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. Audience: This book is intended for mathematicians, engineers, physicists, chemists, and other researchers interested in the issues discussed. 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

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