Numerical analysis Books

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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Book SynopsisDielectric constant and fluctuation formulae for molecular dynamics.- PoissonBoltzmann electrostatics and analytical approximations.- Numerical methods for PoissonBoltzmann equations.- Random walk stochastic methods for boundary value problems.- Deep Neural Network for Solving PDEs.- Fast algorithms for long-range interactions.- Fast multipole methods for long-range interactions in layered media.- Maxwell equations, potentials, and physical/artificial boundary conditions.- Dyadic Green's functions in layered media.- High-order methods for surface electromagnetic integral equations.- High-order hierarchical Nedelec edge elements.- Time-domain methods discontinuous Galerkin method and Yee scheme.- Scattering in periodic structures and surface plasmons.- Schr odinger equations for waveguides and quantum dots.- Quantum electron transport in semiconductors.- Non-equilibrium Green's function (NEGF) methods for transport.- Numerical methods for Wigner quantum transport.- Hydrodynamic electron transport and finite difference methods.- Transport models in plasma media and numerical methods.

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Book SynopsisChapter 1 Introduction.- Chapter 2 Fundamentals.- Chapter 3 Variational Formulation of Boundary Problems.- Chapter 4 Introduction to Finite Elements.- Chapter 5 Non-conforming Methods.- Chapter 6 Nodal Methods.

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Book SynopsisIt can be used for a course that provides an introduction to basic functional analysis, approximation theory, and numerical analysis, while building upon and applying basic techniques of real variable theory.Trade ReviewSecond Edition S.C. Brenner and L.R. Scott The Mathematical Theory of Finite Element Methods "[This is] a well-written book. A great deal of material is covered, and students who have taken the trouble to master at least some of the advanced material in the later chapters would be well placed to embark on research in the area." ZENTRALBLATT MATH From the reviews of the third edition: "An excelent survey of the deep mathematical roots of finite element methods as well as of some of the newest and most formal results concerning these methods. … The approach remains very clear and precise … . A significant number of examples and exercises improve considerably the accessability of the text. The authors also point out different ways the book could be used in various courses. … valuable reference and source for researchers (mainly mathematicians) in the topic." (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1135 (13), 2008)Table of ContentsPreface(3rdEd).- Preface(2ndEd).- Preface(1stED).- Basic Concepts.- Sobolev Spaces.- Variational Formulation of Elliptic Boundary Value Problems.- The Construction of a Finite Element of Space.- Polynomial Approximation Theory in Sobolev Spaces.- n-Dimensional Variational Problems.- Finite Element Multigrid Methods.- Additive Schwarz Preconditioners.- Max-norm Estimates.- Adaptive Meshes.- Variational Crimes.- Applications to Planar Elasticity.- Mixed Methods.- Iterative Techniques for Mixed Methods.- Applications of Operator-Interpolation Theory.- References.- Index.

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Book Synopsis1 Introduction.- 1.1 The Classical Sampling Theorem.- 1.2 Non-Uniform Sampling and Frames.- 1.3 Outline of the Book.- 2 On the Transmission Capacity of the Ether and Wire in Electrocommunications.- I Sampling, Wavelets, and the Uncertainty Principle.- 3 Wavelets and Sampling.- 4 Embeddings and Uncertainty Principles for Generalized Modulation Spaces.- 5 Sampling Theory for Certain Hilbert Spaces of Bandlimited Functions.- 6 Shannon-Type Wavelets and the Convergence of Their Associated Wavelet Series.- II Sampling Topics from Mathematical Analysis.- 7 Non-Uniform Sampling in Higher Dimensions: From Trigonometric Polynomials to Bandlimited Functions.- 8 The Analysis of Oscillatory Behavior in Signals Through Their Samples.- 9 Residue and Sampling Techniques in Deconvolution.- 10 Sampling Theorems from the Iteration of Low Order Differential Operators.- 11 Approximation of Continuous Functions by RogosinskiType Sampling Series.- III Sampling Tools and Applications.- 12 Fast Fourier Transforms for Nonequispaced Data: A Tutorial.- 13 Efficient Minimum Rate Sampling of Signals with Frequency Support over Non-Commensurable Sets.- 14 Finite-and Infinite-Dimensional Models for Oversampled Filter Banks.- 15 Statistical Aspects of Sampling for Noisy and Grouped Data.- 16 Reconstruction of MRI Images from Non-Uniform Sampling and Its Application to Intrascan Motion Correction in Functional MRI.- 17 Efficient Sampling of the Rotation Invariant Radon Transform.- References.Trade Review"The introduction (Chapter 1) gives an excellent overview of the history and development of sampling theory. It shows that the WSK sampling theory has roots in many classical areas of mathematics, such as harmonic analysis, number theory, and interpolation theory. Many famous mathematicians, such as Cauchy, Borel, Hadamard, and de la Vallee-Poussin contributed directly or indirectly to its development. The introduction then proceeds to show how sampling theory is connected to more recent topics in mathematical analysis, such as wavelets, Gabor systems, density theorems, frames, and sampling in locally compact abelian groups." —Mathematical Reviews "Engineers and mathematicians working in wavelets, signal processing, and harmonic analysis, as well as scientists and engineers working on applications as varied as medical imaging and synthetic aperture radar, will find the book to be a modern and authoritative guide to sampling theory." —Publicationes MathematicaeTable of ContentsIntroduction, On the transmission capacity of the 'ether' and wire in electrocommunications, Part I: Sampling, wavelets, and the uncertainty principle, Wavelets and sampling, Embeddings and uncertainty principles for generalized modulation spaces, Sampling theory for certain hilbert spaces of bandlimited functions, Shannon-type wavelets and the convergence of their associated wavelet series, Part II: Sampling topics from mathematical analysis, Non-uniform sampling in higher dimensions: From trigonometric polynomials to bandlimited functions, The analysis of oscillatory behavior in signals through their samples, Residue and sampling techniques in deconvolution, Sampling theorems from the iteration of low order differential operators, Approximation of continuous functions by Rogosinski-Type sampling series, Part III: Sampling tools and applications, Fast fourier transforms for nonequispaced data: A tutorial, Efficient minimum rate sampling of signals with frequency support over non-commensurable sets, Finite and infinite-dimensional models for oversampled filter banks, Statistical aspects of sampling for noisy and grouped data, Reconstruction of MRI images from non-uniform sampling, application to Intrascan motion correction in functional MRI, Efficient sampling of the rotation invariant radon transform

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Book Synopsis1 From brain physiology to brain physics.- 2 Meshing the intracranial compartments: Cerebellum, cerebrum, brainstem and cerebrospinal fluid.- 3 Segmenting, meshing and modeling CSF spaces.- 4 The pulsating brain: An interface-coupled fluid-poroelasticinteraction model of the cranial cavity.- 5 Quantifying cerebrospinal fluid tracer concentration in the brain.- 6 Signal increase ratio prediction with CNNs.- 7 Estimating molecular transport parameters using inverse PDEmodels.- 8 Two-compartment modeling of tracer transport in the brain.- 9 An introduction to identifying velocity fields from contrast imaging via PDE-constrained optimization. 10 An introduction to network models of neurodegenerative diseases.

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Book SynopsisPreface.- Notation.- I Introduction.- II Uniform distribution modulo one.- III QMC integration in reproducing kernel Hilbert spaces.- IV Lattice point sets.- V (t, m, s)-nets and (t, s)-sequences.- VI A short discussion of the discrepancy bounds.- VII Foundations of financial mathematics.- VIII MC and QMC simulation.

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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 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 SynopsisDesigned to present the chief results of the Lebesgue theory of integration, this textbook provides students with detailed coverage of the main ideas of the Lebesgue measure. The approach is particularly well suited for students of analysis, probability and statistics.Table of ContentsTHE ELEMENTS OF INTEGRATION. Measurable Functions. Measures. The Integral. Integrable Functions. The Lebesgue Spaces Lp. Modes of Convergence. Decomposition of Measures. Generation of Measures. Product Measures. THE ELEMENTS OF LEBESGUE MEASURE. Volumes of Cells and Intervals. The Outer Measure. Measurable Sets. Examples of Measurable Sets. Approximation of Measurable Sets. Additivity and Nonadditivity. Nonmeasurable and Non-Borel Sets. References. Index.

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Book SynopsisThis introduction to a wide range of theoretical studies in fluid dynamics, covers a great deal of material and offers updated information on topics such as stability and turbulence. It surveys nearly the entire field of classical fluid dynamics and discusses the various conceptual and analytical models of fluid flow.Trade Review"I know of no other modern book in theoretical fluid dynamics that covers so much material so well." (Physics Today, November 1998)Table of ContentsReview of Basic Concepts and Equations of Fluid Dynamics. Dynamics of Inviscid Incompressible Fluid Flows. Dynamics of Inviscid Compressible Fluid Flows. Dynamics of Viscous Fluid Flows. Hydrodynamic Stability. Dynamics of Turbulence. Bibliography. Index.

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Book SynopsisThis revised and updated volume describes methods fundamental to the theory and explanation of data analysis. This edition includes extensions and devices such as component and component-plus residual plots, cross-verification with a second sample and an index of required x-precision.Trade Review"...a grand historical document for industrial statistics in its glory days, as its selection for the Classics Library implies." --Technometrics Vol. 42, No. 4 May 2001 This book provides an excellent insight into the minds of two master craftsmen at work. I very much applaud the decision to include this in a "classics library" and would encourage more authors to produce statistics books in the same vein, i.e. focused on the practical application of the subject rather than methodology development. Anyone involved in the analysis of unbalanced multifactor dtaa will find this book an extremely useful source of practical advice. --The Statistician 50 (1) 2001.Table of ContentsAssumptions and Methods of Fitting Equations. One Independent Variable. Two or More Independent Variables. Fitting an Equation in Three Independent Variables. Selection of Independent Variables. Some Consequences of the Disposition of the Data Points. Selection of Variables in Nested Data. Nonlinear Least Squares, a Complex Example. Glossary. User's Manual. Bibliography. Index.

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Book SynopsisProvides an exploration of standard numerical analysis topics, as well as non-traditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. This textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering.Trade Review"Distinguishing features are the inclusion of many recent applications of numerical methods and the extensive discussion of methods based on Chebyshev interpolation. This book would be suitable for use in courses aimed at advanced undergraduate students in mathematics, the sciences, and engineering."--Choice "An instructor could assemble several different one-semester courses using this book--numerical linear algebra and interpolation, or numerical solutions of differential equations--or perhaps a two-semester sequence. This is a charming book, well worth consideration for the next numerical analysis course."--William J. Satzer, MAA FocusTable of ContentsPreface xiii Chapter 1: MATHEMATICAL MODELING 1 1.1 Modeling in Computer Animation 2 1.1.1 A Model Robe 2 1.2 Modeling in Physics: Radiation Transport 4 1.3 Modeling in Sports 6 1.4 Ecological Models 8 1.5 Modeling a Web Surfer and Google 11 1.5.1 The Vector Space Model 11 1.5.2 Google's PageRank 13 1.6 Chapter 1 Exercises 14 Chapter 2: BASIC OPERATIONS WITH MATLAB 19 2.1 Launching MATLAB 19 2.2 Vectors 20 2.3 Getting Help 22 2.4 Matrices 23 2.5 Creating and Running .m Files 24 2.6 Comments 25 2.7 Plotting 25 2.8 Creating Your Own Functions 27 2.9 Printing 28 2.10 More Loops and Conditionals 29 2.11 Clearing Variables 31 2.12 Logging Your Session 31 2.13 More Advanced Commands 31 2.14 Chapter 2 Exercises 32 Chapter 3: MONTE CARLO METHODS 41 3.1 A Mathematical Game of Cards 41 3.1.1 The Odds in Texas Holdem 42 3.2 Basic Statistics 46 3.2.1 Discrete Random Variables 48 3.2.2 Continuous Random Variables 51 3.2.3 The Central Limit Theorem 53 3.3 Monte Carlo Integration 56 3.3.1 Buffon's Needle 56 3.3.2 Estimating pi 58 3.3.3 Another Example of Monte Carlo Integration 60 3.4 Monte Carlo Simulation of Web Surfing 64 3.5 Chapter 3 Exercises 67 Chapter 4: SOLUTION OF A SINGLE NONLINEAR EQUATION IN ONE UNKNOWN 71 4.1 Bisection 75 4.2 Taylor's Theorem 80 4.3 Newton's Method 83 4.4 Quasi-Newton Methods 89 4.4.1 Avoiding Derivatives 89 4.4.2 Constant Slope Method 89 4.4.3 Secant Method 90 4.5 Analysis of Fixed Point Methods 93 4.6 Fractals, Julia Sets, and Mandelbrot Sets 98 4.7 Chapter 4 Exercises 102 Chapter 5: FLOATING-POINT ARITHMETIC 107 5.1 Costly Disasters Caused by Rounding Errors 108 5.2 Binary Representation and Base 2 Arithmetic 110 5.3 Floating-Point Representation 112 5.4 IEEE Floating-Point Arithmetic 114 5.5 Rounding 116 5.6 Correctly Rounded Floating-Point Operations 118 5.7 Exceptions 119 5.8 Chapter 5 Exercises 120 Chapter 6: CONDITIONING OF PROBLEMS; STABILITY OF ALGORITHMS 124 6.1 Conditioning of Problems 125 6.2 Stability of Algorithms 126 6.3 Chapter 6 Exercises 129 Chapter 7: DIRECT METHODS FOR SOLVING LINEAR SYSTEMS AND LEAST SQUARES PROBLEMS 131 7.1 Review of Matrix Multiplication 132 7.2 Gaussian Elimination 133 7.2.1 Operation Counts 137 7.2.2 LU Factorization 139 7.2.3 Pivoting 141 7.2.4 Banded Matrices and Matrices for Which Pivoting Is Not Required 144 7.2.5 Implementation Considerations for High Performance 148 7.3 Other Methods for Solving Ax = b 151 7.4 Conditioning of Linear Systems 154 7.4.1 Norms 154 7.4.2 Sensitivity of Solutions of Linear Systems 158 7.5 Stability of Gaussian Elimination with Partial Pivoting 164 7.6 Least Squares Problems 166 7.6.1 The Normal Equations 167 7.6.2 QR Decomposition 168 7.6.3 Fitting Polynomials to Data 171 7.7 Chapter 7 Exercises 175 Chapter 8: POLYNOMIAL AND PIECEWISE POLYNOMIAL INTERPOLATION 181 8.1 The Vandermonde System 181 8.2 The Lagrange Form of the Interpolation Polynomial 181 8.3 The Newton Form of the Interpolation Polynomial 185 8.3.1 Divided Differences 187 8.4 The Error in Polynomial Interpolation 190 8.5 Interpolation at Chebyshev Points and chebfun 192 8.6 Piecewise Polynomial Interpolation 197 8.6.1 Piecewise Cubic Hermite Interpolation 200 8.6.2 Cubic Spline Interpolation 201 8.7 Some Applications 204 8.8 Chapter 8 Exercises 206 Chapter 9: NUMERICAL DIFFERENTIATION AND RICHARDSON EXTRAPOLATION 212 9.1 Numerical Differentiation 213 9.2 Richardson Extrapolation 221 9.3 Chapter 9 Exercises 225 Chapter 10: NUMERICAL INTEGRATION 227 10.1 Newton-Cotes Formulas 227 10.2 Formulas Based on Piecewise Polynomial Interpolation 232 10.3 Gauss Quadrature 234 10.3.1 Orthogonal Polynomials 236 10.4 Clenshaw-Curtis Quadrature 240 10.5 Romberg Integration 242 10.6 Periodic Functions and the Euler-Maclaurin Formula 243 10.7 Singularities 247 10.8 Chapter 10 Exercises 248 Chapter 11: NUMERICAL SOLUTION OF THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS 251 11.1 Existence and Uniqueness of Solutions 253 11.2 One-Step Methods 257 11.2.1 Euler's Method 257 11.2.2 Higher-Order Methods Based on Taylor Series 262 11.2.3 Midpoint Method 262 11.2.4 Methods Based on Quadrature Formulas 264 11.2.5 Classical Fourth-Order Runge-Kutta and Runge-Kutta-Fehlberg Methods 265 11.2.6 An Example Using MATLAB's ODE Solver 267 11.2.7 Analysis of One-Step Methods 270 11.2.8 Practical Implementation Considerations 272 11.2.9 Systems of Equations 274 11.3 Multistep Methods 275 11.3.1 Adams-Bashforth and Adams-Moulton Methods 275 11.3.2 General Linear m-Step Methods 277 11.3.3 Linear Difference Equations 280 11.3.4 The Dahlquist Equivalence Theorem 283 11.4 Stiff Equations 284 11.4.1 Absolute Stability 285 11.4.2 Backward Differentiation Formulas (BDF Methods) 289 11.4.3 Implicit Runge-Kutta (IRK) Methods 290 11.5 Solving Systems of Nonlinear Equations in Implicit Methods 291 11.5.1 Fixed Point Iteration 292 11.5.2 Newton's Method 293 11.6 Chapter 11 Exercises 295 Chapter 12: MORE NUMERICAL LINEAR ALGEBRA: EIGENVALUES AND ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS 300 12.1 Eigenvalue Problems 300 12.1.1 The Power Method for Computing the Largest Eigenpair 310 12.1.2 Inverse Iteration 313 12.1.3 Rayleigh Quotient Iteration 315 12.1.4 The QR Algorithm 316 12.1.5 Google's PageRank 320 12.2 Iterative Methods for Solving Linear Systems 327 12.2.1 Basic Iterative Methods for Solving Linear Systems 327 12.2.2 Simple Iteration 328 12.2.3 Analysis of Convergence 332 12.2.4 The Conjugate Gradient Algorithm 336 12.2.5 Methods for Nonsymmetric Linear Systems 334 12.3 Chapter 12 Exercises 345 Chapter 13: NUMERICAL SOLUTION OF TWO-POINT BOUNDARY VALUE PROBLEMS 350 13.1 An Application: Steady-State Temperature Distribution 350 13.2 Finite Difference Methods 352 13.2.1 Accuracy 354 13.2.2 More General Equations and Boundary Conditions 360 13.3 Finite Element Methods 365 13.3.1 Accuracy 372 13.4 Spectral Methods 374 13.5 Chapter 13 Exercises 376 Chapter 14: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 379 14.1 Elliptic Equations 381 14.1.1 Finite Difference Methods 381 14.1.2 Finite Element Methods 386 14.2 Parabolic Equations 388 14.2.1 Semidiscretization and the Method of Lines 389 14.2.2 Discretization in Time 389 14.3 Separation of Variables 396 14.3.1 Separation of Variables for Difference Equations 400 14.4 Hyperbolic Equations 402 14.4.1 Characteristics 402 14.4.2 Systems of Hyperbolic Equations 403 14.4.3 Boundary Conditions 404 14.4.4 Finite Difference Methods 404 14.5 Fast Methods for Poisson's Equation 409 14.5.1 The Fast Fourier Transform 411 14.6 Multigrid Methods 414 14.7 Chapter 14 Exercises 418 APPENDIX A REVIEW OF LINEAR ALGEBRA 421 A.1 Vectors and Vector Spaces 421 A.2 Linear Independence and Dependence 422 A.3 Span of a Set of Vectors; Bases and Coordinates; Dimension of a Vector Space 423 A.4 The Dot Product; Orthogonal and Orthonormal Sets; the Gram-Schmidt Algorithm 423 A.5 Matrices and Linear Equations 425 A.6 Existence and Uniqueness of Solutions; the Inverse; Conditions for Invertibility 427 A.7 Linear Transformations; the Matrix of a Linear Transformation 431 A.8 Similarity Transformations; Eigenvalues and Eigenvectors 432 APPENDIX B TAYLOR'S THEOREM IN MULTIDIMENSIONS 436 References 439 Index 445

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Book SynopsisDescribes in monograph form important applications in numerical methods of linear algebra. This book studies the behavior of the resolvent of a matrix under the perturbation by low rank matrices. It also introduces the basics of value distribution theory of meromorphic scalar functions.Table of ContentsPrologue Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Epilogue Bibliography.

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Book SynopsisUseful in analyzing electromagnetic problems in a variety of engineering circumstances, the finite element method is a powerful simulation technique. This book explains the method's processes and techniques in careful, meticulous prose. It covers not only essential finite element method theory, but also its latest developments and applications.Table of ContentsPreface xix Preface to the First Edition xxiii Preface to the Second Edition xxvii 1 Basic Electromagnetic Theory 1 1.1 Brief Review of Vector Analysis 2 1.2 Maxwell's Equations 4 1.3 Scalar and Vector Potentials 6 1.4 Wave Equations 7 1.5 Boundary Conditions 8 1.6 Radiation Conditions 11 1.7 Fields in an Infinite Homogeneous Medium 11 1.8 Huygen's Principle 13 1.9 Radar Cross Sections 14 1.10 Summary 15 2 Introduction to the Finite Element Method 17 2.1 Classical Methods for Boundary-Value Problems 17 2.2 Simple Example 21 2.3 Basic Steps of the Finite Element Method 27 2.4 Alternative Presentation of the Finite Element Formulation 34 2.5 Summary 36 3 One-Dimensional Finite Element Analysis 39 3.1 Boundary-Value Problem 39 3.2 Variational Formulation 40 3.3 Finite Element Analysis 42 3.4 Plane-Wave Reflection by a Metal-Backed Dielectric Slab 53 3.5 Scattering by a Smooth, Convex Impedance Cylinder 59 3.6 Higher-Order Elements 62 3.7 Summary 74 4 Two-Dimensional Finite Element Analysis 77 4.1 Boundary-Value Problem 77 4.2 Variational Formulation 79 4.3 Finite Element Analysis 81 4.4 Application to Electrostatic Problems 98 4.5 Application to Magnetostatic Problems 103 4.6 Application to Quasistatic Problems: Analysis of Multiconductor Transmission Lines 105 4.7 Application to Time-Harmonic Problems 109 4.8 Higher-Order Elements 128 4.9 Isoparametric Elements 144 4.10 Summary 149 5 Three-Dimensional Finite Element Analysis 151 5.1 Boundary-Value Problem 151 5.2 Variational Formulation 152 5.3 Finite Element Analysis 153 5.4 Higher-Order Elements 160 5.5 Isoparametric Elements 162 5.6 Application to Electrostatic Problems 168 5.7 Application to Magnetostatic Problems 169 5.8 Application to Time-Harmonic Field Problems 176 5.9 Summary 188 6 Variational Principles for Electromagnetics 191 6.1 Standard Variational Principle 192 6.2 Modified Variational Principle 197 6.3 Generalized Variational Principle 201 6.4 Variational Principle for Anisotrpic Medium 203 6.5 Variational Principle for Resistive Sheets 207 6.6 Concluding Remarks 209 7 Eigenvalue Problems: Waveguides and Cavities 211 7.1 Scalar Formulations for Closed Waveguides 212 7.2 Vector Formulations for Closed Waveguides 225 7.3 Open Waveguides 235 7.4 Three-Dimensional Cavities 238 7.5 Summary 239 8 Vector Finite Elements 243 8.1 Two-Dimensional Edge Elements 244 8.2 Waveguide Problem Revisited 256 8.3 Three-Dimensional Edge Elements 259 8.4 Cavity Problem Revisited 270 8.5 Waveguide Discontinuities 274 8.6 Higher-Order Interpolatory Vector Elements 278 8.7 Higher-Order Hierarchical Vector Elements 293 8.8 Computational Issues 305 8.9 Summary 309 9 Absorbing Boundary Conditions 315 9.1 Two-Dimensional Absorbing Boundary Conditions 316 9.2 Three-Dimensional Absorbing Boundary Conditions 323 9.3 Scattering Analysis Using Absorbing Boundary Conditons 328 9.4 Adaptive Absorbing Boundary Conditons 339 9.5 Fictitious Absorbers 348 9.6 Perfectly Matched Layers 350 9.7 Application of PML to Body-of-Revolutions Problems 368 9.8 Summary 371 10 Finite Element-Boundary Integral Methods 379 10.1 Scattering by Two-Dimensional Cavity-Backed Apertures 381 10.2 Scattering by Two-Dimensional Cylindrical Structures 399 10.3 Scattering by Three-Dimensional Cavity-Backed Apertures 411 10.4 Radiation by Microstrip Patch Antennas in a Cavity 425 10.5 Scattering by General Three-Dimensional Bodies 430 10.6 Solution of the Finite Element-Boundary Integral System 436 10.7 Symmetric Finite Element-Boundary Integral Formulations 447 10.8 Summary 462 11 Finite Element-Eigenfunction Expansion Methods 469 11.1 Waveguide Port Boundary Conditions 470 11.2 Open-Region Scattering 487 11.3 Coupled Basis Functions: The Unimoment Method 494 11.4 Finite Element-Extended Boundary Condition Method 502 11.5 Summary 509 12 Finite Element Analysis in the Time Domain 513 12.1 Finite Element Formulation and Temporal Excitation 514 12.2 Time-Domain Discretization 518 12.3 Stability Analysis 523 12.4 Modeling of Dispersive Media 529 12.5 Truncation via Absorbing Boundary Conditions 538 12.6 Truncation via Perfectly Matched Layers 541 12.7 Truncation via Boundary Integral Equations 551 12.8 Time-Domain Wqaveguide Port Boundary Conditions 562 12.9 Hybrid Field-Circuit Analysis 569 12.10 Dual-Field Domain Decomposition and Element-Level Methods 587 12.11 Discontinuous Galerkin Time-Domain Methods 605 12.12 Summary 625 13 Finite Element Analysis of Periodic Structures 637 13.1 Finite Element Formulation for a Unit Cell 638 13.2 Scattering by One-Dimensional Periodic Structures: Frequency-Domain Analysis 651 13.3 Scattering by One-Dimensional Periodic Structures: Time-Domain Analysis 656 13.4 Scattering by Two-Dimensional Periodic Structures: Frequency-Domain Analysis 663 13.5 Scattering by Two-Dimensonal Periodic Structures: Time-Domain Analysis 670 13.6 Analysis of Angular Periodic Strctures 678 13.7 Summary 682 14 Domain Decompsition for Large-Scale Analysis 687 14.1 Schwarz Methods 688 14.2 Schur Complement Methods 693 14.3 FETI-DP Method for Low-Frequency Problems 705 14.4 FETI-DP Method for High-Frequency Problems 728 14.5 Noncomformal FETI-DP Method Based on Cement Elements 743 14.6 Application of Second-Order Transmission Conditions 753 14.7 Summary 760 15 Solution of Finite Element Equations 767 15.1 Decomposition Methods 769 15.2 Conjugate Gradient Methods 778 15.3 Solution of Eigenvalue Problems 791 15.4 Fast Frequency-Sweep Computation 797 15.5 Summary 803 Appendix A: Basic Vector Identities and Integral Theorems 809 Appendix B: The Ritz Procedure for Complex-Valued Problems 813 Appendix C: Green's Functions 817 Appendix D: Singular Integral Evaluation 825 Appendix E: Some Special Functions 829 Index 837

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Book SynopsisHistory of Experimental Structural Mechanics.- Sensors .- Instrumentation.- Applied Digital Signal Processing.- Basic Measurements.- Structural Measurements.- Environmental Measurements.- Design of Tests.- Modal Parameter Estimation.- Modal Analysis of Rotating Systems.- Operating Modal Analysis.- Computational Methods in Structural Dynamics.- Finite/Boundary Element Modeling and Model Reduction.- FE Model Correlation.- Model Updating.- Damping of Materials and Stuctures.- Model Validation/Verification/Calibration.- Uncertainty Quantification and Statistical Issues.- Nonlinear System Analysis.- Rotating System Analysis.- Structural Health Monitoring and Damage Detection.- System Modeling.- Modal Modeling.- Impedance Modeling.- Acoustics of Structural Systems-VibroAcoustics.- Automotive Structural Testing.- Civil Structural Testing.- Aerospace Structural Testing.- Sports Equipment Testing.Table of ContentsHistory of Experimental Structural Mechanics.- Sensors .- Instrumentation.- Applied Digital Signal Processing.- Basic Measurements.- Structural Measurements.- Environmental Measurements.- Design of Tests.- Modal Parameter Estimation.- Modal Analysis of Rotating Systems.- Operating Modal Analysis.- Computational Methods in Structural Dynamics.- Finite/Boundary Element Modeling and Model Reduction.- FE Model Correlation.- Model Updating.- Damping of Materials and Stuctures.- Model Validation/Verification/Calibration.- Uncertainty Quantification and Statistical Issues.- Nonlinear System Analysis.- Rotating System Analysis.- Structural Health Monitoring and Damage Detection.- System Modeling.- Modal Modeling.- Impedance Modeling.- Acoustics of Structural Systems-VibroAcoustics.- Automotive Structural Testing.- Civil Structural Testing.- Aerospace Structural Testing.- Sports Equipment Testing.

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Book SynopsisPreliminary concepts.- Nonlinear Finite Element Analysis Procedure.- Finite Element Analysis for Nonlinear Elastic Systems.- Finite Element Analysis for Elastoplastic Problems.- Finite Element Analysis for Contact Problems. Table of ContentsPreliminary concepts.- Nonlinear Finite Element Analysis Procedure.- Finite Element Analysis for Nonlinear Elastic Systems.- Finite Element Analysis for Elastoplastic Problems.- Finite Element Analysis for Contact Problems.

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Book SynopsisSystems of linear equations are ubiquitous in numerical analysis and scientific computing. and iterative methods are indispensable for the numerical treatment of such systems. This book offers a rigorous introduction to fundamental iterative methods for systems of linear algebraic equations. The book distinguishes itself from other texts on the topic by providing a straightforward yet comprehensive analysis of the Krylov subspace methods, approaching the development and analysis of algorithms from various perspectives, and going beyond the standard description of iterative methods by connecting them in a natural way to the idea of preconditioning. The book supplements standard texts on numerical mathematics for first-year graduate and advanced undergraduate courses and is suitable for advanced graduate classes covering numerical linear algebra and Krylov subspace and multigrid iterative methods. It will be useful to researchers interested in numerical linear algebra and engineers who use iterative methods for solving large algebraic systems.Table of ContentsList of figures; List of algorithms; Preface; 1. Krylov subspace methods; 2. Toeplitz matrices and preconditioners; 3. Multigrid preconditioners; 4. Preconditioners by space decomposition; 5. Some applications; Bibliography; Index.

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Book Synopsis“If mathematical modeling is the process of turning real phenomena into mathematical abstractions, then numerical computation is largely about the transformation from abstract mathematics to concrete reality. Many science and engineering disciplines have long benefited from the tremendous value of the correspondence between quantitative information and mathematical manipulation.” -from the PrefaceFundamentals of Numerical Computation is an advanced undergraduate-level introduction to the mathematics and use of algorithms for the fundamental problems of numerical computation: linear algebra, finding roots, approximating data and functions, and solving differential equations. The book is organized with simpler methods in the first half and more advanced methods in the second half, allowing use for either a single course or a sequence of two courses. The authors take readers from basic to advanced methods, illustrating them with over 200 self-contained MATLAB functions and examples designed for those with no prior MATLAB experience. Although the text provides many examples, exercises, and illustrations, the aim of the authors is not to provide a cookbook per se, but rather an exploration of the principles of cooking.Professors Driscoll and Braun have developed an online resource that includes well-tested materials related to every chapter. Among these materials are lecture-related slides and videos, ideas for student projects, laboratory exercises, computational examples and scripts, and all the functions presented in the book.

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Book SynopsisThis book provides an elementary yet comprehensive introduction to the numerical solution of partial differential equations (PDEs). Used to model important phenomena, such as the heating of apartments and the behavior of electromagnetic waves, these equations have applications in engineering and the life sciences, and most can only be solved approximately using computers.Numerical Analysis of Partial Differential Equations Using Maple and MATLAB provides detailed descriptions of the four major classes of discretization methods for PDEs (finite difference method, finite volume method, spectral method, and finite element method) and runnable MATLAB® code for each of the discretization methods and exercises. It also gives self-contained convergence proofs for each method using the tools and techniques required for the general convergence analysis but adapted to the simplest setting to keep the presentation clear and complete.This book is intended for advanced undergraduate and early graduate students in numerical analysis and scientific computing and researchers in related fields. It is appropriate for a course on numerical methods for partial differential equations.

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Book SynopsisThe Portable, Extensible Toolkit for Scientific Computation (PETSc) is an open-source library of advanced data structures and methods for solving linear and nonlinear equations and for managing discretizations. This book uses these modern numerical tools to demonstrate how to solve nonlinear partial differential equations (PDEs) in parallel. It starts from key mathematical concepts, such as Krylov space methods, preconditioning, multigrid, and Newton’s method. In PETSc these components are composed at run time into fast solvers.Discretizations are introduced from the beginning, with an emphasis on finite difference and finite element methodologies. The example C programs of the first 12 chapters, listed on the inside front cover, solve (mostly) elliptic and parabolic PDE problems. Discretization leads to large, sparse, and generally nonlinear systems of algebraic equations. For such problems, mathematical solver concepts are explained and illustrated through the examples, with sufficient context to speed further development. PETSc for Partial Differential Equations addresses both discretizations and fast solvers for PDEs, emphasizing practice more than theory. Well-structured examples lead to run-time choices that result in high solver performance and parallel scalability. The last two chapters build on the reader’s understanding of fast solver concepts when applying the Firedrake Python finite element solver library. This textbook, the first to cover PETSc programming for nonlinear PDEs, provides an on-ramp for graduate students and researchers to a major area of high-performance computing for science and engineering. It is suitable as a supplement for courses in scientific computing or numerical methods for differential equations.

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Book SynopsisThis companion piece to the author’s 2018 book, A Software Repository for Orthogonal Polynomials, focuses on Gaussian quadrature and the related Christoffel function. The book makes Gauss quadrature rules of any order easily accessible for a large variety of weight functions and for arbitrary precision. It also documents and illustrates known as well as original approximations for Gauss quadrature weights and Christoffel functions.The repository contains 60 datasets, each dealing with a particular weight function. Included are classical, quasi-classical, and, most of all, nonclassical weight functions and associated orthogonal polynomials.

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Book SynopsisThis book presents the first survey of the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. The authors provide a careful error analysis, including previously unpublished results, and a complete implementation of the method in MATLAB. They also reveal how the LOD method relates to classical homogenization and domain decomposition. Illustrated with numerical experiments that demonstrate the significance of the method, the book is enhanced by a survey of applications including eigenvalue problems and evolution problems.Numerical Homogenization by Localized Orthogonal Decomposition is appropriate for graduate students in applied mathematics, numerical analysis, and scientific computing. Researchers in the field of computational partial differential equations will find this self-contained book of interest, as will applied scientists and engineers interested in multiscale simulation.

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Book SynopsisFit for students just starting to build a background in mathematics, this textbook provides an introduction to numerical methods for linear algebra problems.Introduction to Numerical Linear Algebra is ideal for a flipped classroom, as it provides detailed explanations that allow students to read on their own and instructors to go beyond lecturing, assumes that the reader has taken a course on linear algebra, but reviews background as needed, and covers several topics not commonly addressed in related introductory books, including diffusion, a toy model of computed tomography, global positioning systems, the use of eigenvalues in analyzing stability of equilibria, a detailed derivation and careful motivation of the QR method for eigenvalues starting from power iteration, a discussion of the use of the SVD for assigning grades, and multigrid methods. This textbook is appropriate for undergraduate and beginning graduate students in mathematics and related fields. It can be used in the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory

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Book SynopsisThis expansive volume describes the history of numerical methods proposed to solve linear algebra problems,, from antiquity to the present day. The authors focus on methods for solving linear systems of equations and eigenvalue problems and describe the interplay between numerical methods and the computing tools available at the time. The second part of the book consists of 78 biographies of the main important contributors to the field.A Journey through the History of Numerical Linear Algebra will be of special interest to applied mathematicians, especially researchers in numerical linear algebra, and to applied mathematiciansas well as to and historians of mathematics as well.

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Book SynopsisReduced order modeling is an important, growing field in computational science and engineering, and this is the first book to address the subject in relation to computational fluid dynamics. It focuses on complex parametrization of shapes for their optimization and includes recent developments in advanced topics such as turbulence, stability of flows, inverse problems, optimization, and flow control, as well as applications.This book will be of interest to researchers and graduate students in the field of reduced order modeling.

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Book SynopsisRounding Errors in Algebraic Processes was the first book to give systematic analyses of the effects of rounding errors on a variety of key computations involving polynomials and matrices.A detailed analysis is given of the rounding errors made in the elementary arithmetic operations and inner products, for both floating-point arithmetic and fixed-point arithmetic. The results are then applied in the error analyses of a variety of computations involving polynomials as well as the solution of linear systems, matrix inversion, and eigenvalue computations.The conditioning of these problems is investigated. The aim was to provide a unified method of treatment, and emphasis is placed on the underlying concepts.This book is intended for mathematicians, computer scientists, those interested in the historical development of numerical analysis, and students in numerical analysis and numerical linear algebra.Trade Review[This book] combines a rigorous mathematical analysis with a practicality that stems from an obvious first-hand contact with the actual numerical computation. The well-chosen examples alone show vividly both the importance of the study of rounding errors and the perils of its neglect. A. A. Grau, SIAM Review (1966)

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Book SynopsisDivided into two self-contained parts, this textbook is an introduction to modern real analysis. More than 350 exercises and 100 examples are integrated into the text to help clarify the theoretical considerations and the practical applications to differential geometry, Fourier series, differential equations, and other subjects. The first section of Classical Analysis of Real-Valued Functions covers the theorems of existence of supremum and infimum of bounded sets on the real line and the Lagrange formula for differentiable functions. Applications of these results are crucial for classical mathematical analysis, andmany are threaded through the text. In the second part of the book, the implicit function theorem plays a central role, while the Gauss–Ostrogradskii formula, surface integration, Heine–Borel lemma, the Ascoli–Arzelà theorem, and the one-dimensional indefinite Lebesgue integral are also covered. This book is intended for students in the first and second years of classical universities majoring in pure and applied mathematics, but students of engineering disciplines will also gain important and helpful insights. It is appropriate for courses in mathematical analysis, functional analysis, real analysis, and calculus and can be used for self-study as well.

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Book SynopsisTriangulations, and more precisely meshes, are at the heart of many problems relating to a wide variety of scientific disciplines, and in particular numerical simulations of all kinds of physical phenomena. In numerical simulations, the functional spaces of approximation used to search for solutions are defined from meshes, and in this sense these meshes play a fundamental role. This strong link between the meshes and functional spaces leads us to consider advanced simulation methods in which the meshes are adapted to the behaviors of the underlying physical phenomena. This book presents the basic elements of this meshing vision.Table of ContentsForeword 9 Introduction 11 Chapter 1 Finite Elements and Shape Functions 15 1.1. Basic concepts 15 1.2. Shape functions, complete elements 18 1.2.1. Generic expression of shape functions 18 1.2.2. Explicit expression for degrees 1–3 22 1.3. Shape functions, reduced elements 26 1.3.1. Simplices, triangles and tetrahedra 27 1.3.2. Tensor elements, quadrilateral and hexahedral elements 31 1.3.3. Other elements, prisms and pyramids 48 1.4. Shape functions, rational elements 49 1.4.1. Rational triangle with a degree of 2 or arbitrary degree 49 1.4.2. Rational quadrilateral of an arbitrary degree 50 1.4.3. General case, B-splines or Nurbs elements 50 Chapter 2 Lagrange and Bézier Interpolants 53 2.1. Lagrange–Bézier analogy 54 2.2. Lagrange functions expressed in Bézier forms 55 2.2.1. The case of tensors, natural coordinates 55 2.2.2. Simplicial case, barycentric coordinates 63 2.3. Bézier polynomials expressed in Lagrangian form 66 2.4. Application to curves 66 2.4.1. Bézier expression for a Lagrange curve 67 2.4.2. Lagrangian expression for a Bézier curve 70 2.5. Application to patches 71 2.5.1. Bézier expression for a patch in Lagrangian form 71 2.5.2. Lagrangian expression for a patch in Bézier form 73 2.6. Reduced elements 74 2.6.1. The tensor case, Bézier expression for a reduced Lagrangian patch 74 2.6.2. The tensor case, definition of reduced Bézier patches 82 2.6.3. The tensor case, Lagrangian expression of a reduced Bézier patch 90 2.6.4. The case of simplices 92 Chapter 3 Geometric Elements and Geometric Validity 95 3.1. Two-dimensional elements 96 3.2. Surface elements 105 3.3. Volumetric elements 105 3.4. Control points based on nodes 111 3.5. Reduced elements 115 3.5.1. Simplices, triangles and tetrahedra 115 3.5.2. Tensor elements, quadrilaterals and hexahedra 116 3.5.3. Other elements, prisms and pyramids 120 3.6. Rational elements 121 3.6.1. Shift from Lagrange rationals to Bézier rationals 121 3.6.2. Degree 2, working on the (arc of a) circle 121 3.6.3. Application to the analysis of rational elements 123 3.6.4. On the use of rational elements or more 138 Chapter 4 Triangulation 141 4.1. Triangulation, definitions, basic concepts and natural entities 142 4.1.1. Definitions and basic concepts 142 4.1.2. Natural entities 145 4.1.3. A ball (topological) of a vertex 145 4.1.4 A shell of a k-face 145 4.1.5 The ring of a k-face 146 4.2. Topology and local topological modifications 146 4.2.1. Flipping an edge in two dimensions 148 4.2.2. Flipping a face in three dimensions 148 4.2.3. Flipping an edge in three dimensions 148 4.2.4. Other flips? 150 4.3. Enriched data structures 151 4.3.1. Minimal structure 151 4.3.2. Enriched structure 152 4.4. Construction of natural entities 153 4.5. Triangulation, construction methods 156 4.6. The incremental method, a generic method 159 4.6.1. Naive triangulation 160 4.6.2. Delaunay triangulation 163 Chapter 5 Delaunay Triangulation 165 5.1. History 166 5.2. Definitions and properties 168 5.3. The incremental method for Delaunay 175 5.4. Other methods of construction 181 5.5. Variants 186 5.6. Anisotropy 188 Chapter 6 Triangulation and Constraints 193 6.1. Triangulation of a domain 194 6.1.1. Triangulation of a domain in two dimensions 195 6.1.2. Triangulation of a domain in three dimensions 202 6.2. Delaunay Triangulation “Delaunay admissibility” 214 6.3. Triangulation of a variety 219 6.4. Topological invariants (triangles and tetrahedra) 222 Chapter 7 Geometric Modeling: Methods 233 7.1. Implicit or explicit form (CAD), starting from an analytical definition 234 7.1.1. Modeling an implicit curve, continuous → discrete 234 7.1.2. Modeling a parametric curve 237 7.1.3. Modeling an implicit surface 238 7.1.4. Modeling of a parametric surface 242 7.2. Starting from a discretization or triangulation, discrete → continuous 246 7.2.1. Case of a curve 247 7.2.2. The case of a surface 253 7.3. Starting from a point cloud, discrete → discrete 278 7.3.1. The case of a curve in two dimensions 278 7.3.2. The case of a surface 283 7.4. Extraction of characteristic points and characteristic lines 302 Chapter 8 Geometric Modeling: Examples 305 8.1. Geometric modeling of parametric patches 306 8.2. Characteristic lines of a discrete surface 311 8.3. Parametrization of a surface patch through unfolding 311 8.4. Geometric simplification of a surface triangulation 324 8.5. Geometric support for a discrete surface 325 8.6. Discrete reconstruction of a digitized object or environment 330 Chapter 9 A Few Basic Algorithms and Formulae 343 9.1. Subdivision of an entity (De Casteljau) 344 9.1.1. Subdivision of a curve 344 9.1.2. Subdivision of a patch 345 9.2. Computing control coefficients (higher order elements) 348 9.3. Algorithms for the insertion of a point (Delaunay) 351 9.3.1. Classic algorithm 352 9.3.2. Modified algorithms 355 9.4. Construction of neighboring relationships, balls and shells 357 9.4.1. Neighboring relationships 357 9.4.2. Construction of the ball of a vertex 359 9.4.3. Construction of the shell of an edge 361 9.5. Localization problems 363 9.5.1. Triangulations or simplicial meshes 363 9.5.2. Other meshes 367 9.6. Some formulae 367 Conclusions and Perspectives 369 Bibliography 371 Index 377

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Book SynopsisThree-dimensional surface meshes are the most common discrete representation of the exterior of a virtual shape. Extracting relevant geometric or topological features from them can simplify the way objects are looked at, help with their recognition, and facilitate description and categorization according to specific criteria. This book adopts the point of view of discrete mathematics, the aim of which is to propose discrete counterparts to concepts mathematically defined in continuous terms. It explains how standard geometric and topological notions of surfaces can be calculated and computed on a 3D surface mesh, as well as their use for shape analysis. Several applications are also detailed, demonstrating that each of them requires specific adjustments to fit with generic approaches. The book is intended not only for students, researchers and engineers in computer science and shape analysis, but also numerical geologists, anthropologists, biologists and other scientists looking for practical solutions to their shape analysis, understanding or recognition problems.Table of ContentsPreface ix Introduction xi Chapter 1. Geometric Features based on Curvatures 1 1.1. Introduction 1 1.2. Some mathematical reminders of the differential geometry of surfaces 2 1.2.1. Fundamental forms and normal curvature 2 1.2.2. Principal curvatures and shape index 5 1.2.3. Principal directions and lines of curvature 6 1.2.4. Weingarten equations and shape operator 9 1.2.5. Practical computation of differential parameters 12 1.2.6. Euler’s theorem 13 1.2.7. Meusnier’s theorem 15 1.2.8. Local approximation of the surface 16 1.2.9. Focal surfaces 17 1.3. Computation of differential parameters on a discrete 3D mesh 19 1.3.1. Introduction 19 1.3.2. Some notations 19 1.3.3. Computing normal vectors 20 1.3.4. Locally fitting a parametric surface 22 1.3.5. Discrete differential geometry operators 22 1.3.6. Integrating 2D curvatures 28 1.3.7. Tensor of curvature: Taubin’s formula 28 1.3.8. Tensor of curvature based on the normal cycle theory 30 1.3.9. Integral estimators 34 1.3.10. Processing unstructured 3D point clouds 38 1.3.11. Discussion of the methods 38 1.4. Feature line extraction 46 1.4.1. Introduction 46 1.4.2. Lines of curvature 47 1.4.3. Crest/ridge lines 55 1.4.4. Feature lines based on homotopic thinning 79 1.5. Region-based approaches 84 1.5.1. Mesh segmentation 84 1.5.2. Shape description based on graphs 87 1.6. Conclusion 98 Chapter 2. Topological Features 99 2.1. Mathematical background 99 2.1.1. A topological view on surfaces 100 2.1.2. Algebraic topology 103 2.2. Computation of global topological features 106 2.2.1. Connected components and genus 106 2.2.2. Homology groups 107 2.3. Combining geometric and topological features 111 2.3.1. Persistent homology 112 2.3.2. Reeb graph and Morse–Smale complex 115 2.3.3. Homology generators 118 2.3.4. Measuring holes 121 2.4. Conclusion 128 Chapter 3. Applications 131 3.1. Introduction 131 3.2. Medicine: lines of curvature for polyp detection in virtual colonoscopy 131 3.3. Paleo-anthropology: crest/ridge lines for shape analysis of human fossils 133 3.4. Geology: extraction of fracture lines on virtual outcrops 137 3.5. Planetary science: detection of feature lines for the extraction of impact craters on asteroids and rocky planets 140 3.6. Botany: persistent homology to recover the branching structure of plants 143 Conclusion 145 References 149 Index 169

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Book SynopsisEarthquake occurrence modeling is a rapidly developing research area. This book deals with its critical issues, ranging from theoretical advances to practical applications. The introductory chapter outlines state-of-the-art earthquake modeling approaches based on stochastic models. Chapter 2 presents seismogenesis in association with the evolving stress field. Chapters 3 to 5 present earthquake occurrence modeling by means of hidden (semi-)Markov models and discuss associated characteristic measures and relative estimation aspects. Further comparisons, the most important results and our concluding remarks are provided in Chapters 6 and 7.Table of ContentsList of Abbreviations ix List of Symbols xi Preface xv Introduction xix Chapter 1. Fundamentals on Stress Changes 1 1.1. Introduction 1 1.2. Stress interaction 4 1.3. Stress changes calculation 12 1.4. Modeling of Coulomb stress changes for different faulting types 15 1.4.1.ΔCS for strike-slip faulting 15 1.4.2.ΔCS for dip-slip faulting 16 1.5. Seismicity triggered by stress transfer 21 1.5.1. Triggering of strong earthquakes 21 1.5.2. Aftershock triggering 23 1.5.3. Triggering of mining seismicity 28 1.6. Discussion on stress interaction 31 Chapter 2. Hidden Markov Models 35 2.1. Introduction 35 2.2. Hidden Markov framework 37 2.3. Seismotectonic regime and seismicity data 42 2.4. Application to earthquake occurrences 44 2.4.1. Two hidden states and three observation types 45 2.4.2. Three hidden states and three observation types 48 2.4.3. Model selection and simulation 50 2.4.4. Steps number for the first earthquake occurrence 53 2.5. Conclusion 54 Chapter 3. Hidden Markov Renewal Models 57 3.1. Introduction 57 3.2. Semi-Markov framework 58 3.3. Hidden Markov renewal framework 65 3.4. Modeling earthquakes in Greece 66 3.4.1. Hitting times and earthquake occurrence numbers 69 3.5. Conclusion 73 Chapter 4. Hitting Time Intensity 75 4.1. Introduction 75 4.2. DTIHT for semi-Markov chains 76 4.2.1. Statistical estimation of the DTIHT 78 4.3. DTIHT for hidden Markov renewal chains 83 4.3.1. Statistical estimation of the DTIHT 85 4.4. Conclusion 87 Chapter 5. Models Comparison 89 5.1. Introduction 89 5.2. Markov framework 90 5.2.1. HMM case 92 5.2.2. HMRM case 92 5.3. Markov renewal framework 93 5.3.1. HMM case 95 5.3.2. HMRM case 96 5.4. Conclusion 97 Discussion & Concluding Remarks 99 Appendices 105 Appendix 1 107 Appendix 2 113 Appendix 3 117 References 119 Index 137

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Book SynopsisThe purpose of this book is to introduce and study numerical methods basic and advanced ones for scientific computing. This last refers to the implementation of appropriate approaches to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or of engineering (mechanics of structures, mechanics of fluids, treatment signal, etc.). Each chapter of this book recalls the essence of the different methods resolution and presents several applications in the field of engineering as well as programs developed under Matlab software.Table of ContentsPreface ix Part 1. Solving Equations 1 Chapter 1. Solving Nonlinear Equations 3 1.1 Introduction 3 1.2 Separating the roots 3 1.3 Approximating a separated root 4 1.3.1 Bisection method (or dichotomy method) 4 1.3.2 Fixed-point method 6 1.3.3 First convergence criterion 7 1.3.4 Iterative stopping criteria.8 1.3.5 Second convergence criterion (local criterion) 9 1.3.6 Newton’s method (or the method of tangents) 10 1.3.7 Secant method 12 1.3.8 Regula falsi method (or false position method) 17 1.4 Order of an iterative process.19 1.5 Using Matlab 19 1.5.1 Finding the roots of polynomials 19 1.5.2 Bisection method 21 1.5.3 Newton’s method 22 Chapter 2. Numerically Solving Differential Equations 25 2.1 Introduction 25 2.2 Cauchy problem and discretization 27 2.3 Euler’s method 30 2.3.1 Interpretation 30 2.3.2 Convergence 30 2.4 One-step Runge–Kutta method 31 2.4.1 Second-order Runge–Kutta method 32 2.4.2 Fourth-order Runge–Kutta method 33 2.5 Multi-step Adams methods 36 2.5.1 Open Adams methods 36 2.5.2 Closed Adams formulas 39 2.6 Predictor–Corrector method.41 2.7 Using Matlab 43 Part 2. Solving PDEs 47 Chapter 3. Finite Difference Methods 49 3.1 Introduction 49 3.2 Presentation of the finite difference method 51 3.2.1 Convergence, consistency and stability 53 3.2.2 Courant–Friedrichs–Lewy condition 56 3.2.3 Von Neumann stability analysis 57 3.3 Hyperbolic equations 58 3.3.1 Key results 59 3.3.2 Numerical schemes for solving the transport equation 63 3.3.3 Wave equation 66 3.3.4 Burgers equation 68 3.4 Elliptic equations 72 3.4.1 Poisson equation 72 3.5 Parabolic equations 74 3.5.1 Heat equation 74 3.6 Using Matlab 76 Chapter 4. Finite Element Method 83 4.1 Introduction 83 4.2 One-dimensional finite element methods 83 4.3 Two-dimensional finite element methods 88 4.4 General procedure of the method 93 4.5 Finite element method for computing elastic structures 93 4.5.1 Linear elasticity 93 4.5.2 Variational formulation of the linear elasticity problem 97 4.5.3 Planar linear elasticity problems 99 4.5.4 Applying the finite element method to planar problems 101 4.5.5 Axisymmetric problems.105 4.5.6 Three-dimensional problems 107 4.6 Using Matlab 107 4.6.1 Solving Poisson’s equation 108 4.6.2 Solving the heat equation.111 4.6.3 Computing structures 112 Chapter 5. Finite Volume Methods 117 5.1 Introduction 117 5.2 Finite volume method (FVM) 118 5.2.1 Conservation properties of the method 118 5.2.2 The stages of the method.119 5.2.3 Convergence 120 5.2.4 Consistency 120 5.2.5 Stability 120 5.3 Advection schemes 121 5.3.1 Two-dimensional FVM. 126 5.3.2 Convection-diffusion equation 129 5.3.3 Central differencing scheme 131 5.3.4 Upwind (decentered) scheme 133 5.3.5 Hybrid scheme 136 5.3.6 Power-law scheme 136 5.3.7 QUICK scheme 137 5.3.8 Higher-order schemes 139 5.3.9 Unsteady one-dimensional convection-diffusion Equation 140 5.3.10 Explicit scheme 142 5.3.11 Crank–Nicolson scheme.142 5.3.12 Implicit scheme 143 5.4 Using Matlab 144 Chapter 6. Meshless Methods. 147 6.1 Introduction 147 6.2 Limitations of the FEM and motivation of meshless methods 148 6.3 Examples of meshless methods148 6.3.1 Advantages of meshless methods 149 6.3.2 Disadvantages of meshless methods150 6.3.3 Comparison of the finite element method and meshless methods 151 6.4 Basis of meshless methods 151 6.4.1 Approximations 151 6.4.2 Kernel (weight) functions.152 6.4.3 Completeness 152 6.4.4 Partition of unity 152 6.5 Meshless method (EFG) 153 6.5.1 Theory 153 6.5.2 Moving Least-Squares Approximation 153 6.6 Application of the meshless method to elasticity 163 6.6.1 Formulation of static linear elasticity 163 6.6.2 Imposing essential boundary conditions 165 6.7 Numerical examples 170 6.7.1 Fixed-free beam 170 6.7.2 Compressed block 171 6.8 Using Matlab 173 Part 3. Appendices 179 Appendix 1181 Appendix 2189 Bibliography 195 Index 199

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Book SynopsisThis book studies methods to concretely address inverse problems. An inverse problem arises when the causes that produced a given effect must be determined or when one seeks to indirectly estimate the parameters of a physical system. The author uses practical examples to illustrate inverse problems in physical sciences. He presents the techniques and specific methods chosen to solve inverse problems in a general domain of application, choosing to focus on a small number of methods that can be used in most applications. This book is aimed at readers with a mathematical and scientific computing background. Despite this, it is a book with a practical perspective. The methods described are applicable, have been applied, and are often illustrated by numerical examples.Trade Review"The book is very carefully written, in a reader-friendly style. It can be considered as an introductory textbook for the theory of ill-posed problems and their numerical solution." (Mathematical Reviews/MathSciNet 11/05/2017)Table of ContentsPreface ix Part 1. Introduction and Examples 1 Chapter 1. Overview of Inverse Problems 3 1.1. Direct and inverse problems 3 1.2. Well-posed and ill-posed problems 4 Chapter 2. Examples of Inverse Problems 9 2.1. Inverse problems in heat transfer 10 2.2. Inverse problems in hydrogeology 13 2.3. Inverse problems in seismic exploration 16 2.4. Medical imaging 21 2.5. Other examples 25 Part 2. Linear Inverse Problems 29 Chapter 3. Integral Operators and Integral Equations 31 3.1. Definition and first properties 31 3.2. Discretization of integral equations 36 3.2.1. Discretization by quadrature–collocation 36 3.2.2. Discretization by the Galerkin method 39 3.3. Exercises 42 Chapter 4. Linear Least Squares Problems – Singular Value Decomposition 45 4.1. Mathematical properties of least squares problems 45 4.1.1. Finite dimensional case 50 4.2. Singular value decomposition for matrices 52 4.3. Singular value expansion for compact operators 57 4.4. Applications of the SVD to least squares problems 60 4.4.1. The matrix case 60 4.4.2. The operator case 63 4.5. Exercises 65 Chapter 5. Regularization of Linear Inverse Problems 71 5.1. Tikhonov’s method 72 5.1.1. Presentation 72 5.1.2. Convergence 73 5.1.3. The L-curve 81 5.2. Applications of the SVE 83 5.2.1. SVE and Tikhonov’s method 84 5.2.2. Regularization by truncated SVE 85 5.3. Choice of the regularization parameter 88 5.3.1. Morozov’s discrepancy principle 88 5.3.2. The L-curve 91 5.3.3. Numerical methods 92 5.4. Iterative methods 94 5.5. Exercises 98 Part 3. Nonlinear Inverse Problems 103 Chapter 6. Nonlinear Inverse Problems – Generalities 105 6.1. The three fundamental spaces 106 6.2. Least squares formulation 111 6.2.1. Difficulties of inverse problems 114 6.2.2. Optimization, parametrization, discretization 114 6.3. Methods for computing the gradient – the adjoint state method 116 6.3.1. The finite difference method 116 6.3.2. Sensitivity functions 118 6.3.3. The adjoint state method 119 6.3.4. Computation of the adjoint state by the Lagrangian 120 6.3.5. The inner product test 123 6.4. Parametrization and general organization 123 6.5. Exercises 125 Chapter 7. Some Parameter Estimation Examples 127 7.1. Elliptic equation in one dimension 127 7.1.1. Computation of the gradient 128 7.2. Stationary diffusion: elliptic equation in two dimensions 129 7.2.1. Computation of the gradient: application of the general method 132 7.2.2. Computation of the gradient by the Lagrangian 134 7.2.3. The inner product test 135 7.2.4. Multiscale parametrization 135 7.2.5. Example 136 7.3. Ordinary differential equations 137 7.3.1. An application example 144 7.4. Transient diffusion: heat equation 147 7.5. Exercises 152 Chapter 8. Further Information 155 8.1. Regularization in other norms 155 8.1.1. Sobolev semi-norms 155 8.1.2. Bounded variation regularization norm 157 8.2. Statistical approach: Bayesian inversion 157 8.2.1. Least squares and statistics 158 8.2.2. Bayesian inversion 160 8.3. Other topics 163 8.3.1. Theoretical aspects: identifiability 163 8.3.2. Algorithmic differentiation . 163 8.3.3. Iterative methods and large-scale problems 164 8.3.4. Software 164 Appendices 167 Appendix 1 169 Appendix 2 183 Appendix 3 193 Bibliography 205 Index 213

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Book SynopsisThis textbook offers an accessible introduction to the theory and numerics of approximation methods, combining classical topics of approximation with recent advances in mathematical signal processing, and adopting a constructive approach, in which the development of numerical algorithms for data analysis plays an important role. The following topics are covered: * least-squares approximation and regularization methods * interpolation by algebraic and trigonometric polynomials * basic results on best approximations * Euclidean approximation * Chebyshev approximation * asymptotic concepts: error estimates and convergence rates * signal approximation by Fourier and wavelet methods * kernel-based multivariate approximation * approximation methods in computerized tomography Providing numerous supporting examples, graphical illustrations, and carefully selected exercises, this textbook is suitable for introductory courses, seminars, and distance learning programs on approximation for undergraduate students.Trade Review“This book is an excellent first course in approximation theory, covering all the aspects from theoretical results to practical methods, from discrete to continuous approximation, from univariate to multivariate. … The book is an excellent text for an undergraduate course in approximation methods. … this book is a very important textbook on approximation theory and its methods.” (Ana Cristina Matos, Mathematical Reviews, August, 2019)Table of Contents1 Introduction.- 2 Basic Methods and Numerical Analysis.- 3 Best Approximations.- 4 Euclidean Approximations.- 5 Chebyshev Approximations.- 6 Asymptotic Results.- 7 Basic Concepts of Signal Approximation.- 8 Kernel-Based Approximation.- 9 Computational Topology.- References.- Subject Index.- Name Index.

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Book SynopsisThis textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.Trade Review“The book … is intended ‘for a wide audience ranging from upper undergraduate to advanced graduate students in mathematics, physics, and more broadly engineering students, especially in computer science.’ … The text’s coverage is extensive, its exposition clear throughout, and the color illustrations helpful. The authors are also familiar with many texts at a comparable level and have drawn on them in several places to include some of the most insightful proofs already in the literature.” (Jer-Chin Chuang, MAA Reviews, October 4, 2021)“The book is intended for incremental study and covers both basic concepts and more advanced ones. The former are thoroughly supported with theory and examples, and the latter are backed up with extensive reading lists and references. … Thanks to its design and approach style this is a timely and much needed addition that enables interdisciplinary bridges and the discovery of new applications for differential geometry.” (Corina Mohorian, zbMATH 1453.53001, 2021)Table of Contents1. The Matrix Exponential; Some Matrix Lie Groups.- 2. Adjoint Representations and the Derivative of exp.- 3. Introduction to Manifolds and Lie Groups.- 4. Groups and Group Actions.- 5. The Lorentz Groups ⊛.- 6. The Structure of O(p,q) and SO(p, q).- 7. Manifolds, Tangent Spaces, Cotangent Spaces.- 8. Construction of Manifolds From Gluing Data ⊛.- 9. Vector Fields, Integral Curves, Flows.- 10. Partitions of Unity, Covering Maps ⊛.- 11. Basic Analysis: Review of Series and Derivatives.- 12. A Review of Point Set Topology.-13. Riemannian Metrics, Riemannian Manifolds.- 14. Connections on Manifolds.- 15. Geodesics on Riemannian Manifolds.- 16. Curvature in Riemannian Manifolds.- 17. Isometries, Submersions, Killing Vector Fields.- 18. Lie Groups, Lie Algebra, Exponential Map.- 19. The Derivative of exp and Dynkin's Formula ⊛.- 20. Metrics, Connections, and Curvature of Lie Groups.- 21. The Log-Euclidean Framework.- 22. Manifolds Arising from Group Actions.

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Book SynopsisThe book originates from the mini-symposium "Mathematical descriptions of traffic flow: micro, macro and kinetic models" organised by the editors within the ICIAM 2019 Congress held in Valencia, Spain, in July 2019. The book is composed of five chapters, which address new research lines in the mathematical modelling of vehicular traffic, at the cutting edge of contemporary research, including traffic automation by means of autonomous vehicles. The contributions span the three most representative scales of mathematical modelling: the microscopic scale of particles, the mesoscopic scale of statistical kinetic description and the macroscopic scale of partial differential equations.The work is addressed to researchers in the field.Table of ContentsM. Herty et al., Reconstruction of traffic speed distributions from kinetic models with uncertainties.- M. Herty et al., From kinetic to macroscopic models and back.- R. Ramadan et al., Structural Properties of the Stability of Jamitons.- C. Balzotti and E. Iacomini, Stop-and-go waves: A Microscopic and a Macroscopic Description.- F. A. Chiarello, An overview of non-local traffic flow models.

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Book SynopsisThis volume includes contributions from the 9th Parallel-in-Time (PinT) workshop, an annual gathering devoted to the field of time-parallel methods, aiming to adapt existing computer models to next-generation machines by adding a new dimension of scalability. As the latest supercomputers advance in microprocessing ability, they require new mathematical algorithms in order to fully realize their potential for complex systems. The use of parallel-in-time methods will provide dramatically faster simulations in many important areas, including biomedical (e.g., heart modeling), computational fluid dynamics (e.g., aerodynamics and weather prediction), and machine learning applications. Computational and applied mathematics is crucial to this progress, as it requires advanced methodologies from the theory of partial differential equations in a functional analytic setting, numerical discretization and integration, convergence analyses of iterative methods, and the development and implementation of new parallel algorithms. Therefore, the workshop seeks to bring together an interdisciplinary group of experts across these fields to disseminate cutting-edge research and facilitate discussions on parallel time integration methods. Table of ContentsTight two-level convergence of linear Parareal and MGRIT: Extensions and implications in practice (Southworth et al.).- A Parallel algorithm for solving linear parabolic evolution equations (van Venetië et al.).- Using performance analysis tools for a parallel-in-time integrator (Speck et al.).- Twelve Ways to Fool the Masses When Giving Parallel-In-Time Results (Götschel et al.).- IMEX Runge-Kutta Parareal for Non-Diffusive Equations (Buvoli et al.).

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