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

Häftad, Engelska, 2014

Av Josef Málek, Zdeněk Strakoš, Zdeněk Strakos, Zden& Strakos

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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.

Produktinformation

Utgivningsdatum2014-12-30
Mått170 x 251 x 7 mm
Vikt223 g
FormatHäftad
SpråkEngelska
SerieSIAM Spotlights
Antal sidor113
FörlagSociety for Industrial & Applied Mathematics,U.S.
ISBN9781611973839

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Josef Málek is a Professor at the Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. He is a director of the Nečas 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.

Chapter 1: IntroductionChapter 2: Linear elliptic partial differential equationsChapter 3: Elements of functional analysisChapter 4: Riesz map and operator preconditioningChapter 5: Conjugate gradient method in Hilbert spacesChapter 6: Finite-dimensional Hilbert spaces and the matrix formulation of the conjugate gradient methodChapter 7: Comments on the Galerkin discretizationChapter 8: Preconditioning of the algebraic system as transformation of the discretization basisChapter 9: Fundamental theorem on discretizationChapter 10: Local and global information in discretization and in computationChapter 11: Limits of the condition number-based descriptionsChapter 12: Inexact computations, a posteriori error analysis and stopping criteriaChapter 13: Summary and outlook