Preconditioning And The Conjugate Gradient Method In The Context Of Solving Pdes
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Preconditioning And The Conjugate Gradient Method In The Context Of Solving Pdes
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Author : Josef Malek
language : en
Publisher: SIAM
Release Date : 2014-12-22
Preconditioning And The Conjugate Gradient Method In The Context Of Solving Pdes written by Josef Malek and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-22 with Mathematics categories.
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.?
Preconditioning And The Conjugate Gradient Method In The Context Of Solving Pdes
DOWNLOAD
Author : Josef Malek
language : en
Publisher: SIAM
Release Date : 2014-12-22
Preconditioning And The Conjugate Gradient Method In The Context Of Solving Pdes written by Josef Malek and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-22 with Mathematics categories.
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
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Author : Gérard Meurant
language : en
Publisher: SIAM
Release Date : 2024-01-30
Error Norm Estimation In The Conjugate Gradient Algorithm written by Gérard Meurant and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-01-30 with Mathematics categories.
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.
Saddle Point Problems And Their Iterative Solution
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Author : Miroslav Rozložník
language : en
Publisher: Springer
Release Date : 2018-11-19
Saddle Point Problems And Their Iterative Solution written by Miroslav Rozložník and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-19 with Mathematics categories.
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.
Spectral And High Order Methods For Partial Differential Equations Icosahom 2020 1
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Author : Jens M. Melenk
language : en
Publisher: Springer Nature
Release Date : 2023-06-30
Spectral And High Order Methods For Partial Differential Equations Icosahom 2020 1 written by Jens M. Melenk and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-06-30 with Mathematics categories.
The volume features high-quality papers based on the presentations at the ICOSAHOM 2020+1 on spectral and high order methods. The carefully reviewed articles cover state of the art topics in high order discretizations of partial differential equations. The volume presents a wide range of topics including the design and analysis of high order methods, the development of fast solvers on modern computer architecture, and the application of these methods in fluid and structural mechanics computations.
Advanced Numerical Methods In Applied Sciences
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Author : Luigi Brugnano
language : en
Publisher: MDPI
Release Date : 2019-06-20
Advanced Numerical Methods In Applied Sciences written by Luigi Brugnano and has been published by MDPI this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-06-20 with Juvenile Nonfiction categories.
The use of scientific computing tools is currently customary for solving problems at several complexity levels in Applied Sciences. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and better performing numerical methods that are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.
Pole Swapping Algorithms For The Eigenvalue Problem
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Author : Daan Camps
language : en
Publisher: SIAM
Release Date : 2025-05-01
Pole Swapping Algorithms For The Eigenvalue Problem written by Daan Camps and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-05-01 with Mathematics categories.
Matrix eigenvalue problems arise in a wide variety of fields in science and engineering, so it is important to have reliable and efficient methods for solving them. Of the methods devised, bulge-chasing algorithms, such as the famous QR and QZ algorithms, are the most important. This book focuses on pole-swapping algorithms, a new class of methods that are generalizations of bulge-chasing algorithms and a bit faster and more accurate owing to their inherent flexibility. The pole-swapping theory developed by the authors sheds light on the functioning of the whole class of algorithms, including QR and QZ. Pole-Swapping Algorithms for the Eigenvalue Problem is the only book on the topic. It describes the state of the art on eigenvalue methods and provides an improved understanding and explanation of why these important algorithms work. This book is for researchers and students in the field of matrix computations, software developers, and anyone in academia or industry who needs to understand how to solve eigenvalue problems, which are ubiquitous in science and engineering.
Iterative Solution Of Symmetric Quasi Definite Linear Systems
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Author : Dominique Orban
language : en
Publisher: SIAM
Release Date : 2017-04-07
Iterative Solution Of Symmetric Quasi Definite Linear Systems written by Dominique Orban and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-04-07 with Mathematics categories.
Numerous applications, including computational optimization and fluid dynamics, give rise to block linear systems of equations said to have the quasi-definite structure. In practical situations, the size or density of those systems can preclude a factorization approach, leaving only iterative methods as the solution technique. Known iterative methods, however, are not specifically designed to take advantage of the quasi-definite structure. This book discusses the connection between quasi-definite systems and linear least-squares problems, the most common and best understood problems in applied mathematics, and explains how quasi-definite systems can be solved using tailored iterative methods for linear least squares (with half as much work!). To encourage researchers and students to use the software, it is provided in MATLAB, Python, and Julia. The authors provide a concise account of the most well-known methods for symmetric systems and least-squares problems, research-level advances in the solution of problems with specific illustrations in optimization and fluid dynamics, and a website that hosts software in three languages.
Inside Finite Elements
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Author : Martin Weiser
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2016-05-10
Inside Finite Elements written by Martin Weiser and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-10 with Mathematics categories.
All relevant implementation aspects of finite element methods are discussed in this book. The focus is on algorithms and data structures as well as on their concrete implementation. Theory is covered only as far as it gives insight into the construction of algorithms. In the exercises, a complete FE-solver for stationary 2D problems is implemented in Matlab/Octave. Contents: Finite Element Fundamentals Grids and Finite Elements Assembly Solvers Error Estimation Mesh Refinement Multigrid Elastomechanics Fluid Mechanics Grid Data Structure Function Reference
Numerical Homogenization By Localized Decomposition
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Author : Axel Målqvist
language : en
Publisher: SIAM
Release Date : 2020-11-23
Numerical Homogenization By Localized Decomposition written by Axel Målqvist and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-11-23 with Mathematics categories.
This book presents the first survey of the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. The authors provide a careful error analysis, including previously unpublished results, and a complete implementation of the method in MATLAB. They also reveal how the LOD method relates to classical homogenization and domain decomposition. Illustrated with numerical experiments that demonstrate the significance of the method, the book is enhanced by a survey of applications including eigenvalue problems and evolution problems. Numerical Homogenization by Localized Orthogonal Decomposition is appropriate for graduate students in applied mathematics, numerical analysis, and scientific computing. Researchers in the field of computational partial differential equations will find this self-contained book of interest, as will applied scientists and engineers interested in multiscale simulation.