Numerical analysis Books

Book SynopsisThis monograph grew out of the authors' efforts to provide a natural geometric description for the class of maps invariant under parabolic renormalization and for the Inou-Shishikura fixed point itself as well as to carry out a computer-assisted study of the parabolic renormalization operator. It introduces a renormalization-invariant class of analytic maps with a maximal domain of analyticity and rigid covering properties and presents a numerical scheme for computing parabolic renormalization of a germ, which is used to compute the Inou-Shishikura renormalization fixed point. Inside, readers will find a detailed introduction into the theory of parabolic bifurcation, Fatou coordinates, Écalle-Voronin conjugacy invariants of parabolic germs, and the definition and basic properties of parabolic renormalization. The systematic view of parabolic renormalization developed in the book and the numerical approach to its study will be interesting to both experts in the field as well as graduate students wishing to explore one of the frontiers of modern complex dynamics.Trade Review“The book under review is devoted to the study of parabolic renormalization. … The book is very well written and self-contained … and most results are stated together with their proofs.” (Jasmin Raissy, zbMATH 1342.37051, 2016)Table of Contents​1 Introduction.- 2 Local dynamics of a parabolic germ.- 3 Global theory.- 4 Numerical results.- 5 For dessert: several amusing examples.- Index.

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Book SynopsisThis book presents a comprehensive set of guidelines and applications of DIgSILENT PowerFactory, an advanced power system simulation software package, for different types of power systems studies. Written by specialists in the field, it combines expertise and years of experience in the use of DIgSILENT PowerFactory with a deep understanding of power systems analysis. These complementary approaches therefore provide a fresh perspective on how to model, simulate and analyse power systems. It presents methodological approaches for modelling of system components, including both classical and non-conventional devices used in generation, transmission and distribution systems, discussing relevant assumptions and implications on performance assessment. This background is complemented with several guidelines for advanced use of DSL and DPL languages as well as for interfacing with other software packages, which is of great value for creating and performing different types of steady-state and dynamic performance simulation analysis. All employed test case studies are provided as supporting material to the reader to ease recreation of all examples presented in the book as well as to facilitate their use in other cases related to planning and operation studies. Providing an invaluable resource for the formal instruction of power system undergraduate/postgraduate students, this book is also a useful reference for engineers working in power system operation and planning.Table of ContentsLoad Flow Calculation and its Application.- Modelling of Transmission Systems under Unsymmetrical Conditions and contingency analysis using DIgSILENT PowerFactory.- Probabilistic load flow module for PowerFactory.- Unbalanced Power Flow in Distribution Systems using TRX Matrix: Implementation using DIgSILENT Programming Language.- Primal-dual interior point algorithm applied to DC optimal power flow using DIgSILENT Programming Language.- Indices to Assess the Integration of Renewable Energy Resources on Standard Test Networks through DIgSILENT’s Programming Language.- Modelling of automatic generation control in power systems.- Gas Turbine Modelling for Power System Dynamic Simulation Studies.- Implementation of Simplified Models of DFG-Based Wind Turbines for RMS-Type Simulation in DIgSILENT Power Factory.- Parameterized modal analysis using DIgSILENT Programming Language.- Probabilistic Approach for Risk Evaluation of Power System Oscillatory Stability.- Mean-Variance Mapping Optimization Algorithm for Power System Applications in DIgSILENT Power Factory.- Application and Requirement of DIgSILENT PowerFactory to Matlab/Simulink Interface.- Advanced applications of DPL language: Simulation automation and management of results.- Interfacing PowerFactory: co-simulation, real-time simulation and controller hardware-in-the-loop applications.- PowerFactory as a software stand-in for Hardware-In-Loop Testing.- Programming of simplified models of Flexible Alternating Current Transmission System (FACTS) devices using DIgSILENT Simulation Language.- Active and Reactive Power Control of Wind Farm based on Integrated Platform of PowerFactory and Matlab.- Implementation of Simplified Models of Local Controller for Muti-terminal HVDC Systems in DIgSILENT Power Factory.- Estimation of Equivalent Model for Clusters of Induction Generators based on PMU Measurements.

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Book SynopsisThis book offers readers a primer on the theory and applications of Ordinary Differential Equations. The style used is simple, yet thorough and rigorous. Each chapter ends with a broad set of exercises that range from the routine to the more challenging and thought-provoking. Solutions to selected exercises can be found at the end of the book. The book contains many interesting examples on topics such as electric circuits, the pendulum equation, the logistic equation, the Lotka-Volterra system, the Laplace Transform, etc., which introduce students to a number of interesting aspects of the theory and applications. The work is mainly intended for students of Mathematics, Physics, Engineering, Computer Science and other areas of the natural and social sciences that use ordinary differential equations, and who have a firm grasp of Calculus and a minimal understanding of the basic concepts used in Linear Algebra. It also studies a few more advanced topics, such as Stability Theory and Boundary Value Problems, which may be suitable for more advanced undergraduate or first-year graduate students. The second edition has been revised to correct minor errata, and features a number of carefully selected new exercises, together with more detailed explanations of some of the topics.A complete Solutions Manual, containing solutions to all the exercises published in the book, is available. Instructors who wish to adopt the book may request the manual by writing directly to one of the authors.Trade Review“This is the second edition of an undergraduate introduction to ordinary differential equations suitable for mathematicians and engineers. … The style is clean and concise with many examples and exercises. Basic results are proven, more involved results are only stated. The new edition features some new exercises and better explanations at various points. So if you are looking for an application oriented introduction which is still concise and rigorous, this book might be just right for you.” (G. Teschl, Monatshefte für Mathematik, 2016)Table of Contents1 First order linear differential equations.- 2 Theory of first order differential equations.- 3 First order nonlinear differential equations.- 4 Existence and uniqueness for systems and higher order equations.- 5 Second order equations.- 6 Higher order linear equations.- 7 Systems of first order equations.- 8 Qualitative analysis of 2x2 systems and nonlinear second order equations.- 9 Sturm Liouville eigenvalue theory.- 10 Solutions by infinite series and Bessel functions.- 11 Laplace transform.- 12 Stability theory.- 13 Boundary value problems.- 14 Appendix A. Numerical methods.- 15 Answers to selected exercises.

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Book SynopsisThis textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation of the FVM numerics and algorithms used for the simulation of incompressible and compressible fluid flows, along with a detailed examination of the components needed for the development of a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The first is uFVM, a three-dimensional unstructured pressure-based finite volume academic CFD code, implemented within Matlab. The second is OpenFOAM®, an open source framework used in the development of a range of CFD programs for the simulation of industrial scale flow problems.With over 220 figures, numerous examples and more than one hundred exercise on FVM numerics, programming, and applications, this textbook is suitable for use in an introductory course on the FVM, in an advanced course on numerics, and as a reference for CFD programmers and researchers. Trade Review“Directed towards future practitioners such as engineers the authors first provide an introduction to fluid dynamics presupposing but a modicum of mathematical and physical knowledge. … . A number of exercises plus special chapters on modelling incompressible and compressible flow make the book very useful for its purpose.” (H. Muthsam, Monatshefte für Mathematik, Vol. 187 (1), September, 2018)“The book is very attractive, carefully written and easy to read by those interested in learning about finite volume methods for fluid dynamics. The authors have made an important effort to bridge the gap between classroom material and actual model development questions. The text is well illustrated by means of quality figures helping to understand the described concepts. Furthermore, the book contains pieces of academic codes in MATLAB … . It is certainly a useful, practical and valuable book.” (Pilar Garcia-Navarro, Mathematical Reviews, May, 2016)Table of ContentsFoundation1 Introduction2 Review of Vector Calculus3 Mathematical Description of Physical Phenomena4 The Discretization Process5 The Finite Volume Method6 The Finite Volume Mesh7 The Finite Volume Mesh in OpenFOAM® and uFVMDiscretization8 Spatial Discretization: The Diffusion Term9 Gradient Computation10 Solving the System of Algebraic Equations11 Discretization of the Convection Term12 High Resolution Schemes13 Temporal Discretization: The Transient Term14 Discretization of the Source Term, Relaxation, and Other DetailsAlgorithms15 Fluid Flow Computation: Incompressible Flows16 Fluid Flow Computation: Compressible FlowsApplications17 Turbulence Modeling18 Boundary Conditions in OpenFOAM® and uFVM19 An OpenFOAM® Turbulent Flow Application 20 Closing RemarksAppendices<20 Closing RemarksAppendices20 Closing RemarksAppendices

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Book SynopsisPresenting topics that have not previously been contained in a single volume, this book offers an up-to-date review of computational methods in electromagnetism, with a focus on recent results in the numerical simulation of real-life electromagnetic problems and on theoretical results that are useful in devising and analyzing approximation algorithms. Based on four courses delivered in Cetraro in June 2014, the material covered includes the spatial discretization of Maxwell’s equations in a bounded domain, the numerical approximation of the eddy current model in harmonic regime, the time domain integral equation method (with an emphasis on the electric-field integral equation) and an overview of qualitative methods for inverse electromagnetic scattering problems.Assuming some knowledge of the variational formulation of PDEs and of finite element/boundary element methods, the book is suitable for PhD students and researchers interested in numerical approximation of partial differential equations and scientific computing.Table of ContentsPreface, Ralf Hiptmair: Maxwell's Equations: Continuous and Discrete Peter Monk: Numerical Methods for Maxwell's Equations, Rodolfo Rodriguez: Numerical Approximation of Low-Frequency Problems; Houssem Haddar: Inverse Electromagnetic Scattering Problems.

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Book SynopsisThis volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these; the qualitative behaviour of solutions being established using classical concepts: maximum principles and energy methods. Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the approximation of multidimensional advection–diffusion problems.The underlying theory is illustrated by numerous examples and there are around 300 exercises, designed to promote and test understanding. They are starred according to level of difficulty. Solutions to odd-numbered exercises are available to all readers while even-numbered solutions are available to authorised instructors.Written in an informal yet rigorous style, Essential Partial Differential Equations is designed for mathematics undergraduates in their final or penultimate year of university study, but will be equally useful for students following other scientific and engineering disciplines in which PDEs are of practical importance. The only prerequisite is a familiarity with the basic concepts of calculus and linear algebra.Trade Review“The book is written in an engaging and lively style that will appeal to students. … aim of the Springer SUMS series is to take a ‘fresh and modern approach’ to core foundational material through to final year topics. This book delivers on that promise with great success. ... As a first text that is set at the appropriate level … which recognizes and incorporates numerical computation as an essential tool for learning and understanding, it looks hard to beat.” (Mark Blyth, SIAM Review, Vol. 59 (1), March, 2017)“UK mathematicians Griffiths (Univ. of Dundee) and Dold and Silvester (both, Univ. of Manchester) introduce undergraduates to partial differential equations (PDEs) from both the analytical and numerical points of view. … Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners.” (D. P. Turner, Choice, Vol. 53 (11), July, 2016)“This introduction to partial differential equations is designed for upper level undergraduates in mathematics. … The writing is lively, the authors make appealing use of computational examples and visualization, and they are very successful at conveying and integrating physical intuition. … This is probably the best introductory book on PDEs that I have seen in some time. It is well worth a look.” (William J. Satzer, MAA Reviews, maa.org, April, 2016)“This textbook offers a nice introduction to analytical and numerical methods for partial differential equations. … The book is self-contained and the prerequisites is a standard course in calculus and linear algebra. The textbook appeals to undergraduate students in both scientific and engineering programs in which PDEs are of practical importance.” (Marius Ghergu, zbMATH 1330.35001, 2016)Table of ContentsSetting the scene.- Boundary and initial data.- The origin of PDEs.- Classification of PDEs.- Boundary value problems in R1.- Finite difference methods in R1.- Maximum principles and energy methods.- Separation of variables.- The method of characteristics.- Finite difference methods for elliptic PDEs.- Finite difference methods for parabolic PDEs.- Finite difference methods for hyperbolic PDEs.- Projects.

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Book SynopsisFocusing on special matrices and matrices which are in some sense `near’ to structured matrices, this volume covers a broad range of topics of current interest in numerical linear algebra. Exploitation of these less obvious structural properties can be of great importance in the design of efficient numerical methods, for example algorithms for matrices with low-rank block structure, matrices with decay, and structured tensor computations. Applications range from quantum chemistry to queuing theory. Structured matrices arise frequently in applications. Examples include banded and sparse matrices, Toeplitz-type matrices, and matrices with semi-separable or quasi-separable structure, as well as Hamiltonian and symplectic matrices. The associated literature is enormous, and many efficient algorithms have been developed for solving problems involving such matrices. The text arose from a C.I.M.E. course held in Cetraro (Italy) in June 2015 which aimed to present this fast growing field to young researchers, exploiting the expertise of five leading lecturers with different theoretical and application perspectives.Table of Contents Preface.-Charles F. Van Loan: Structured Matrix Problems from Tensors.-Dario A. Bini: Matrix Structures in Queuing Models.-. Jonas Ballani and Daniel Kressner: Matrices with Hierarchical Low-Rank Structures.-Michele Benzi: Localization in Matrix Computations: Theory and Applications.-Munthe-Kaas: Groups and Symmetries in Numerical Linear Algebra.

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Book SynopsisThis book, now in a carefully revised second edition, provides an up-to-date account of Oka theory, including the classical Oka-Grauert theory and the wide array of applications to the geometry of Stein manifolds.Oka theory is the field of complex analysis dealing with global problems on Stein manifolds which admit analytic solutions in the absence of topological obstructions. The exposition in the present volume focuses on the notion of an Oka manifold introduced by the author in 2009. It explores connections with elliptic complex geometry initiated by Gromov in 1989, with the Andersén-Lempert theory of holomorphic automorphisms of complex Euclidean spaces and of Stein manifolds with the density property, and with topological methods such as homotopy theory and the Seiberg-Witten theory.Researchers and graduate students interested in the homotopy principle in complex analysis will find this book particularly useful. It is currently the only work that offers a comprehensive introduction to both the Oka theory and the theory of holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with large automorphism groups.Table of ContentsPart I Stein Manifolds.- 1 Preliminaries.- 2 Stein Manifolds.- 3 Stein Neighborhoods and Approximation.- 4 Automorphisms of Complex Euclidean Spaces.- Part II Oka Theory.- 5 Oka Manifolds.- 6 Elliptic Complex Geometry and Oka Theory.- 7 Flexibility Properties of Complex Manifolds and Holomorphic Maps.- Part III Applications.- 8 Applications of Oka Theory and its Methods.- 9 Embeddings, Immersions and Submersions.- 10 Topological Methods in Stein Geometry.- References.- Index.

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Book SynopsisThis new edition is a concise introduction to the basic methods of computational physics. Readers will discover the benefits of numerical methods for solving complex mathematical problems and for the direct simulation of physical processes. The book is divided into two main parts: Deterministic methods and stochastic methods in computational physics. Based on concrete problems, the first part discusses numerical differentiation and integration, as well as the treatment of ordinary differential equations. This is extended by a brief introduction to the numerics of partial differential equations. The second part deals with the generation of random numbers, summarizes the basics of stochastics, and subsequently introduces Monte-Carlo (MC) methods. Specific emphasis is on MARKOV chain MC algorithms. The final two chapters discuss data analysis and stochastic optimization. All this is again motivated and augmented by applications from physics. In addition, the book offers a number of appendices to provide the reader with information on topics not discussed in the main text. Numerous problems with worked-out solutions, chapter introductions and summaries, together with a clear and application-oriented style support the reader. Ready to use C++ codes are provided online.Table of ContentsSome Basic Remarks.- Part I Deterministic Methods.- Numerical Differentiation.- Numerical Integration.- The KEPLER Problem.- Ordinary Differential Equations – Initial Value Problems.- The Double Pendulum.- Molecular Dynamics.- Numerics of Ordinary Differential Equations - Boundary Value Problems.- The One-Dimensional Stationary Heat Equation.- The One-Dimensional Stationary SCHRÖDINGER Equation.- Partial Differential Equations.- Part II Stochastic Methods.- Pseudo Random Number Generators.- Random Sampling Methods.- A Brief Introduction to Monte-Carlo Methods.- The ISING Model.- Some Basics of Stochastic Processes.- The Random Walk and Diffusion Theory.- MARKOV-Chain Monte Carlo and the POTTS Model.- Data Analysis.- Stochastic Optimization.- Appendix: The Two-Body Problem.- Solving Non-Linear Equations. The NEWTON Method.- Numerical Solution of Systems of Equations.- Fast Fourier Transform.- Basics of Probability Theory.- Phase Transitions.- Fractional Integrals and Derivatives in 1D.- Least Squares Fit.- Deterministic Optimization.

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Book Synopsis​This book provides a complete and comprehensive guide to Pyomo (Python Optimization Modeling Objects) for beginning and advanced modelers, including students at the undergraduate and graduate levels, academic researchers, and practitioners. Using many examples to illustrate the different techniques useful for formulating models, this text beautifully elucidates the breadth of modeling capabilities that are supported by Pyomo and its handling of complex real-world applications. This second edition provides an expanded presentation of Pyomo’s modeling capabilities, providing a broader description of the software that will enable the user to develop and optimize models. Introductory chapters have been revised to extend tutorials; chapters that discuss advanced features now include the new functionalities added to Pyomo since the first edition including generalized disjunctive programming, mathematical programming with equilibrium constraints, and bilevel programming.Pyomo is an open source software package for formulating and solving large-scale optimization problems. The software extends the modeling approach supported by modern AML (Algebraic Modeling Language) tools. Pyomo is a flexible, extensible, and portable AML that is embedded in Python, a full-featured scripting language. Python is a powerful and dynamic programming language that has a very clear, readable syntax and intuitive object orientation. Pyomo includes Python classes for defining sparse sets, parameters, and variables, which can be used to formulate algebraic expressions that define objectives and constraints. Moreover, Pyomo can be used from a command-line interface and within Python's interactive command environment, which makes it easy to create Pyomo models, apply a variety of optimizers, and examine solutions.Trade Review“This book provides a detailed guide to Pyomo for beginners and advanced users from undergraduate students to academic researchers to practitioners. … the book is a good software guide which I strongly recommend to anybody interested in looking for an alternative to commercial modeling languages in general or in learning or intensifying their Pyomo skills in particular.” (Christina Schenk, SIAM Review, Vol. 61 (1), March, 2019)Table of Contents1. Introduction.- Part I. An Introduction to Pyomo.- 2. Mathematical Modeling and Optimization.- 3. Pyomo Overview.- 4. Pyomo Models and Components.- 5. The Pyomo Command.- 6. Data Command Files.- Part II. Advanced Features and Extensions.- 7. Nonlinear Programming with Pyomo.- 8. Structured Modeling with Blocks.- 9. Generalized Disjunctive Programming.- 10. Stochastic Programming Extensions.- 11. Differential Algebraic Equations.- 12. Mathematical Programs with Equilibrium Constraints.- 13. Bilevel Programming.- 14. Scripting.- A. A Brief Python Tutorial.- Index.

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Book SynopsisThis book takes readers on a multi-perspective tour through state-of-the-art mathematical developments related to the numerical treatment of PDEs based on splines, and in particular isogeometric methods. A wide variety of research topics are covered, ranging from approximation theory to structured numerical linear algebra. More precisely, the book provides (i) a self-contained introduction to B-splines, with special focus on approximation and hierarchical refinement, (ii) a broad survey of numerical schemes for control problems based on B-splines and B-spline-type wavelets, (iii) an exhaustive description of methods for computing and analyzing the spectral distribution of discretization matrices, and (iv) a detailed overview of the mathematical and implementational aspects of isogeometric analysis. The text is the outcome of a C.I.M.E. summer school held in Cetraro (Italy), July 2017, featuring four prominent lecturers with different theoretical and application perspectives. The book may serve both as a reference and an entry point into further research.Table of ContentsFoundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement.- Adaptive Multiscale Methods for the Numerical Treatment of Systems of PDEs.- Generalized Locally Toeplitz Sequences: A Spectral Analysis Tool for Discretized Differential Equations.- Isogeometric Analysis: Mathematical and Implementational Aspects, with Applications.

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Book SynopsisThis book is a guide to numerical methods for solving fluid dynamics problems. The most widely used discretization and solution methods, which are also found in most commercial CFD-programs, are described in detail. Some advanced topics, like moving grids, simulation of turbulence, computation of free-surface flows, multigrid methods and parallel computing, are also covered. Since CFD is a very broad field, we provide fundamental methods and ideas, with some illustrative examples, upon which more advanced techniques are built. Numerical accuracy and estimation of errors are important aspects and are discussed in many examples. Computer codes that include many of the methods described in the book can be obtained online. This 4th edition includes major revision of all chapters; some new methods are described and references to more recent publications with new approaches are included. Former Chapter 7 on solution of the Navier-Stokes equations has been split into two Chapters to allow for a more detailed description of several variants of the Fractional Step Method and a comparison with SIMPLE-like approaches. In Chapters 7 to 13, most examples have been replaced or recomputed, and hints regarding practical applications are made. Several new sections have been added, to cover, e.g., immersed-boundary methods, overset grids methods, fluid-structure interaction and conjugate heat transfer.Table of ContentsBasic Concepts of Fluid Flow.- Introduction to Numerical Methods.- Finite Difference Methods.- Finite Volume Methods.- Solution of Linear Equation Systems.-Methods for Unsteady Problems.- Solution of the Navier-Stokes Equations.- Complex Geometries.- Turbulent Flows.- Compressible Flows.- Efficiency, Accuracy and Grid Quality.- Special Topics.

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Book SynopsisTable of Contents1. Einleitung.- 2. Algebraische Grundlagen.- 3. Geringte Räume.- 4. Supermannigfaltigkeiten.- 5. Analysis auf Supergebieten.- 6. Anwendungen.- 7. Lie—Algebren und Grundbegriffe der Darstellungstheorie.- 8. Höchstgewichtsdarstellungen der Virasoro-Algebra.- 9. Vertexoperatoren.- 10. Beweis der Kac’schen Determinantenformel.- 11. Konstruktion singulärer Vektoren im Fockraum.- 12.Unitäre Höchstgewichtsdarstellungen der Virasoro-Algebra.

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Book SynopsisDas Buch ist ein Hilfsmittel zur Bearbeitung von Übungsaufgaben bzw. zur Vorbereitung auf die Prüfung zur Vorlesung in Numerischer Mathematik bzw. wissenschaftliches Rechnen. Es ist unabhängig von einem bestimmten Lehrbuch konzipiert und umfaßt den Standardstoff einer einführenden Vorlesung in Numerik oder in das wissenschaftliche Rechnen. Es ist in Paragraphen gegliedert, welche jeweils einem bestimmten Sachgebiet der Numerik entsprechen. Jedem Paragraphen ist in ein bis zwei Seiten eine kurze Zusammenstellung der wichtigsten Tatsachen und Formeln vorangestellt, die als Arbeitsgrundlage und Verweismöglichkeit bei den Aufgaben dient. Jeder Paragraph enthält eine Sammlung für das Gebiet typischer Übungsaufgaben mit mustergültig ausgearbeiteten Lösungen.Table of ContentsRechnerarithmetik - Polynome und Interpolation - Numerische Differentiation und RICHARDSON-Extrapolation - BANACHscher Fixpunktsatz und Konvergenzordnung - Nichtlineare Gleichungen - Matrixanalyse und Normen - Lineare Gleichungssysteme

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Book SynopsisThe authors consider applications of singularity theory and computer algebra to bifurcations of Hamiltonian dynamical systems. They restrict themselves to the case were the following simplification is possible. Near the equilibrium or (quasi-) periodic solution under consideration the linear part allows approximation by a normalized Hamiltonian system with a torus symmetry. It is assumed that reduction by this symmetry leads to a system with one degree of freedom. The volume focuses on two such reduction methods, the planar reduction (or polar coordinates) method and the reduction by the energy momentum mapping. The one-degree-of-freedom system then is tackled by singularity theory, where computer algebra, in particular, Gröbner basis techniques, are applied. The readership addressed consists of advanced graduate students and researchers in dynamical systems.Table of ContentsIntroduction.- I. Applications: Methods I: Planar reduction; Method II: The energy-momentum map.- II. Theory: Birkhoff Normalization; Singularity Theory; Gröbner bases and Standard bases; Computing normalizing transformations.- Appendix A.1. Classification of term orders; Appendix A.2. Proof of Proposition 5.8.- References.- Index.

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Book SynopsisThese notes are an elaboration of the first part of a course on foliations which I have given at Strasbourg in 1976 and at Tunis in 1977. They are concerned mostly with dynamical sys­ tems in dimensions one and two, in particular with a view to their applications to foliated manifolds. An important chapter, however, is missing, which would have been dealing with structural stability. The publication of the French edition was re­ alized by-the efforts of the secretariat and the printing office of the Department of Mathematics of Strasbourg. I am deeply grateful to all those who contributed, in particular to Mme. Lambert for typing the manuscript, and to Messrs. Bodo and Christ for its reproduction. Strasbourg, January 1979. Table of Contents I. VECTOR FIELDS ON MANIFOLDS 1. Integration of vector fields. 1 2. General theory of orbits. 13 3. Irlvariant and minimaI sets. 18 4. Limit sets. 21 5. Direction fields. 27 A. Vector fields and isotopies. 34 II. THE LOCAL BEHAVIOUR OF VECTOR FIELDS 39 1. Stability and conjugation. 39 2. Linear differential equations. 44 3. Linear differential equations with constant coefficients. 47 4. Linear differential equations with periodic coefficients. 50 5. Variation field of a vector field. 52 6. Behaviour near a singular point. 57 7. Behaviour near a periodic orbit. 59 A. Conjugation of contractions in R. 67 III. PLANAR VECTOR FIELDS 75 1. Limit sets in the plane. 75 2. Periodic orbits. 82 3. Singular points. 90 4. The Poincare index.Table of ContentsI. Vector Fields on Manifolds.- 1. Integration of vector fields.- 2. General theory of orbits.- 3. Invariant and minimal sets.- 4. Limit sets.- 5. Direction fields.- A. Vector fields and isotopies.- II. The Local Behaviour of Vector Fields.- 1. Stability and conjugation.- 2. Linear differential equations.- 3. Linear differential equations with constant coefficients.- 4. Linear differential equations with periodic coefficients.- 5. Variation field of a vector field.- 6. Behaviour near a singular point.- 7. Behaviour near a periodic orbit.- A. Conjugation of contractions in R.- III. Planar Vector Fields.- 1. Limit sets in the plane.- 2. Periodic orbits.- 3. Singular points.- 4. The Poincaré index.- 5. Planar direction fields.- 6. Direction, fields on cylinders and Moebius strips.- A. Singular generic foliations of a disc.- IV. Direction Fields on the Torus and Homeomorphisms of the Circle.- 1. Direction fields on the torus.- 2. Direction fields on a Klein bottle.- 3. Homeomorphisms of the circle without periodic point.- 4. Rotation number of Poincaré.- 5. Conjugation of circle homeomorphisms to rotations.- A. Homeomorphism groups of an interval.- B. Homeomorphism groups of the circle.- V. Vector Fields on Surfaces.- 1. Classification of compact surfaces.- 2. Vector fields on surfaces.- 3. The index theorem.- A. Elements of differential geometry of surfaces.

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Book SynopsisHigh resolution upwind and centered methods are a mature generation of computational techniques. They are applicable to a wide range of engineering and scientific disciplines, Computational Fluid Dynamics (CFD) being the most prominent up to now. This textbook gives a comprehensive, coherent and practical presentation of this class of techniques. For its third edition the book has been thoroughly revised to contain new material.Table of ContentsThe Equations of Fluid Dynamics.- Notions on Hyperbolic Partial Differential Equations.- Some Properties of the Euler Equations.- The Riemann Problem for the Euler Equations.- Notions on Numerical Methods.- The Method of Godunov for Non#x2014;linear Systems.- Random Choice and Related Methods.- Flux Vector Splitting Methods.- Approximate#x2014;State Riemann Solvers.- The HLL and HLLC Riemann Solvers.- The Riemann Solver of Roe.- The Riemann Solver of Osher.- High#x2013;Order and TVD Methods for Scalar Equations.- High#x2013;Order and TVD Schemes for Non#x2013;Linear Systems.- Splitting Schemes for PDEs with Source Terms.- Methods for Multi#x2013;Dimensional PDEs.- Multidimensional Test Problems.- FORCE Fluxes in Multiple Space Dimensions.- The Generalized Riemann Problem.- The ADER Approach.- Concluding Remarks.

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Book SynopsisWave propagation is an important topic in engineering sciences, especially, in the field of solid mechanics. A description of wave propagation phenomena is given by Graff [98]: The effect of a sharply applied, localized disturbance in a medium soon transmits or 'spreads' to other parts of the medium. These effects are familiar to everyone, e.g., transmission of sound in air, the spreading of ripples on a pond of water, or the transmission of radio waves. From all wave types in nature, here, attention is focused only on waves in solids. Thus, solely mechanical disturbances in contrast to electro-magnetic or acoustic disturbances are considered. of waves - the compression wave similar to the In solids, there are two types pressure wave in fluids and, additionally, the shear wave. Due to continual reflec­ tions at boundaries and propagation of waves in bounded solids after some time a steady state is reached. Depending on the influence of the inertia terms, this state is governed by a static or dynamic equilibrium in frequency domain. However, if the rate of onset of the load is high compared to the time needed to reach this steady state, wave propagation phenomena have to be considered.Table of Contents1. Introduction.- 2. Convolution quadrature method.- 2.1 Basic theory of the convolution quadrature method.- 2.2 Numerical tests.- 2.2.1 Series expansion of the test functions f1 and f2.- 2.2.2 Computing the integration weights ?n.- 2.2.3 Numerical convolution.- 3. Viscoelastically supported Euler-Bernoulli beam.- 3.1 Integral equation for a beam resting on viscoelastic foundation.- 3.1.1 Fundamental solutions.- 3.1.2 Integral equation.- 3.2 Numerical example.- 3.2.1 Fixed-simply supported beam.- 3.2.2 Fixed-free viscoelastic supported beam.- 4. Time domain boundary element formulation.- 4.1 Integral equation for elastodynamics.- 4.2 Boundary element formulation for elastodynamics.- 4.3 Validation of proposed method: Wave propagation in a rod.- 4.3.1 Influence of the spatial and time discretization.- 4.3.2 Comparison with the “classical” time domain BE formulation.- 5. Viscoelastodynamic boundary element formulation.- 5.1 Viscoelastic constitutive equation.- 5.2 Boundary integral equation.- 5.3 Boundary element formulation.- 5.4 Validation of the method and parameter study.- 5.4.1 Three-dimensional rod.- 5.4.2 Elastic foundation on viscoelastic half space.- 6. Poroelastodynamic boundary element formulation.- 6.1 Biot’s theory of poroelasticity.- 6.1.1 Elastic skeleton.- 6.1.2 Viscoelastic skeleton.- 6.2 Fundamental solutions.- 6.3 Poroelastic Boundary Integral Formulation.- 6.3.1 Boundary integral equation.- 6.3.2 Boundary element formulation.- 6.4 Numerical studies.- 6.4.1 Influence of time step size and mesh size.- 6.4.2 Poroelastic half space.- 7. Wave propagation.- 7.1 Wave propagation in poroelastic one-dimensional column.- 7.1.1 Analytical solution.- 7.1.2 Poroelastic results.- 7.1.3 Poroviscoelastic results.- 7.2 Waves in half space.- 7.2.1 Rayleigh surface wave.- 7.2.2 Slow compressional wave in poroelastic half space.- 8. Conclusions — Applications.- 8.1 Summary.- 8.2 Outlook on further applications.- A. Mathematic preliminaries.- A.1 Distributions or generalized functions.- A.2 Convolution integrals.- A.3 Laplace transform.- A.4 Linear multistep method.- B. BEM details.- B.1 Fundamental solutions.- B.1.1 Visco- and elastodynamic fundamental solutions.- B.1.2 Poroelastodynamic fundamental solutions.- B.2 “Classical” time domain BE formulation.- Notation Index.- References.

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Book SynopsisThis work provides an introduction to four central problems in analytic number theory. These are (1) the problems of estimating the number of integer points in planar domains, (2) the problem of the distribution of prime numbers in the sequence of all natural numbers and in arithmetic progressions, (3) Goldbach's problems on sums of primes, and (4) Waring's problem on sums of k-th powers. The following fundamental methods of analytic number theory are used to solve these problems: complex integration, I.M. Vinogradov's method of trigonometric sums, and the circle method of G.H. Hardy, J.E. Littlewood, and S. Ramanujan. There are numerous exercises at the end of each chapter. These exercises either refine the theorems proved in the text, or lead to new ideas in number theory. The author also includes a section of hints for the solution of the exercises.

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Book SynopsisThe main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.Trade ReviewFrom the reviews:"The book under review is an introduction to the field of linear partial differential equations and to standard methods for their numerical solution. … The balanced combination of mathematical theory with numerical analysis is an essential feature of the book. … The book is easily accessible and concentrates on the main ideas while avoiding unnecessary technicalities. It is therefore well suited as a textbook for a beginning graduate course in applied mathematics." (A. Ostermann, IMN - Internationale Mathematische Nachrichten, Vol. 59 (198), 2005)"This book, which is aimed at beginning graduate students of applied mathematics and engineering, provides an up to date synthesis of mathematical analysis, and the corresponding numerical analysis, for elliptic, parabolic and hyperbolic partial differential equations. … This widely applicable material is attractively presented in this impeccably well-organised text. … Partial differential equations with numerical methods covers a lot of ground authoritatively and without ostentation and with a constant focus on the needs of practitioners." (Nick Lord, The Mathematical Gazette, March, 2005)"Larsson and Thomée … discuss numerical solution methods of linear partial differential equations. They explain finite difference and finite element methods and apply these concepts to elliptic, parabolic, and hyperbolic partial differential equations. … The text is enhanced by 13 figures and 150 problems. Also included are appendixes on mathematical analysis preliminaries and a connection to numerical linear algebra. Summing Up: Recommended. Upper-division undergraduates through faculty." (D. P. Turner, CHOICE, March, 2004)"This book presents a very well written and systematic introduction to the finite difference and finite element methods for the numerical solution of the basic types of linear partial differential equations (PDE). … the book is very well written, the exposition is clear, readable and very systematic." (Emil Minchev, Zentralblatt MATH, Vol. 1025, 2003)"The author’s purpose is to give an elementary, relatively short, and readable account of the basic types of linear partial differential equations, their properties, and the most commonly used methods for their numerical solution. … We warmly recommend it to advanced undergraduate and beginning graduate students of applied mathematics and/or engineering at every university of the world." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 71, 2005)"The presentation of the book is smart and very classical; it is more a reference book for applied mathematicians … . The convergence results, error estimates, variation formulations, all the theorems proofs, are very clear and well presented, the annexes A and B summary the necessary background for the understanding, without redundant generalisation or forgotten matter. The bibliography is presented by theme, well targeted on the topic of the book." (Anne Lemaitre, Physicalia Magazine, Vol. 28 (1), 2006)“Offers basic theory of linear partial differential equations and discusses the most commonly used numerical methods to solve these equations. … There are two appendices providing some extra basic material, useful to help understanding some of the theoretical principles that might be unfamiliar to unexperienced readers and students. The text is elementary and meant for students in mathematics, physics, engineering. … The bibliography is well arranged according to the important issues, which makes it easy to get informed about possible references for further study.” (Paula Bruggen, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)Table of ContentsA Two-Point Boundary Value Problem.- Elliptic Equations.- Finite Difference Methods for Elliptic Equations.- Finite Element Methods for Elliptic Equations.- The Elliptic Eigenvalue Problem.- Initial-Value Problems for Ordinary Differential Equations.- Parabolic Equations.- Finite Difference Methods for Parabolic Problems.- The Finite Element Method for a Parabolic Problem.- Hyperbolic Equations.- Finite Difference Methods for Hyperbolic Equations.- The Finite Element Method for Hyperbolic Equations.- Some Other Classes of Numerical Methods.

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Book SynopsisFrom the reviews: "The account is quite detailed and is written in a manner that will appeal to analysts and numerical practitioners alike...they contain everything from rigorous proofs to tables of numerical calculations.... one of the strong features of these books...that they are designed not for the expert, but for those who whish to learn the subject matter starting from little or no background...there are numerous examples, and counter-examples, to back up the theory...To my knowledge, no other authors have given such a clear geometric account of convex analysis." "This innovative text is well written, copiously illustrated, and accessible to a wide audience"Trade ReviewFrom the reviews: "The account is quite detailed and is written in a manner that will appeal to analysts and numerical practitioners alike...they contain everything from rigorous proofs to tables of numerical calculations.... one of the strong features of these books. . . [is] that they are designed not for the expert, but for those who wish to learn the subject matter starting from little or no background...there are numerous examples, and counter-examples, to back up the theory...To my knowledge, no other authors have given such a clear geometric account of convex analysis." "This innovative text is well written, copiously illustrated, and accessible to a wide audience"Table of ContentsIX. Inner Construction of the Subdifferential.- X. Conjugacy in Convex Analysis.- XI. Approximate Subdifferentials of Convex Functions.- XII. Abstract Duality for Practitioners.- XIII. Methods of ?-Descent.- XIV. Dynamic Construction of Approximate Subdifferentials: Dual Form of Bundle Methods.- XV. Acceleration of the Cutting-Plane Algorithm: Primal Forms of Bundle Methods.- Bibliographical Comments.- References.

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Book SynopsisBackward stochastic differential equations (BSDEs) provide a general mathematical framework for solving pricing and risk management questions of financial derivatives. They are of growing importance for nonlinear pricing problems such as CVA computations that have been developed since the crisis. Although BSDEs are well known to academics, they are less familiar to practitioners in the financial industry. In order to fill this gap, this book revisits financial modeling and computational finance from a BSDE perspective, presenting a unified view of the pricing and hedging theory across all asset classes. It also contains a review of quantitative finance tools, including Fourier techniques, Monte Carlo methods, finite differences and model calibration schemes. With a view to use in graduate courses in computational finance and financial modeling, corrected problem sets and Matlab sheets have been provided. Stéphane Crépey’s book starts with a few chapters on classical stochastic processes material, and then... fasten your seatbelt... the author starts traveling backwards in time through backward stochastic differential equations (BSDEs). This does not mean that one has to read the book backwards, like a manga! Rather, the possibility to move backwards in time, even if from a variety of final scenarios following a probability law, opens a multitude of possibilities for all those pricing problems whose solution is not a straightforward expectation. For example, this allows for framing problems like pricing with credit and funding costs in a rigorous mathematical setup. This is, as far as I know, the first book written for several levels of audiences, with applications to financial modeling and using BSDEs as one of the main tools, and as the song says: "it's never as good as the first time".Damiano Brigo, Chair of Mathematical Finance, Imperial College LondonWhile the classical theory of arbitrage free pricing has matured, and is now well understood and used by the finance industry, the theory of BSDEs continues to enjoy a rapid growth and remains a domain restricted to academic researchers and a handful of practitioners. Crépey’s book presents this novel approach to a wider community of researchers involved in mathematical modeling in finance. It is clearly an essential reference for anyone interested in the latest developments in financial mathematics. Marek Musiela, Deputy Director of the Oxford-Man Institute of Quantitative FinanceTable of ContentsPart I: An Introductory Course in Stochastic Processes.- 1.Some classes of Discrete-Time Stochastic Processes.-2.Some Classes of Continuous-Time Stochastic Processes.- 3.Elements of Stochastic Analysis.- Part II: Pricing Equations.- 4.Martingale Modeling.- 5.Benchmark Models.- Part III: Numerical Solutions.- 6.Monte Carlo Methods.- 7.Tree Methods.- 8.Finite Differences.- 9.Callibration Methods.- Part IV: Applications.- 10.Simulation/ Regression Pricing Schemes in Diffusive Setups.- 11.Simulation/ Regression Pricing Schemes in Pure Jump Setups.- Part V: Jump-Diffusion Setup with Regime Switching (**).- 12.Backward Stochastic Differential Equations.- 13.Analytic Approach.- 14.Extensions.- Part VI: Appendix.- A.Technical Proofs (**).- B.Exercises.- C.Corrected Problem Sets.​

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Book SynopsisThis book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.​ Trade ReviewFrom the reviews:“The authors give an introduction to the finite element method as a general computational method for solving partial differential equations (PDEs) approximately. … The book should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed.” (Răzvan Răducanu, Zentralblatt MATH, Vol. 1263, 2013)Table of Contents1. Piecewise Polynomial Approximation in 1D.- 2. The Finite Element Method in 1D.- 3. Piecewise Polynomial Approximation in 2D.- 4. The Finite Element Method in 2D.- 5. Time-dependent Problems.- 6. Solving Large Sparse Linear Systems.- 7. Abstract Finite Element Analysis.- 8. The Finite Element.- 9. Non-linear Problems.- 10. Transport Problems.- 11. Solid Mechanics.- 12. Fluid Mechanics.- 13. Electromagnetics.- 14. Discontinuous Galerkin Methods.- A. Some Additional Matlab Code.- References.

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Book SynopsisI. In this second volume, we continue at first the study of non­ homogeneous boundary value problems for particular classes of evolu­ tion equations. 1 In Chapter 4 , we study parabolic operators by the method of Agranovitch-Vishik [lJ; this is step (i) (Introduction to Volume I, Section 4), i.e. the study of regularity. The next steps: (ii) transposition, (iii) interpolation, are similar in principle to those of Chapter 2, but involve rather considerable additional technical difficulties. In Chapter 5, we study hyperbolic operators or operators well­ defined in thesense of Petrowski or Schroedinger. Our regularity results (step (i)) seem to be new. Steps (ii) and (iii) are all3.logous to those of the parabolic case, except for certain technical differences. In Chapter 6, the results of Chapter'> 4 and 5 are applied to the study of optimal control problems for systems governed by evolution equations, when the control appears in the boundary conditions (so that non-homogeneous boundary value problems are the basic tool of this theory). Another type of application, to the characterization of "all" well-posed problems for the operators in question, is given in the Ap­ pendix. Still other applications, for example to numerical analysis, will be given in Volume 3.Table of Contents4 Parabolic Evolution Operators. Hilbert Theory.- 1. Notation and Hypotheses. First Regularity Theorem.- 1.1 Notation.- 1.2 Statement of the Problems.- 1.3 (Formal) Green’s Formulas.- 1.4 First Existence and Uniqueness Theorem (Statement).- 1.5 Orientation.- 2. The Spaces Hr, s(Q). Trace Theorems. Compatibility Relations.- 2.1 Hr, s-Spaces.- 2.2 First Trace Theorem.- 2.3 Local Compatibility Relations.- 2.4 Global Compatibility Relations for a Particular Case.- 2.5 General Compatibility Relations.- 3. Evolution Equations and the Laplace Transform.- 3.1 Vector Distribution Solutions.- 3.2 L2-Solutions.- 4. The Case of Operators Independent of t.- 4.1 Hypotheses.- 4.2 Basic Inequalities.- 4.3 Solution of the Problem.- 5. Regularity.- 5.1 Preliminaries.- 5.2 Basic Inequalities.- 5.3 An Abstract Result.- 5.4 Solution of the Boundary Value Problem.- 6. Case of Time-Dependent Operators. Existence of Solutions in the Spaces H2r m, m(Q), Real r ? 1.- 6.1 Hypotheses. Statement of the Result.- 6.2 Local Result in t.- 6.3 Proof of Theorem 6.1.- 6.4 Regular Non-Homogeneous Problems.- Adjoint Isomorphism of Order r.- 7.1 The Adjoint Problem.- 7.2 Adjoint Isomorphism of Order r.- 8. Transposition of the Adjoint Isomorphism of Order r. (I): Generalities.- 8.1 Transposition.- 8.2 Orientation.- 8.3 The Spaces H??, ??(Q), H??, ??(?), ?, ? ? 0.- 8.4 (Formal) Choice of L.- 9. Choice of f. The Spaces ?2rm,r(Q).- 9.1 The Space ?2rm,r(Q).- 9.2 The Space ??2rm,?r(Q).- 9.3 Choice of f. The Space D?(r?1)(P)(Q).- 10. Trace Theorems for the Spaces D?(r?1)(P)(Q), r ? 1.- 10.1 Density Theorem.- 10.2 Trace Theorem on ?.- 10.3 Continuity of the Trace on Surfaces Neighbouring ?.- 10.4 Trace Theorem on ?0.- 10.5 Continuity of the Trace on Sections Neighbouring ?.- 11. Choice of gj and uo. The Spaces H2?m ??(?).- 11.1 The Spaces H2?m ??(?).- 11.2 Choice of gj.- 11.3 Choice of uo.- 12. Transposition of the Adjoint Isomorphism of Order ?. (II): Results; Existence of Solutions in H2mr,r(Q)-Spaces, Real r ? 0.- 12.1 Final Choice of L.- 12.2 Results.- 12.3 Complements.- 13. State of the Problem. Complements on the Transposition of the Adjoint Isomorphism of Order 1.- 13.1 State of the Problem.- 13.2 Complements on the Transposition of the Adjoint Isomorphism of Order 1.- 13.3 Orientation.- 14. Some Interpolation Theorems.- 14.1 Notation. Statement of the Main Result.- 14.2 Outline of the Proof.- 14.3 First Auxiliary Interpolation Theorem.- 14.4 Second Auxiliary Interpolation Theorem.- 14.5 Third Auxiliary Interpolation Theorem.- 14.6 Proof of Theorem 14.1.- 15. Final Results; Existence of Solutions in the Spaces H2mr,r(Q), 0 < r < 1. Applications.- 15.1 Application of the Results of Section 14.- 15.2 Examples; Generalities.- 15.3 Examples (I).- 15.4 Examples (II).- 15.5 Some Complements on the Dirichlet Problem.- 16. Comments.- 17. Problems.- 5 Hyperbolic Evolution Operators, of Petrowski and of Schroedinger. Hilbert Theory.- 1. Application of the Results of Chapter 3 and General Remarks.- 1.1 Notation. Hypotheses.- 1.2 Application of the Results of Chapter 3.- 1.3 A Counter-Example.- 2. A Regularity Theorem (I).- 3. Regular Non-Homogeneous Problems.- 3.1 Statement of the Problem.- 3.2 The Compatibility Relations.- 3.3 The Case of the Dirichlet Problem.- 4. Transposition.- 4.1 Adjoint Isomorphism.- 4.2 Transposition.- 4.3 Choice of L.- 4.4 Conclusion.- 5. Interpolation.- 5.1 Statement of the Problem.- 5.2 Some Interpolation Results.- 5.3 Consequences.- 5.4 The Case of the Dirichlet Problem.- 6. Applications and Examples.- 6.1 General Results.- 6.2 Examples.- 7. Regularity Theorem (II).- 7.1 Statement.- 7.2 Proof of Theorem 7.1.- 8. Non-Integer Order Regularity Theorem.- 8.1 Orientation.- 8.2 Interpolation in r.- 8.3 Interpretation of the Space V(2r?1)m,2r(Q), r ? 1.- 9. Adjoint Isomorphism of Order r and Transposition.- 9.1 Adjoint Isomorphism of Order r.- 9.2 Transposition.- 9.3 Formal Choice of L.- 10. Choice of f, $$ \vec g $$, u0, u1.- 10.1 Choice of f.- 10.2 The Space $$ D_{A + D_t^2}^{ - \left( {2r - 1} \right)}\left( Q \right) $$.- 10.3 Choice of gj.- 10.4 Choice of u0, u1.- 10.5 Conclusion.- 11. Trace Theorems in the Space $$ D_{A + D_t^2}^{ - \left( {2r - 1} \right)}\left( Q \right) $$.- 11.1 Density Theorem.- 11.2 Traces on ?.- 11.3 Continuity of the Trace on Neighbouring Surfaces.- 11.4 Traces on ?0.- 11.5 Continuity of the Trace on Sections Neighbouring ?0.- 11.6 Remark.- 12. Schroedinger Type Equations.- 12.1 Notation.- 12.2 First Regularity Theorem. Parabolic Regularization.- 12.3 Second Regularity Theorem.- 12.4 r-Isomorphism Theorem.- 12.5 Choice of L.- 12.6 Trace Theorem.- 13. Comments.- 14. Problems.- 6 Applications to Optimal Control Problems.- 1. Statement of the Problems for the Linear Parabolic Case.- 1.1 Notation.- 1.2 Optimization Problems.- 2. Choice of the Norms in the Cost Function.- 2.1 Reminder. Condition on K1(Q).- 2.2 Space Described by $$ \vec S\,y $$. Conditions on K2(?).- 2.3 Space Described by y(x, T; u). Condition on K3(?).- 3. Optimality Condition for Quadratic Cost Functions.- 3.1 Notation.- 3.2 Optimality Condition.- 4. Optimality Condition and Green’s Formula.- 4.1 Optimality Condition. Application of Section 3.2.- 4.2 The Isomorphisms ?i.- 4.3 The “Adjoint” Problem.- 4.4 New Form of the Optimality Condition.- 5. The Particular Case $$ \mu \,\, = \,\,m\,\, + \,\,\frac{1}{2} $$, E3 ? 0.- 5.1 Properties of y.- 5.2 Choice of K1(Q).- 5.3 Choice of K2(?) and K3(?).- 5.4 Adjoint Problem and Optimality Condition.- 6. Consequences of the Optimality Condition (I).- 6.1 Generalities.- 6.2 Consequences of Theorem 6.1.- 7. Consequences of the Optimality Condition (II).- 7.1 Additional Hypotheses.- 7.2 Optimality Condition.- 8. Complements on the Choice of the Spaces Ki.- 8.1 Orientation.- 8.2 Choice of K1(Q).- 8.3 Choice of K2(?).- 8.4 Choice of K3(?).- 9. Examples.- 10. Non-Parabolic Cases. Statement of the Problems. Generalities.- 10.1 Notation.- 10.2 Cost Function.- 10.3 Optimality Condition (I).- 10.4 Adjoint Problem.- 10.5 Green’s Formula.- 10.6 Optimality Condition (II).- 10.7 Consequences.- 11. Applications. Examples.- 11.1 Control in the Boundary Conditions.- 11.2 Choice of K1.- 11.3 Choice of K2.- 11.4 Examples.- 12. Comments.- 13. Problems.- Boundary Value Problems and Operator Extensions.- 1. Statement of the Problem. Well-Posed Spaces.- 1.1 Notation.- 2. Abstract Boundary Conditions.- 2.1 Boundary Spaces and Operators.- 2.2 Characterization of Well-Posed Spaces.- 3. Example 1. Elliptic Operators.- 3.1 Notation.- 3.2 The Boundary Operators and Spaces.- 3.3 Consequences.- 3.4 Various Remarks.- 4. Example 2. Parabolic Operators.- 4.1 Notation.- 4.2 The Boundary Operators and Spaces.- 4.3 Consequences.- 5.1 Notation.- 5.2 Formal Results.- 6. Comments and Problems.

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Book SynopsisThis self-contained monograph presents matrix algorithms and their analysis. The new technique enables not only the solution of linear systems but also the approximation of matrix functions, e.g., the matrix exponential. Other applications include the solution of matrix equations, e.g., the Lyapunov or Riccati equation. The required mathematical background can be found in the appendix.The numerical treatment of fully populated large-scale matrices is usually rather costly. However, the technique of hierarchical matrices makes it possible to store matrices and to perform matrix operations approximately with almost linear cost and a controllable degree of approximation error. For important classes of matrices, the computational cost increases only logarithmically with the approximation error. The operations provided include the matrix inversion and LU decomposition.Since large-scale linear algebra problems are standard in scientific computing, the subject of hierarchical matrices is of interest to scientists in computational mathematics, physics, chemistry and engineering.Trade Review“Every line of the book reflects that the author is the leading expert for hierarchical matrices. … Hierarchical matrices: algorithms and analysis is without a doubt a beautiful, comprehensive introduction to hierarchical matrices that can serve as both a graduate level textbook and a valuable resource for future research.” (Thomas Mach, Mathematical Reviews, April, 2017)“The book ‘Hierarchical matrices: algorithms and analysis’ is a self-contained monograph which presents an efficient possibility to handle the numerical treatment of fully populated large scale matrices appearing in scientific computations, and therefore it is of interest to scientists in computational mathematics, physics, chemistry and engineering.” (Constantin Popa, zbMATH 1336.65041, 2016)Table of ContentsPreface.- Part I: Introductory and Preparatory Topics.- 1. Introduction.- 2. Rank-r Matrices.- 3. Introductory Example.- 4. Separable Expansions and Low-Rank Matrices.- 5. Matrix Partition.- Part II: H-Matrices and Their Arithmetic.- 6. Definition and Properties of Hierarchical Matrices.- 7. Formatted Matrix Operations for Hierarchical Matrices.- 8. H2-Matrices.- 9. Miscellaneous Supplements.- Part III: Applications.- 10. Applications to Discretised Integral Operators.- 11. Applications to Finite Element Matrices.- 12. Inversion with Partial Evaluation.- 13. Eigenvalue Problems.- 14. Matrix Functions.- 15. Matrix Equations.- 16. Tensor Spaces.- Part IV: Appendices.- A. Graphs and Trees.- B. Polynomials.- C. Linear Algebra and Functional Analysis.- D. Sinc Functions and Exponential Sums.- E. Asymptotically Smooth Functions.- References.- Index.

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Book SynopsisEinleitung.- Teil I Numerische Methoden der linearen Algebra. Grundlagen der linearen Algebra.- Lineare Gleichungssysteme.- Orthogonalisierungsverfahren und die QR-Zerlegung.- Berechnung von Eigenwerten.- Teil II Numerische Methoden der Analysis. Nullstellenbestimmung.- Numerische Iterationsverfahren für lineare Gleichungssysteme.- Interpolation und Approximation.- Verzeichnis der Exkurse.- Literaturverzeichnis.- Sachverzeichnis.

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Book SynopsisEste livro é uma introdução ao Cálculo Científico. O seu objectivo consiste em apresentar vários métodos numéricos para resolver no computador certos problemas matemáticos que não podem ser tratados de maneira mais simples. São abordadas questões clássicas como o cálculo de zeros ou de integrais de funções contínuas, a resolução de sistemas lineares, a aproximação de funções por polinómios e a construção de aproximações precisas de soluções de equações diferenciais. Todos os algoritmos são apresentados nas linguagens de programação MATLAB e Octave, cujos comandos e instruções principais se introduzem de forma gradual, visando em particular a sua compatibilidade nas duas linguagens. O leitor pode assim verificar experimentalmente propriedades teóricas como a estabilidade, a precisão e a complexidade. O livro inclui ainda a resolução de problemas através de numerosos exercícios e exemplos, frequentemente ligados a aplicações concretas. No fim de cada capítulo encontra-se uma secção específica que apresenta assuntos não abordados e as referências bibliográficas que permitem ao leitor aprofundar os conhecimentos adquiridos.Table of ContentsO que não se pode ignorar.- Equações não lineares.- Aproximação de funções e de dados.- Derivação e integração numéricas.- Sistemas lineares.- Valores próprios e vectores próprios.- Equações diferenciais ordinárias.- Métodos numéricos para problemas de valores iniciais e na fronteira.- Soluções dos exercícios.

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Book SynopsisUnderstanding earth systems and its dynamic behavior requires objective insights into the complex observational data sets and their interrelationships. Drawing meaningful inferences from such data is not always an easy task as the deterministic relationships between various geological variables often remain obscured. These interrelationships need to be determined empirically through the analysis of a large set of data and validated through numerical simulations. The ever widening horizon of techniques of numerical analysis and simulation now provides a good number of tools to aid the interpretation. However, due to the inherent complexity of earth science data, expert supervision is required at all stages of analysis from collection to dissemination. This ensures that the most appropriate methodology is adopted and the results remain consistent with the geological principles. Discussions on these practical issues often lie beyond the scope of textbooks and this is precisely where this book is placed. In this book eminent geoscientists present their experiences in analyzing and managing earth science data as well as in designing numerical models to simulate earth processes. Apart from giving a discourse of their own approach towards a particular research problem they also discuss at length the relative merits of alternative methodologies. These seven authoritative articles, richly illustrated, will be a valuable resource for research students and professionals interested in research and teaching in various branches of earth science like, tectonics, GPS geodesy, sedimentology, geographical information science, and evolutionary biology.Table of Contents1. Active deformation in the Darjiling-Sikkim Himalaya based on 2000-2004 geodetic global positioning system measurements. 2. Preliminary results of a study of crustal deformation in the Himalayan Frontal zone in North Bengal using GPS Geodesy. 3. Spatial and temporal variations of the strain fields in orogenic belts: an analysis based on kinematic models. 4. Grain size distribution patterns of some rivers in the light of three model distributions. 5. Estimation of shape changes of skull roof bones in Benthosuchus Sushkini, a Temnospondyl amphibian from the Triassic of Russia. 6. Spatial mathematical models for mineral potential mapping. 7. Enterprise GIS in Geological Survey of India.

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Book SynopsisThis book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors’ interests in incompressible vortex-dominated ?ows and compressible ?ows at low Mach numbers. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the rich mathematical foundation associated with probability and measure spaces. Despite the authors’ fascination with this foundation, the discussion only - ludes to those theoretical aspects needed to set the stage for subsequent applications. The book is authored by practitioners, and is primarily intended for researchers or graduate students in computational mathematics, physics, or ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid dynamics (CFD) would require familiarity with key, and in some cases delicate, features of the associated numerical methods. Besides these shortcomings, our aim is to treat algorithmic and computational aspects of spectral stochastic methods with details suf?cient to address and reconstruct all but those highly elaborate examples.Table of ContentsIntroduction: Uncertainty Quantification and Propagation.- Basic Formulations.- Spectral Expansions.- Non-intrusive Methods.- Galerkin Methods.- Detailed Elementary Applications.- Application to Navier-Stokes Equations.- Advanced topics.- Solvers for Stochastic Galerkin Problems.- Wavelet and Multiresolution Analysis Schemes.- Adaptive Methods.- Epilogue.

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Book SynopsisAddressing students and researchers as well as Computational Fluid Dynamics practitioners, this book is the most comprehensive review of high-resolution schemes based on the principle of Flux-Corrected Transport (FCT). The foreword by J.P. Boris and historical note by D.L. Book describe the development of the classical FCT methodology for convection-dominated transport problems, while the design philosophy behind modern FCT schemes is explained by S.T. Zalesak. The subsequent chapters present various improvements and generalizations proposed over the past three decades. In this new edition, recent results are integrated into existing chapters in order to describe significant advances since the publication of the first edition. Also, 3 new chapters were added in order to cover the following topics: algebraic flux correction for finite elements, iterative and linearized FCT schemes, TVD-like flux limiters, acceleration of explicit and implicit solvers, mesh adaptation, failsafe limiting for systems of conservation laws, flux-corrected interpolation (remapping), positivity preservation in RANS turbulence models, and the use of FCT as an implicit subgrid scale model for large eddy simulations.Table of ContentsThe conception, gestation, birth and infancy of FCT.- The design of flux-corrected transport (FCT) algorithms for structured grids.- On monotonically integrated large eddy simulation of tubulent flows based on FCT algorithms.- Large scale urban simulations with FCT.- 30 years of FCT.- Algebraic flux corretion I.- Algebraic flux correction II.- Algebraic flux correction III.- Algebraic flux correction IV.- An evaluation of the FCT method for high-speed flows.- Flux-corrected and optimization-based remap.

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