Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs
Published on 30. December 2014
Book
Paperback/Softback
113 pages
978-1-61197-383-9 (ISBN)
Description
This first title in SIAM's Spotlights book series 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.
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.
More details
Series
Language
English
Place of publication
New York
United States
Target group
College/higher education
Product notice
Paperback (trade)
Unsewn / adhesive bound
Dimensions
Height: 254 mm
Width: 177 mm
Thickness: 15 mm
Weight
222 gr
ISBN-13
978-1-61197-383-9 (9781611973839)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Persons
Josef Malek is a Professor at the Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. He is a director of the Necas Center for Mathematical Modeling and head of the Department of Mathematical Modeling, comprising researchers with diverse backgrounds focused upon the graduate study program 'Mathematical modeling in science and technology'. His research primarily concerns mathematical analysis of nonlinear PDEs stemming from non-Newtonian fluid mechanics, and he has contributed to the constitutive theory in the fluid and solid mechanics and in the theory of mixtures. His research approach emphasizes the need for interactions between modeling, analysis, scientific computing, and experiments. Zdenek Strakos is a Professor at the Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. He has been an active member of several professional committees and journal editorial boards. He served on the ERC Advanced Grants Evaluation Panel for Computer Science and Informatics and was named its Chair in 2014. He was awarded the SIAM Activity Group on Linear Algebra Prize (1994), the Annual Prize of the Academy of Sciences of the Czech Republic (2007), and the Bernard Bolzano Medal of the Academy of Sciences of the Czech Republic for Merits in Mathematical Sciences (2013). In 2014 he was selected as a SIAM Fellow for advances in numerical linear algebra, especially iterative methods. He is interested in looking for interconnections between problems and disciplines, and in viewing particular questions in a wide context.
Content
Chapter 1: Introduction
Chapter 2: Linear elliptic partial differential equations
Chapter 3: Elements of functional analysis
Chapter 4: Riesz map and operator preconditioning
Chapter 5: Conjugate gradient method in Hilbert spaces
Chapter 6: Finite-dimensional Hilbert spaces and the matrix formulation of the conjugate gradient method
Chapter 7: Comments on the Galerkin discretization
Chapter 8: Preconditioning of the algebraic system as transformation of the discretization basis
Chapter 9: Fundamental theorem on discretization
Chapter 10: Local and global information in discretization and in computation
Chapter 11: Limits of the condition number-based descriptions
Chapter 12: Inexact computations, a posteriori error analysis and stopping criteria
Chapter 13: Summary and outlook
Chapter 2: Linear elliptic partial differential equations
Chapter 3: Elements of functional analysis
Chapter 4: Riesz map and operator preconditioning
Chapter 5: Conjugate gradient method in Hilbert spaces
Chapter 6: Finite-dimensional Hilbert spaces and the matrix formulation of the conjugate gradient method
Chapter 7: Comments on the Galerkin discretization
Chapter 8: Preconditioning of the algebraic system as transformation of the discretization basis
Chapter 9: Fundamental theorem on discretization
Chapter 10: Local and global information in discretization and in computation
Chapter 11: Limits of the condition number-based descriptions
Chapter 12: Inexact computations, a posteriori error analysis and stopping criteria
Chapter 13: Summary and outlook