- ホーム
- > 洋書
- > 英文書
- > Science / Mathematics
Full 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.
Contents
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
