISBN 13 :
Total Pages : 352 pages
Book Rating : 4.:/5 (89 download)
Download or read book Theoretical and Experimental Results for a Variety of Multigrid Algorithms written by David L. Kamowitz and published by . This book was released on 1986 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : James Lottes
Publisher : Springer
ISBN 13 : 3319563068
Total Pages : 138 pages
Book Rating : 4.3/5 (195 download)
Download or read book Towards Robust Algebraic Multigrid Methods for Nonsymmetric Problems written by James Lottes and published by Springer. This book was released on 2017-03-24 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis presents a rigorous, abstract analysis of multigrid methods for positive nonsymmetric problems, particularly suited to algebraic multigrid, with a completely new approach to nonsymmetry which is based on a new concept of absolute value for nonsymmetric operators. Multigrid, and in particular algebraic multigrid, has become an indispensable tool for the solution of discretizations of partial differential equations. While used in both the symmetric and nonsymmetric cases, the theory for the nonsymmetric case has lagged substantially behind that for the symmetric case. This thesis closes some of this gap, presenting a major and highly original contribution to an important problem of computational science. The new approach to nonsymmetry will be of interest to anyone working on the analysis of discretizations of nonsymmetric operators, even outside the context of multigrid. The presentation of the convergence theory may interest even those only concerned with the symmetric case, as it sheds some new light on and extends existing results.
Author : Wolfgang Joppich
Publisher : Springer Science & Business Media
ISBN 13 : 3709192536
Total Pages : 327 pages
Book Rating : 4.7/5 (91 download)
Download or read book Multigrid Methods for Process Simulation written by Wolfgang Joppich and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 327 pages. Available in PDF, EPUB and Kindle. Book excerpt: It was about 1985 when both of the authors started their work using multigrid methods for process simulation problems. This happened in dependent from each other, with a completely different background and different intentions in mind. At this time, some important monographs appeared or have been in preparation. There are the three "classical" ones, from our point of view: the so-called "1984 Guide" [12J by Brandt, the "Multi-Grid Methods and Applications" [49J by Hackbusch and the so-called "Fundamentals" [132J by Stiiben and Trottenberg. Stiiben and Trottenberg in [132J state a "delayed acceptance, resent ments" with respect to multigrid algorithms. They complain: "Nevertheless, even today's situation is still unsatisfactory in several respects. If this is true for the development of standard methods, it applies all the more to the area of really difficult, complex applications." In spite of all the above mentioned publications and without ignoring important theoretical and practical improvements of multigrid, this situa tion has not yet changed dramatically. This statement is made under the condition that a numerical principle like multigrid is "accepted", if there exist "professional" programs for research and production purposes. "Professional" in this context stands for "solving complex technical prob lems in an industrial environment by a large community of users". Such a use demands not only for fast solution methods but also requires a high robustness with respect to the physical parameters of the problem.
Author : Michael Holst
Publisher :
ISBN 13 :
Total Pages : 26 pages
Book Rating : 4.:/5 (31 download)
Download or read book Multigrid Solution of the Poisson-Boltzmann Equation written by Michael Holst and published by . This book was released on 1992 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "A multigrid method is presented for the numerical solution of the linearized Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the finite volume method, and the numerical solution of the discrete equations is accomplished with multiple grid techniques originally developed for two-dimensional interface problems occurring in reactor physics. A detailed analysis of the resulting method is presented for several computer architectures, including comparisons to diagonally scaled CG, ICCG, vectorized ICCG and MICCG, and to SOR provided with an optimal relaxation parameter. Our results indicate that the multigrid method is superior to the preconditioned CG methods and SOR, and that the advantage of multigrid grows with the problem size."
Author : Gunes Senel
Publisher :
ISBN 13 :
Total Pages : 79 pages
Book Rating : 4.:/5 (13 download)
Download or read book Multigrid Method written by Gunes Senel and published by . This book was released on 2017 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt: The multigrid method is an algorithm aiming to provide a numerical solution to a system of equations. In this thesis we focus on its original purpose: solving differential equations. By discretizing a differential equation, one obtains a system of linear equations, which can be solved using multigrid methods. We will start our discussion in one dimension, considering ordinary differential equations. First, we will review standard pointwise relaxation methods, since they are a fundamental part of any multigrid method. We will compare them in different settings and see their shortcomings. This part also provides an insight of the need for a better method, and one of the better methods is the multigrid method. Then we will discuss elements of multigrid method in one dimension. We will apply the method on different problems, compare and discuss the results, providing proofs and calculations for easy but well-known phenomena, such as the reduction rate of the error and the exact outcome of the method on the homogeneous Poisson equation when damped Jacobi relaxation is chosen to be used in the multigrid method, and proof of the fact that the error for the same equation can be totally eliminated in one single iteration using the multigrid method with the choice of red-black Gauss-Seidel relaxation. We will consider a nontrivial ordinary differential equation in order to be convinced that the problems that the multigrid method can deal with and the conclusions we have arrived can be generalized. Afterwards we discuss the multigrid method in two dimensions, where it is mostly used in real life applications. As we did in one dimension, we shortly review the relaxation methods in two dimensions. Then we introduce the elements of the multigrid method in two dimensions. We apply the multigrid method to some problems and tabulate the results, leaving the redundant cases out. We discuss the results and compare them with the one dimensional results.
Author : Wolfgang Hackbusch
Publisher : Springer Science & Business Media
ISBN 13 : 3662024276
Total Pages : 391 pages
Book Rating : 4.6/5 (62 download)
Download or read book Multi-Grid Methods and Applications written by Wolfgang Hackbusch and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multi-grid methods are the most efficient tools for solving elliptic boundary value problems. The reader finds here an elementary introduction to multi-grid algorithms as well as a comprehensive convergence analysis. One section describes special applications (convection-diffusion equations, singular perturbation problems, eigenvalue problems, etc.). The book also contains a complete presentation of the multi-grid method of the second kind, which has important applications to integral equations (e.g. the "panel method") and to numerous other problems. Readers with a practical interest in multi-grid methods will benefit from this book as well as readers with a more theoretical interest.
Author : Klaus STUEBEN
Publisher :
ISBN 13 :
Total Pages : 176 pages
Book Rating : 4.:/5 (439 download)
Download or read book Multigrid Methods written by Klaus STUEBEN and published by . This book was released on 1984 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Wolfgang Hackbusch
Publisher : Springer
ISBN 13 : 3540473726
Total Pages : 342 pages
Book Rating : 4.5/5 (44 download)
Download or read book Multigrid Methods II written by Wolfgang Hackbusch and published by Springer. This book was released on 2006-11-14 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt:
