I. S. Duff, A. M. Erisman, and J. K. Reid
- Published in print:
- 2017
- Published Online:
- April 2017
- ISBN:
- 9780198508380
- eISBN:
- 9780191746420
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508380.001.0001
- Subject:
- Mathematics, Numerical Analysis
Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and ...
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Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer architectures are now much more complex, requiring new ways of adapting algorithms to parallel environments with memory hierarchies. Because the area is such an important one to all of computational science and engineering, a huge amount of research has been done since the first edition, some of it by the authors. This new research is integrated into the text with a clear explanation of the underlying mathematics and algorithms. New research that is described includes new techniques for scaling and error control, new orderings, new combinatorial techniques for partitioning both symmetric and unsymmetric problems, and a detailed description of the multifrontal approach to solving systems that was pioneered by the research of the authors and colleagues. This includes a discussion of techniques for exploiting parallel architectures and new work for indefinite and unsymmetric systems.Less
Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer architectures are now much more complex, requiring new ways of adapting algorithms to parallel environments with memory hierarchies. Because the area is such an important one to all of computational science and engineering, a huge amount of research has been done since the first edition, some of it by the authors. This new research is integrated into the text with a clear explanation of the underlying mathematics and algorithms. New research that is described includes new techniques for scaling and error control, new orderings, new combinatorial techniques for partitioning both symmetric and unsymmetric problems, and a detailed description of the multifrontal approach to solving systems that was pioneered by the research of the authors and colleagues. This includes a discussion of techniques for exploiting parallel architectures and new work for indefinite and unsymmetric systems.
Gerald W Johnson, Michel L. Lapidus, and Lance Nielsen
- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198702498
- eISBN:
- 9780191772160
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198702498.001.0001
- Subject:
- Mathematics, Mathematical Physics, Numerical Analysis
This book provides an abstract theory of Feynman’s operational calculus for functions of (typically) noncommuting operators. Although it is inspired by Feynman’s original heuristic suggestions and ...
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This book provides an abstract theory of Feynman’s operational calculus for functions of (typically) noncommuting operators. Although it is inspired by Feynman’s original heuristic suggestions and time-ordering (or disentangling) rules in his seminal 1951 paper, as is made clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman’s work. In particular, the work presented in this volume is oriented towards dealing with abstract and (typically) noncommuting linear operators acting on some Banach space, rather than operators arising from some variety of path integration. Some of the key structures developed in this volume enable us to obtain, in some sense, an appropriate abstract substitute for a generalized functional integral associated with the Feynman operational calculus attached to a given n-tuple of pairs {(Aj,μj)}j=1n of typically noncommuting bounded operators Aj and probability measures μj, for j = 1, …, n and n ≥ 2.Less
This book provides an abstract theory of Feynman’s operational calculus for functions of (typically) noncommuting operators. Although it is inspired by Feynman’s original heuristic suggestions and time-ordering (or disentangling) rules in his seminal 1951 paper, as is made clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman’s work. In particular, the work presented in this volume is oriented towards dealing with abstract and (typically) noncommuting linear operators acting on some Banach space, rather than operators arising from some variety of path integration. Some of the key structures developed in this volume enable us to obtain, in some sense, an appropriate abstract substitute for a generalized functional integral associated with the Feynman operational calculus attached to a given n-tuple of pairs {(Aj,μj)}j=1n of typically noncommuting bounded operators Aj and probability measures μj, for j = 1, …, n and n ≥ 2.
Peter Monk
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.001.0001
- Subject:
- Mathematics, Numerical Analysis
Since the middle of the last century, computing power has increased sufficiently that the direct numerical approximation of Maxwell’s equations is now an increasingly important tool in science and ...
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Since the middle of the last century, computing power has increased sufficiently that the direct numerical approximation of Maxwell’s equations is now an increasingly important tool in science and engineering. Parallel to the increasing use of numerical methods in computational electromagnetism, there has also been considerable progress in the mathematical understanding of the properties of Maxwell’s equations relevant to numerical analysis. The aim of this book is to provide an up-to-date and sound theoretical foundation for finite element methods in computational electromagnetism. The emphasis is on finite element methods for scattering problems that involve the solution of Maxwell’s equations on infinite domains. Suitable variational formulations are developed and justified mathematically. An error analysis of edge finite element methods that are particularly well suited to Maxwell’s equations is the main focus of the book. The analysis involves a complete justification of the discrete de Rham diagram and discrete compactness of edge elements. The numerical methods are justified for Lipschitz polyhedral domains that can cause strong singularities in the solution. The book ends with a short introduction to inverse problems in electromagnetism.Less
Since the middle of the last century, computing power has increased sufficiently that the direct numerical approximation of Maxwell’s equations is now an increasingly important tool in science and engineering. Parallel to the increasing use of numerical methods in computational electromagnetism, there has also been considerable progress in the mathematical understanding of the properties of Maxwell’s equations relevant to numerical analysis. The aim of this book is to provide an up-to-date and sound theoretical foundation for finite element methods in computational electromagnetism. The emphasis is on finite element methods for scattering problems that involve the solution of Maxwell’s equations on infinite domains. Suitable variational formulations are developed and justified mathematically. An error analysis of edge finite element methods that are particularly well suited to Maxwell’s equations is the main focus of the book. The analysis involves a complete justification of the discrete de Rham diagram and discrete compactness of edge elements. The numerical methods are justified for Lipschitz polyhedral domains that can cause strong singularities in the solution. The book ends with a short introduction to inverse problems in electromagnetism.
Howard Elman, David Silvester, and Andy Wathen
- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780199678792
- eISBN:
- 9780191780745
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199678792.001.0001
- Subject:
- Mathematics, Numerical Analysis, Computational Mathematics / Optimization
The subject of this book is the efficient solution of partial differential equations (PDEs) that arise when modelling incompressible fluid flow. The first part (Chapters 1 through 5) covers the ...
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The subject of this book is the efficient solution of partial differential equations (PDEs) that arise when modelling incompressible fluid flow. The first part (Chapters 1 through 5) covers the Poisson equation and the Stokes equations. For each PDE, there is a chapter concerned with finite element discretization and a companion chapter concerned with efficient iterative solution of the algebraic equations obtained from discretization. Chapter 5 describes the basics of PDE-constrained optimization. The second part of the book (Chapters 6 to 11) is a more advanced introduction to the numerical analysis of incompressible flows. It starts with four chapters on the convection–diffusion equation and the steady Navier–Stokes equations, organized by equation with a chapter describing discretization coupled with a companion concerned with iterative solution algorithms. The book concludes with two chapters describing discretization and solution methods for models of unsteady flow and buoyancy-driven flow.Less
The subject of this book is the efficient solution of partial differential equations (PDEs) that arise when modelling incompressible fluid flow. The first part (Chapters 1 through 5) covers the Poisson equation and the Stokes equations. For each PDE, there is a chapter concerned with finite element discretization and a companion chapter concerned with efficient iterative solution of the algebraic equations obtained from discretization. Chapter 5 describes the basics of PDE-constrained optimization. The second part of the book (Chapters 6 to 11) is a more advanced introduction to the numerical analysis of incompressible flows. It starts with four chapters on the convection–diffusion equation and the steady Navier–Stokes equations, organized by equation with a chapter describing discretization coupled with a companion concerned with iterative solution algorithms. The book concludes with two chapters describing discretization and solution methods for models of unsteady flow and buoyancy-driven flow.
Alfredo Bellen and Marino Zennaro
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506546
- eISBN:
- 9780191709609
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506546.001.0001
- Subject:
- Mathematics, Numerical Analysis
The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary ...
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The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary differential equations (ODEs) are made using examples illustrating some unexpected and often surprising behaviours of the true and numerical solutions. The book briefly reviews the various approaches existing in the literature and develops an error and well-posedness analysis for general one-step and multistep methods. The continuous extensions of Runge-Kutta methods are presented in detail, which are useful for more general problems such as dense output and discontinuous equations. Some deeper insight into convergence and superconvergence is then carried out for DDEs with various kinds of delays. The stepsize control mechanism is developed on a firm mathematical basis. Classical results and an unconventional analysis of stability with respect to forcing term are reviewed for ODEs in view of the subsequent stability analysis for DDEs. Moreover, an exhaustive description of stability domains for some test DDEs is carried out and the corresponding investigations for the numerical methods are made. Reformulations of DDEs as partial differential equations and subsequent semi-discretization are described and compared with the classical approach. A list of available codes is provided.Less
The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary differential equations (ODEs) are made using examples illustrating some unexpected and often surprising behaviours of the true and numerical solutions. The book briefly reviews the various approaches existing in the literature and develops an error and well-posedness analysis for general one-step and multistep methods. The continuous extensions of Runge-Kutta methods are presented in detail, which are useful for more general problems such as dense output and discontinuous equations. Some deeper insight into convergence and superconvergence is then carried out for DDEs with various kinds of delays. The stepsize control mechanism is developed on a firm mathematical basis. Classical results and an unconventional analysis of stability with respect to forcing term are reviewed for ODEs in view of the subsequent stability analysis for DDEs. Moreover, an exhaustive description of stability domains for some test DDEs is carried out and the corresponding investigations for the numerical methods are made. Reformulations of DDEs as partial differential equations and subsequent semi-discretization are described and compared with the classical approach. A list of available codes is provided.
Dario A. Bini, Guy Latouche, and Beatrice Meini
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.001.0001
- Subject:
- Mathematics, Numerical Analysis
The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1-type Markov chains, QBD processes, non-skip-free queues, and tree-like stochastic processes and ...
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The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1-type Markov chains, QBD processes, non-skip-free queues, and tree-like stochastic processes and has a wide applicability in queueing theory and stochastic modeling. It presents in a unified language the most up to date algorithms, which are so far scattered in diverse papers, written with different languages and notation. It contains a thorough treatment of numerical algorithms to solve these problems, from the simplest to the most advanced and most efficient. Nonlinear matrix equations are at the heart of the analysis of structured Markov chains, they are analysed both from the theoretical, from the probabilistic, and from the computational point of view. The set of methods for solution contains functional iterations, doubling methods, logarithmic reduction, cyclic reduction, and subspace iteration, all are described and analysed in detail. They are also adapted to interesting specific queueing models coming from applications. The book also offers a comprehensive and self-contained treatment of the structured matrix tools which are at the basis of the fastest algorithmic techniques for structured Markov chains. Results about Toeplitz matrices, displacement operators, and Wiener-Hopf factorizations are reported to the extent that they are useful for the numerical treatment of Markov chains. Every and all solution methods are reported in detailed algorithmic form so that they can be coded in a high-level language with minimum effort.Less
The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1-type Markov chains, QBD processes, non-skip-free queues, and tree-like stochastic processes and has a wide applicability in queueing theory and stochastic modeling. It presents in a unified language the most up to date algorithms, which are so far scattered in diverse papers, written with different languages and notation. It contains a thorough treatment of numerical algorithms to solve these problems, from the simplest to the most advanced and most efficient. Nonlinear matrix equations are at the heart of the analysis of structured Markov chains, they are analysed both from the theoretical, from the probabilistic, and from the computational point of view. The set of methods for solution contains functional iterations, doubling methods, logarithmic reduction, cyclic reduction, and subspace iteration, all are described and analysed in detail. They are also adapted to interesting specific queueing models coming from applications. The book also offers a comprehensive and self-contained treatment of the structured matrix tools which are at the basis of the fastest algorithmic techniques for structured Markov chains. Results about Toeplitz matrices, displacement operators, and Wiener-Hopf factorizations are reported to the extent that they are useful for the numerical treatment of Markov chains. Every and all solution methods are reported in detailed algorithmic form so that they can be coded in a high-level language with minimum effort.
Rüdiger Verfürth
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199679423
- eISBN:
- 9780191758485
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199679423.001.0001
- Subject:
- Mathematics, Applied Mathematics, Numerical Analysis
Self-adaptive discretization methods nowadays are an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to ...
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Self-adaptive discretization methods nowadays are an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation. In this monograph we review the most frequently used a posteriori error estimation techniques and apply them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. Our main goal is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems. Chapters 1 and 2 are quite elementary and present various error indicators and their use for mesh adaptation in the framework of a simple model problem. The intention here is to present the basic principles using a minimal amount of notation and techniques. Chapters 4–6, on the other hand, are more advanced and present a posteriori error estimates within a general framework using the technical tools collected in Chapter 3. Most sections close with a bibliographical remark which indicates the historical development and hints at further results.Less
Self-adaptive discretization methods nowadays are an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation. In this monograph we review the most frequently used a posteriori error estimation techniques and apply them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. Our main goal is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems. Chapters 1 and 2 are quite elementary and present various error indicators and their use for mesh adaptation in the framework of a simple model problem. The intention here is to present the basic principles using a minimal amount of notation and techniques. Chapters 4–6, on the other hand, are more advanced and present a posteriori error estimates within a general framework using the technical tools collected in Chapter 3. Most sections close with a bibliographical remark which indicates the historical development and hints at further results.
George Karniadakis and Spencer Sherwin
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528692
- eISBN:
- 9780191713491
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528692.001.0001
- Subject:
- Mathematics, Numerical Analysis
Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computational domains has historically been much more ...
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Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computational domains has historically been much more limited. More recently, the need to find accurate solutions to the viscous flow equations around complex configurations has led to the development of high-order discretization procedures on unstructured meshes, which are also recognized as more efficient for solution of time-dependent oscillatory solutions over long time periods. This book, an updated edition on the original text, presents the recent and significant progress in multi-domain spectral methods at both the fundamental and application level. Containing material on discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilization and filtering techniques, this text introduces the use of spectral/hp element methods with particular emphasis on their application to unstructured meshes. It provides a detailed explanation of the key concepts underlying the methods along with practical examples of their derivation and application.Less
Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computational domains has historically been much more limited. More recently, the need to find accurate solutions to the viscous flow equations around complex configurations has led to the development of high-order discretization procedures on unstructured meshes, which are also recognized as more efficient for solution of time-dependent oscillatory solutions over long time periods. This book, an updated edition on the original text, presents the recent and significant progress in multi-domain spectral methods at both the fundamental and application level. Containing material on discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilization and filtering techniques, this text introduces the use of spectral/hp element methods with particular emphasis on their application to unstructured meshes. It provides a detailed explanation of the key concepts underlying the methods along with practical examples of their derivation and application.
