
Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complexgeometry 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 highorder discretization procedures on unstructured meshes, which are also recognized as more efficient for solution of timedependent oscillatory solutions over long time periods. This book, an updated edition on the original text, presents the recent and significant progress in multidomain spectral methods at both the fundamental and application level. Containing material on discontinuous Galerkin methods, nontensorial 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.

Pattern recognition prowess served our ancestors well. However, today we are confronted by a deluge of data that are far more abstract, complicated, and difficult to interpret than were annual seasons and the sounds of predators. The number of possible patterns that can be identified relative to the number that are genuinely useful has grown exponentially—which means that the chances that a discovered pattern is useful is rapidly approaching zero. Coincidental streaks, clusters, and correlations are the norm—not the exception. Our challenge is to overcome our inherited inclination to think that all patterns are meaningful.Computer algorithms can easily identify an essentially unlimited number of phantom patterns and relationships that vanish when confronted with fresh data. The paradox of big data is that the more data we ransack for patterns, the more likely it is that what we find will be worthless. Our challenge is to overcome our inherited inclination to think that all patterns are meaningful.

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace–Beltrami operators, as well as an improved version of the Weyl formula, the DuistermaatGuillemin theorem under natural assumptions on the geodesic flow. The book shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. It begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. The book avoids the use of Tauberian estimates and instead relies on supnorm estimates for eigenfunctions. It also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the DuistermaatGuillemin theorem. Turning to the related topic of quantum ergodicity, the book demonstrates that if the longterm geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1type Markov chains, QBD processes, nonskipfree queues, and treelike 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 selfcontained 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 WienerHopf 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 highlevel language with minimum effort.

Scientific rigor and critical thinking skills are indispensable in this age of big data because machine learning and artificial intelligence are often led astray by meaningless patterns. The 9 Pitfalls of Data Science is loaded with entertaining realworld examples of both successful and misguided approaches to interpreting data, both grand successes and epic failures. Anyone can learn to distinguish between good data science and nonsense. We are confident that readers will learn how to avoid being duped by data, and make better, more informed decisions. Whether they want to be effective creators, interpreters, or users of data, they need to know the nine pitfalls of data science.

This book goes further than the exploration of the general structure of pseudoreductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the automorphism schemes of these groups. The book also gives a TitsWitt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudosplit forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in this book will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.

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 wellposedness analysis for general onestep and multistep methods. The continuous extensions of RungeKutta 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 semidiscretization are described and compared with the classical approach. A list of available codes is provided.

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 uptodate 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.

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.

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 timeordering (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 ntuple of pairs {(Aj,μj)}j=1n of typically noncommuting bounded operators Aj and probability measures μj, for j = 1, …, n and n ≥ 2.

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze–Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein's contributions to harmonic analysis and related topics, this book gathers papers from internationally renowned mathematicians, many of whom have been Stein's students. The book also includes expository papers on Stein's work and its influence.

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 PDEconstrained 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 buoyancydriven flow.

Selfadaptive 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.