Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs
Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs
$39.00
USDThe route from an applied problem to its numerical solution involves modeling, analysis, discretization, and the solution of the discretized problem. This book concerns the interplay of these stages and the challenges that arise. The authors link analysis of PDEs, functional analysis, and calculus of variations with iterative matrix computation using Krylov subspace methods. While preconditioning of the conjugate gradient method is traditionally developed algebraically using the preconditioned finite-dimensional algebraic system, the authors develop connections between preconditioning and PDEs. Additionally, links between the infinite-dimensional formulation of the conjugate gradient method, its discretization and preconditioning are explored. The book is intended for mathematicians, engineers, physicists, chemists, and any other researchers interested in the issues discussed. Aiming to improve understanding between researchers working on different solution stages, the book challenges commonly held views, addresses widespread misunderstandings, and formulates thought-provoking open questions for further research.
- A novel approach, connecting preconditioning with PDE analysis and the infinite-dimensional formulation of the conjugate gradient method
- Challenges commonly held views, addresses widespread misunderstandings
- Formulates thought-provoking open questions for further research
Product details
October 2016Paperback
9781611973839
114 pages
155 × 175 × 7 mm
0.22kg
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Table of Contents
- Preface
- 1. Introduction
- 2. Linear elliptic partial differential equations
- 3. Elements of functional analysis
- 4. Riesz map and operator preconditioning
- 5. Conjugate gradient method in Hilbert spaces
- 6. Finite dimensional Hilbert spaces and the matrix formulation of the conjugate gradient method
- 7. Comments on the Galerkin discretization
- 8. Preconditioning of the algebraic system as transformation of the discretization basis
- 9. Fundamental theorem on discretization
- 10. Local and global information in discretization and in computation
- 11. Limits of the condition number-based descriptions
- 12. Inexact computations, a posteriori error analysis, and stopping criteria
- 13. Summary and outlook
- Bibliography
- Index.
