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Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs

2014-12-22 Mathematics
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Release : 2014-12-22
Genre : Mathematics
Kind : eBook
Book Rating : 848/5 ( reviews)

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Book Synopsis Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs by : Josef Malek

Download or read book Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs written by Josef Malek. This book was released on 2014-12-22. Available in PDF, EPUB and Kindle. Book excerpt: Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs is about the interplay between modeling, analysis, discretization, matrix computation, and model reduction. The authors link PDE analysis, functional analysis, and calculus of variations with matrix iterative computation using Krylov subspace methods and address the challenges that arise during formulation of the mathematical model through to efficient numerical solution of the algebraic problem. The book?s central concept, preconditioning of the conjugate gradient method, is traditionally developed algebraically using the preconditioned finite-dimensional algebraic system. In this text, however, preconditioning is connected to the PDE analysis, and the infinite-dimensional formulation of the conjugate gradient method and its discretization and preconditioning are linked together. This text challenges commonly held views, addresses widespread misunderstandings, and formulates thought-provoking open questions for further research.

Error Norm Estimation in the Conjugate Gradient Algorithm

2024-01-30 Mathematics
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Release : 2024-01-30
Genre : Mathematics
Kind : eBook
Book Rating : 86X/5 ( reviews)

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Book Synopsis Error Norm Estimation in the Conjugate Gradient Algorithm by : Gérard Meurant

Download or read book Error Norm Estimation in the Conjugate Gradient Algorithm written by Gérard Meurant . This book was released on 2024-01-30. Available in PDF, EPUB and Kindle. Book excerpt: The conjugate gradient (CG) algorithm is almost always the iterative method of choice for solving linear systems with symmetric positive definite matrices. This book describes and analyzes techniques based on Gauss quadrature rules to cheaply compute bounds on norms of the error. The techniques can be used to derive reliable stopping criteria. How to compute estimates of the smallest and largest eigenvalues during CG iterations is also shown. The algorithms are illustrated by many numerical experiments, and they can be easily incorporated into existing CG codes. The book is intended for those in academia and industry who use the conjugate gradient algorithm, including the many branches of science and engineering in which symmetric linear systems have to be solved.

Preconditioning Methods

1983-01-01 Mathematics
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Release : 1983-01-01
Genre : Mathematics
Kind : eBook
Book Rating : 208/5 ( reviews)

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Book Synopsis Preconditioning Methods by : David J. Evans

Download or read book Preconditioning Methods written by David J. Evans. This book was released on 1983-01-01. Available in PDF, EPUB and Kindle. Book excerpt:

Conjugate Gradient Algorithms and Finite Element Methods

2004-06-11 Computers
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Release : 2004-06-11
Genre : Computers
Kind : eBook
Book Rating : 192/5 ( reviews)

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Book Synopsis Conjugate Gradient Algorithms and Finite Element Methods by : M. Křížek

Download or read book Conjugate Gradient Algorithms and Finite Element Methods written by M. Křížek. This book was released on 2004-06-11. Available in PDF, EPUB and Kindle. Book excerpt: The position taken in this collection of pedagogically written essays is that conjugate gradient algorithms and finite element methods complement each other extremely well. Via their combinations practitioners have been able to solve complicated, direct and inverse, multidemensional problems modeled by ordinary or partial differential equations and inequalities, not necessarily linear, optimal control and optimal design being part of these problems. The aim of this book is to present both methods in the context of complicated problems modeled by linear and nonlinear partial differential equations, to provide an in-depth discussion on their implementation aspects. The authors show that conjugate gradient methods and finite element methods apply to the solution of real-life problems. They address graduate students as well as experts in scientific computing.

Saddle-Point Problems and Their Iterative Solution

2018-11-19 Mathematics
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Release : 2018-11-19
Genre : Mathematics
Kind : eBook
Book Rating : 312/5 ( reviews)

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Book Synopsis Saddle-Point Problems and Their Iterative Solution by : Miroslav Rozložník

Download or read book Saddle-Point Problems and Their Iterative Solution written by Miroslav Rozložník. This book was released on 2018-11-19. Available in PDF, EPUB and Kindle. Book excerpt: This book provides essential lecture notes on solving large linear saddle-point systems, which arise in a wide range of applications and often pose computational challenges in science and engineering. The focus is on discussing the particular properties of such linear systems, and a large selection of algebraic methods for solving them, with an emphasis on iterative methods and preconditioning. The theoretical results presented here are complemented by a case study on potential fluid flow problem in a real world-application. This book is mainly intended for students of applied mathematics and scientific computing, but also of interest for researchers and engineers working on various applications. It is assumed that the reader has completed a basic course on linear algebra and numerical mathematics.

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