Bücher der Reihe Applied Mathematical Sciences

This book is devoted to the study of the acoustic wave equation and of the Maxwell system, the two most common wave equations encountered in physics or in engineering. The main goal is to present a detailed analysis of their mathematical and physical properties. Wave equations are time dependent. However, use of the Fourier trans- form reduces their study to that of harmonic systems: the harmonic Helmholtz equation, in the case of the acoustic equation, or the har- monic Maxwell system. This book concentrates on the study of these harmonic problems, which are a first step toward the study of more general time-dependent problems. In each case, we give a mathematical setting that allows us to prove existence and uniqueness theorems. We have systematically chosen the use of variational formulations related to considerations of physical energy. We study the integral representations of the solutions. These representa- tions yield several integral equations. We analyze their essential properties. We introduce variational formulations for these integral equations, which are the basis of most numerical approximations. Different parts of this book were taught for at least ten years by the author at the post-graduate level at Ecole Poly technique and the University of Paris 6, to students in applied mathematics. The actual presentation has been tested on them. I wish to thank them for their active and constructive participation, which has been extremely useful, and I apologize for forcing them to learn some geometry of surfaces.

In the last decades, algorithmic advances as well as hardware and software improvements have provided an excellent environment to create and develop solving methods to hard optimization problems.

This monograph is devoted to Eulerian models for fluid-structure interaction by applying the original point of view of level set methods.In the last 15 years, Eulerian models have become popular tools for studying fluid-structure interaction problems. One major advantage compared to more conventional methods such as ALE methods is that they allow the use of a single grid and a single discretization method for the different media. Level set methods in addition provide a general framework to follow the fluid-solid interfaces, to represent the elastic stresses of solids, and to model the contact forces between solids.This book offers a combination of mathematical modeling, aspects of numerical analysis, elementary codes and numerical illustrations, providing the reader with insights into the applications and performance of these models.Assuming background at the level of a Master's degree, Level Set Methods for Fluid-Structure Interaction provides researchers in the fields of numerical analysis of PDEs, theoretical and computational mechanics with a basic reference on the topic. Its pedagogical style and organization make it particularly suitable for graduate students and young researchers.

This book is intended to provide graduate students and researchers in graph theory with an overview of the elementary methods of graph Ramsey theory. It is especially targeted towards graduate students in extremal graph theory, graph Ramsey theory, and related fields, as the included contents allow the text to be used in seminars. It is structured in thirteen chapters which are application-focused and largely independent, enabling readers to target specific topics and information to focus their study. The first chapter includes a true beginner's overview of elementary examples in graph Ramsey theory mainly using combinatorial methods. The following chapters progress through topics including the probabilistic methods, algebraic construction, regularity method, but that's not all. Many related interesting topics are also included in this book, such as the disproof for a conjecture of Borsuk on geometry, intersecting hypergraphs, Turan numbers and communication channels, etc.

This book discusses numerical methods for solving time-fractional evolution equations. The approach is based on first discretizing in the spatial variables by the Galerkin finite element method, using piecewise linear trial functions, and then applying suitable time stepping schemes, of the type either convolution quadrature or finite difference. The main concern is on stability and error analysis of approximate solutions, efficient implementation and qualitative properties, under various regularity assumptions on the problem data, using tools from semigroup theory and Laplace transform. The book provides a comprehensive survey on the present ideas and methods of analysis, and it covers most important topics in this active area of research. It is recommended for graduate students and researchers in applied and computational mathematics, particularly numerical analysis.

Inverse scattering theory is an important area of applied mathematics due to its central role in such areas as medical imaging , nondestructive testing and geophysical exploration. Until recently all existing algorithms for solving inverse scattering problems were based on using either a weak scattering assumption or on the use of nonlinear optimization techniques. The limitations of these methods have led in recent years to an alternative approach to the inverse scattering problem which avoids the incorrect model assumptions inherent in the use of weak scattering approximations as well as the strong a priori information needed in order to implement nonlinear optimization techniques. These new methods come under the general title of qualitative methods in inverse scattering theory and seek to determine an approximation to the shape of the scattering object as well as estimates on its material properties without making any weak scattering assumption and using essentially no a priori information on the nature of the scattering object. This book is designed to be an introduction to this new approach in inverse scattering theory focusing on the use of sampling methods and transmission eigenvalues. In order to aid the reader coming from a discipline outside of mathematics we have included background material on functional analysis, Sobolev spaces, the theory of ill posed problems and certain topics in in the theory of entire functions of a complex variable. This book is an updated and expanded version of an earlier book by the authors published by Springer titled Qualitative Methods in Inverse Scattering Theory Review of Qualitative Methods in Inverse Scattering Theory All in all, the authors do exceptionally well in combining such a wide variety of mathematical material and in presenting it in a well-organized and easy-to-follow fashion. This text certainly complements the growing body of work in inverse scattering and should well suit both new researchers to the field as well as those who could benefit from such a nice codified collection of profitable results combined in one bound volume. SIAM Review, 2006

The once esoteric idea of embedding scientific computing into a probabilistic framework, mostly along the lines of the Bayesian paradigm, has recently enjoyed wide popularity and found its way into numerous applications. This book provides an insider¿s view of how to combine two mature fields, scientific computing and Bayesian inference, into a powerful language leveraging the capabilities of both components for computational efficiency, high resolution power and uncertainty quantification ability. The impact of Bayesian scientific computing has been particularly significant in the area of computational inverse problems where the data are often scarce or of low quality, but some characteristics of the unknown solution may be available a priori. The ability to combine the flexibility of the Bayesian probabilistic framework with efficient numerical methods has contributed to the popularity of Bayesian inversion, with the prior distribution being the counterpart of classical regularization. However, the interplay between Bayesian inference and numerical analysis is much richer than providing an alternative way to regularize inverse problems, as demonstrated by the discussion of time dependent problems, iterative methods, and sparsity promoting priors in this book. The quantification of uncertainty in computed solutions and model predictions is another area where Bayesian scientific computing plays a critical role. This book demonstrates that Bayesian inference and scientific computing have much more in common than what one may expect, and gradually builds a natural interface between these two areas.

The first edition of this book was published in 1994. Since then considerable progress has been made in both theoretical developments of percolation theory, and in its applications. The 2nd edition of this book is a response to such developments. Not only have all of the chapters of the 1st edition been completely rewritten, reorganized, and updated all the way to 2022, but also 8 new chapters have been added that describe extensive new applications, including biological materials, networks and graphs, directed percolation, earthquakes, geochemical processes, and large-scale real world problems, from spread of technology to ad-hoc mobile networks.

This is a book on nonlinear dynamical systems and their bifurcations under parameter variation. It provides a reader with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems. Special attention is given to efficient numerical implementations of the developed techniques. Several examples from recent research papers are used as illustrations. The book is designed for advanced undergraduate or graduate students in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the previous editions, while updating the context to incorporate recent theoretical and software developments and modern techniques for bifurcation analysis. From reviews of earlier editions: "I know of no other book that so clearly explains the basic phenomena of bifurcation theory." - Math Reviews "The book is a fine addition to the dynamical systems literature. It is good to see, in our modern rush to quick publication, that we, as a mathematical community, still have time to bring together, and in such a readable and considered form, the important results on our subject." - Bulletin of the AMS "It is both a toolkit and a primer" - UK Nonlinear News

This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.

This book introduces the reader to the study of Hamiltonian systems, focusing on the stability of autonomous and periodic systems and expanding to topics that are usually not covered by the canonical literature in the field. It emerged from lectures and seminars given at the Federal University of Pernambuco, Brazil, known as one of the leading research centers in the theory of Hamiltonian dynamics.This book starts with a brief review of some results of linear algebra and advanced calculus, followed by the basic theory of Hamiltonian systems. The study of normal forms of Hamiltonian systems is covered by Ch.3, while Chapters 4 and 5 treat the normalization of Hamiltonian matrices. Stability in non-linear and linear systems are topics in Chapters 6 and 7. This work finishes with a study of parametric resonance in Ch. 8. All the background needed is presented, from the Hamiltonian formulation of the laws of motion to the application of the Krein-Gelfand-Lidskii theory of stronglystable systems.With a clear, self-contained exposition, this work is a valuable help to advanced undergraduate and graduate students, and to mathematicians and physicists doing research on this topic.