This book addresses the need for a fundamental understanding of the physical origin, the mathematical behavior and the numerical treatment of models which include microstructure. Leading scientists present their efforts involving mathematical analysis, numerical analysis, computational mechanics, material modelling and experiment. The mathematical analyses are based on methods from the calculus of variations, while in the numerical implementation global optimization algorithms play a central role. The modeling covers all length scales, from the atomic structure up to macroscopic samples. The development of the models ware guided by experiments on single and polycrystals and results will be checked against experimental data.

Feller Semigroups, Bernstein type Operators and Generalized Convexity Associated with Positive Projections.- Gregory's Rational Cubic Splines in Interpolation Subject to Derivative Obstacles.- Interpolation by Splines on Triangulations Oleg Davydov.- On the Use of Quasi-Newton Methods in DAE-Codes.- On the Regularity of Some Differential Operators.- Some Inequalities for Trigonometric Polynomials and their Derivatives.- Inf-Convolution and Radial Basis Functions.- On a Special Property of the Averaged Modulus for Functions of Bounded Variation.- A Simple Approach to the Variational Theory for Interpolation on Spheres.- Constants in Comonotone Polynomial Approximation - A Survey.- Will Ramanujan kill Baker-Gammel-Wills? (A Selective Survey of Pade Approximation).- Approximation Operators of Binomial Type.- Certain Results involving Gammaoperators.- Recent research at Cambridge on radial basis functions.- Representation of quasi-interpolants as differential operators and applications.- Native Hilbert Spaces for Radial Basis Functions I.- Adaptive Approximation with Walsh-similar Functions.- Dual Recurrence and Christoffel-Darboux-Type Formulas for Orthogonal Polynomials.- On Some Problems of Weighted Polynomial Approximation and Interpolation.- Asymptotics of derivatives of orthogonal polynomials based on generalized Jacobi weights. Some new theorems and applications.- List of participants.

This book introduces and analyzes the multigrid approach for the numerical solution of large sparse linear systems arising from the discretization of elliptic partial differential equations. Special attention is given to the powerful matrix-based-multigrid approach, which is particularly useful for problems with variable coefficients and nonsymmetric and indefinite problems. This approach applies not only to model problems on rectangular grids but also to more realistic applications with complicated grids and domains and discontinuous coefficients. Matrix-Based Multigrid can be used as a textbook in courses in numerical analysis, numerical linear algebra, and numerical PDEs at the advanced undergraduate and graduate levels in computer science, math, and applied math departments. The theory is written in simple algebraic terms and therefore requires preliminary knowledge in basic linear algebra and calculus only. Because it is self contained and includes useful exercises, the book is also suitable for self study by research students, researchers, engineers, and others interested in the numerical solution of partial differential equations.

Any financial asset that is openly traded has a market price. Except for extreme market conditions, market price may be more or less than a fair value. Fair value is likely to be some complicated function of the current intrinsic value of tangible or intangible assets underlying the claim and our assessment of the characteristics of the underlying assets with respect to the expected rate of growth, future dividends, volatility, and other relevant market factors. Some of these factors that affect the price can be measured at the time of a transaction with reasonably high accuracy. Most factors, however, relate to expectations about the future and to subjective issues, such as current management, corporate policies and market environment, that could affect the future financial performance of the underlying assets. Models are thus needed to describe the stochastic factors and environment, and their implementations inevitably require computational finance tools.

Computational Electromagnetics is a young and growing discipline, expanding as a result of the steadily increasing demand for software for the design and analysis of electrical devices. This book introduces three of the most popular numerical methods for simulating electromagnetic fields: the finite difference method, the finite element method and the method of moments. In particular it focuses on how these methods are used to obtain valid approximations to the solutions of Maxwell's equations, using, for example, "staggered grids" and "edge elements." The main goal of the book is to make the reader aware of different sources of errors in numerical computations, and also to provide the tools for assessing the accuracy of numerical methods and their solutions. To reach this goal, convergence analysis, extrapolation, von Neumann stability analysis, and dispersion analysis are introduced and used frequently throughout the book. Another major goal of the book is to provide students with enough practical understanding of the methods so they are able to write simple programs on their own. To achieve this, the book contains several MATLAB programs and detailed description of practical issues such as assembly of finite element matrices and handling of unstructured meshes. Finally, the book aims at making the students well-aware of the strengths and weaknesses of the different methods, so they can decide which method is best for each problem.

In thissecond edition, extensive computer projects are added as well as new material throughout.

Reviews of previous edition:

"The well-written monograph is devoted to students at the undergraduate level, but is also useful for practising engineers." (Zentralblatt MATH, 2007)"

This volume contains a collection of research and survey papers written by some of the most eminent mathematicians in the international community and is dedicated to Helmut Maier, whose own research has been groundbreaking and deeply influential to the field. Specific emphasis is given to topics regarding exponential and trigonometric sums and their behavior in short intervals, anatomy of integers and cyclotomic polynomials, small gaps in sequences of sifted prime numbers, oscillation theorems for primes in arithmetic progressions, inequalities related to the distribution of primes in short intervals, the Moebius function, Euler's totient function, the Riemann zeta function and the Riemann Hypothesis. Graduate students, research mathematicians, as well as computer scientists and engineers who are interested in pure and interdisciplinary research, will find this volume a useful resource. Contributors to this volume: Bill Allombert, Levent Alpoge, Nadine Amersi, Yuri Bilu, Regis de la Breteche, Christian Elsholtz, John B. Friedlander, Kevin Ford, Daniel A. Goldston, Steven M. Gonek, Andrew Granville, Adam J. Harper, Glyn Harman, D. R. Heath-Brown, Aleksandar Ivic, Geoffrey Iyer, Jerzy Kaczorowski, Daniel M. Kane, Sergei Konyagin, Dimitris Koukoulopoulos, Michel L. Lapidus, Oleg Lazarev, Andrew H. Ledoan, Robert J. Lemke Oliver, Florian Luca, James Maynard, Steven J. Miller, Hugh L. Montgomery, Melvyn B. Nathanson, Ashkan Nikeghbali, Alberto Perelli, Amalia Pizarro-Madariaga, Janos Pintz, Paul Pollack, Carl Pomerance, Michael Th. Rassias, Maksym Radziwill, Joel Rivat, Andras Sarkoezy, Jeffrey Shallit, Terence Tao, Gerald Tenenbaum, Laszlo Toth, Tamar Ziegler, Liyang Zhang.

This edited book aims at presenting current research activities in the field of robust variable-structure systems. The scope equally comprises highlighting novel methodological aspects as well as presenting the use of variable-structure techniques in industrial applications including their efficient implementation on hardware for real-time control. The target audience primarily comprises research experts in the field of control theory and nonlinear dynamics but the book may also be beneficial for graduate students.

Infotext]((Kurztext))These are the proceedings of the 7th International Conference on Hyperbolic Problems, held in Zurich in February 1998. The speakers and contributors have been rigorously selected and present the state of the art in this field. The articles, both theoretical and numerical, encompass a wide range of applications, such as nonlinear waves in solids, various computational fluid dynamics from small-scale combustion to relativistic astrophysical problems, multiphase phenomena and geometrical optics.

((Volltext))These proceedings contain, in two volumes, approximately one hundred papers presented at the conference on hyperbolic problems, which has focused to a large extent on the laws of nonlinear hyperbolic conservation. Two-fifths of the papers are devoted to mathematical aspects such as global existence, uniqueness, asymptotic behavior such as large time stability, stability and instabilities of waves and structures, various limits of the solution, the Riemann problem and so on. Roughly the same number of articles are devoted to numerical analysis, for example stability and convergence of numerical schemes, as well as schemes with special desired properties such as shock capturing, interface fitting and high-order approximations to multidimensional systems. The results in these contributions, both theoretical and numerical, encompass a wide range of applications such as nonlinear waves in solids, various computational fluid dynamics from small-scale combustion to relativistic astrophysical problems, multiphase phenomena and geometrical optics."

The approximation of a continuous function by either an algebraic polynomial, a trigonometric polynomial, or a spline, is an important issue in application areas like computer-aided geometric design and signal analysis. This book is an introduction to the mathematical analysis of such approximation, and, with the prerequisites of only calculus and linear algebra, the material is targeted at senior undergraduate level, with a treatment that is both rigorous and self-contained. The topics include polynomial interpolation; Bernstein polynomials and the Weierstrass theorem; best approximations in the general setting of normed linear spaces and inner product spaces; best uniform polynomial approximation; orthogonal polynomials; Newton-Cotes, Gauss and Clenshaw-Curtis quadrature; the Euler-Maclaurin formula; approximation of periodic functions; the uniform convergence of Fourier series; spline approximation, with an extensive treatment of local spline interpolation, and its application in quadrature. Exercises are provided at the end of each chapter

Investigation of vortex wakes behind various aircraft, especially behind wide bodied and heavy cargo ones, is of both scientific and practical in terest. The vortex wakes shed from the wing's trailing edge are long lived and attenuate only atdistances of10-12kmbehindthe wake generating aircraft. The encounter of other aircraft with the vortex wake of a heavy aircraft is open to catastrophic hazards. For example, air refueling is adangerous operationpartly due to thepossibility of the receiver aircraft's encountering the trailing wake of the tanker aircraft. It is very important to know the behavior of vortex wakes of aircraft during theirtakeoff andlanding operations whenthe wakes canpropagate over the airport's ground surface and be a serious hazard to other depart ing or arriving aircraft. This knowledge can help in enhancing safety of aircraft's movements in the terminal areas of congested airports where the threat of vortex encounters limits passenger throughput. Theoreticalinvestigations of aircraft vortex wakes arebeingintensively performedinthe major aviationnations.Usedforthispurpose are various methods for mathematical modeling of turbulent flows: direct numerical simulation based on the Navier-Stokes equations, large eddy simulation using the Navier-Stokes equations in combination with subrigid scale modeling, simulation based on the Reynolds equations closed with a differential turbulence model. These approaches are widely used in works of Russian and other countries' scientists. It should be emphasized that the experiments in wind tunnels and studies of natural vortex wakes behind heavy and light aircraft in flight experiments are equally important.

This book explores the impact of augmenting novel architectural designs with hardware-based application accelerators. The text covers comprehensive aspects of the applications in Geographic Information Science, remote sensing and deploying Modern Accelerator Technologies (MAT) for geospatial simulations and spatiotemporal analytics. MAT in GIS applications, MAT in remotely sensed data processing and analysis, heterogeneous processors, many-core and highly multi-threaded processors and general purpose processors are also presented. This book includes case studies and closes with a chapter on future trends. Modern Accelerator Technologies for GIS is a reference book for practitioners and researchers working in geographical information systems and related fields. Advanced-level students in geography, computational science, computer science and engineering will also find this book useful.

This text is a self-contained Second Edition, providing an introductory account of the main topics in numerical analysis. The book emphasizes both the theorems which show the underlying rigorous mathematics andthe algorithms which define precisely how to program the numerical methods. Both theoretical and practical examples are included.

* a unique blend of theory and applications
* two brand new chapters on eigenvalues and splines
* inclusion of formal algorithms
* numerous fully worked examples
* a large number of problems, many with solutions

This book introduces the concepts and methodologies related to the modelling of the complex phenomena occurring in materials processing. After a short reminder of conservation laws and constitutive relationships, the authors introduce the main numerical methods: finite differences, finite volumes and finite elements. These techniques are developed in three main chapters of the book that tackle more specific problems: phase transformation, solid mechanics and fluid flow. The two last chapters treat inverse methods to obtain the boundary conditions or the material properties and stochastic methods for microstructural simulation. This book is intended for undergraduate and graduate students in materials science and engineering, mechanical engineering and physics and for engineering professionals or researchers who want to get acquainted with numerical simulation to model and compute materials processing.

This book provides a snapshot of the state of the art of the rapidly evolving field of integration of geometric data in finite element computations. The contributions to this volume, based on research presented at the UCL workshop on the topic in January 2016, include three review papers on core topics such as fictitious domain methods for elasticity, trace finite element methods for partial differential equations defined on surfaces, and Nitsche's method for contact problems. Five chapters present original research articles on related theoretical topics, including Lagrange multiplier methods, interface problems, bulk-surface coupling, and approximation of partial differential equations on moving domains. Finally, two chapters discuss advanced applications such as crack propagation or flow in fractured poroelastic media. This is the first volume that provides a comprehensive overview of the field of unfitted finite element methods, including recent techniques such as cutFEM, traceFEM, ghost penalty, and augmented Lagrangian techniques. It is aimed at researchers in applied mathematics, scientific computing or computational engineering.

This volume presents original research contributed to the 3rd Annual International Conference on Computational Mathematics and Computational Geometry (CMCGS 2014), organized and administered by Global Science and Technology Forum (GSTF). Computational Mathematics and Computational Geometry are closely related subjects, but are often studied by separate communities and published in different venues. This volume is unique in its combination of these topics. After the conference, which took place in Singapore, selected contributions chosen for this volume and peer-reviewed. The section on Computational Mathematics contains papers that are concerned with developing new and efficient numerical algorithms for mathematical sciences or scientific computing. They also cover analysis of such algorithms to assess accuracy and reliability. The parts of this project that are related to Computational Geometry aim to develop effective and efficient algorithms for geometrical applications such as representation and computation of surfaces. Other sections in the volume cover Pure Mathematics and Statistics ranging from partial differential equations to matrix analysis, finite difference or finite element methods and function approximation. This volume will appeal to advanced students and researchers in these areas.

Stochastic numerical methods play an important role in large scale computations in the applied sciences. The first goal of this book is to give a mathematical description of classical direct simulation Monte Carlo (DSMC) procedures for rarefied gases, using the theory of Markov processes as a unifying framework. The second goal is a systematic treatment of an extension of DSMC, called stochastic weighted particle method. This method includes several new features, which are introduced for the purpose of variance reduction (rare event simulation). Rigorous convergence results as well as detailed numerical studies are presented.

With contributions by specialists in optimization and practitioners in the fields of aerospace engineering, chemical engineering, and fluid and solid mechanics, the major themes include an assessment of the state of the art in optimization algorithms as well as challenging applications in design and control, in the areas of process engineering and systems with partial differential equation models.

Inverse problems and optimal design have come of age as a consequence of the availability of better, more accurate, and more efficient simulation packages. Many of these simulators, which can run on small workstations, can capture the complicated behavior of the physical systems they are modeling, and have become commonplace tools in engineering and science. There is a great desire to use them as part of a process by which measured field data are analyzed or by which design of a product is automated. A major obstacle in doing precisely this is that one is ultimately confronted with a large-scale optimization problem. This volume contains expository articles on both inverse problems and design problems formulated as optimization. Each paper describes the physical problem in some detail and is meant to be accessible to researchers in optimization as well as those who work in applied areas where optimization is a key tool. What emerges in the presentations is that there are features about the problem that must be taken into account in posing the objective function, and in choosing an optimization strategy. In particular there are certain structures peculiar to the problems that deserve special treatment, and there is ample opportunity for parallel computation. THIS IS BACK COVER TEXT Inverse problems and optimal design have come of age as a consequence of the availability of better, more accurate, and more efficient, simulation packages. The problem of determining the parameters of a physical system from

In this book the author sets out to answer two important questions: 1. Which numerical methods may be combined together? 2. How can different numerical methods be matched together? In doing so the author presents a number of useful combinations, for instance, the combination of various FEMs, the combinations of FEM-FDM, REM-FEM, RGM-FDM, etc. The combined methods have many advantages over single methods: high accuracy of solutions, less CPU time, less computer storage, easy coupling with singularities as well as the complicated boundary conditions. Since coupling techniques are essential to combinations, various matching strategies among different methods are carefully discussed. The author provides the matching rules so that optimal convergence, even superconvergence, and optimal stability can be achieved, and also warns of the matching pitfalls to avoid. Audience: The book is intended for both mathematicians and engineers and may be used as text for advanced students.

This volume, dedicated to the eminent mathematician Vladimir Arnold, presents a collection of research and survey papers written on a large spectrum of theories and problems that have been studied or introduced by Arnold himself. Emphasis is given to topics relating to dynamical systems, stability of integrable systems, algebraic and differential topology, global analysis, singularity theory and classical mechanics. A number of applications of Arnold's groundbreaking work are presented. This publication will assist graduate students and research mathematicians in acquiring an in-depth understanding and insight into a wide domain of research of an interdisciplinary nature.

This specialized and authoritative book contains an overview of modern approaches to constructing approximations to solutions of ill-posed operator equations, both linear and nonlinear. These approximation schemes form a basis for implementable numerical algorithms for the stable solution of operator equations arising in contemporary mathematical modeling, and in particular when solving inverse problems of mathematical physics. The book presents in detail stable solution methods for ill-posed problems using the methodology of iterative regularization of classical iterative schemes and the techniques of finite dimensional and finite difference approximations of the problems under study. Special attention is paid to ill-posed Cauchy problems for linear operator differential equations and to ill-posed variational inequalities and optimization problems. The readers are expected to have basic knowledge in functional analysis and differential equations. The book will be of interest to applied mathematicians and specialists in mathematical modeling and inverse problems, and also to advanced students in these fields. Contents Introduction Regularization Methods For Linear Equations Finite Difference Methods Iterative Regularization Methods Finite-Dimensional Iterative Processes Variational Inequalities and Optimization Problems

With contributions by specialists in optimization and practitioners in the fields of aerospace engineering, chemical engineering, and fluid and solid mechanics, the major themes include an assessment of the state of the art in optimization algorithms as well as challenging applications in design and control, in the areas of process engineering and systems with partial differential equation models.

This book uses numerical analysis as the main tool to investigate methods in machine learning and neural networks. The efficiency of neural network representations for general functions and for polynomial functions is studied in detail, together with an original description of the Latin hypercube method and of the ADAM algorithm for training. Furthermore, unique features include the use of Tensorflow for implementation session, and the description of on going research about the construction of new optimized numerical schemes.

The Bia owie a workshops on Geometric Methods in Physics, taking place in the unique environment of the Bia owie a natural forest in Poland, are among the important meetings in the field. Every year some 80 to 100 participants both from mathematics and physics join to discuss new developments and to interchange ideas. The current volume was produced on the occasion of the XXXI meeting in 2012. For the first time the workshop was followed by a School on Geometry and Physics, which consisted of advanced lectures for graduate students and young researchers. Selected speakers of the workshop were asked to contribute, and additional review articles were added. The selection shows that despite its now long tradition the workshop remains always at the cutting edge of ongoing research. The XXXI workshop had as a special topic the works of the late Boris Vasilievich Fedosov (1938 2011) who is best known for a simple and very natural construction of a deformation quantization for any symplectic manifold, and for his contributions to index theory.

The theory of Vector Optimization is developed by a systematic usage of infimum and supremum. In order to get existence and appropriate properties of the infimum, the image space of the vector optimization problem is embedded into a larger space, which is a subset of the power set, in fact, the space of self-infimal sets. Based on this idea we establish solution concepts, existence and duality results and algorithms for the linear case. The main advantage of this approach is the high degree of analogy to corresponding results of Scalar Optimization. The concepts and results are used to explain and to improve practically relevant algorithms for linear vector optimization problems.