الصفحة 1
الصفحة 1
Geometric numerical integration : Structure-preserving algorithms for ordinary differential equations
Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches.
Galerkin Finite Element Methods for Parabolic Problems
This book provides insight in the mathematics of Galerkin finite element method as applied to parabolic equations. The approach is based on first discretizing in the spatial variables by Galerkin's method, using piecewise polynomial trial functions, and then applying some single step or multistep time stepping method. The concern is stability and error analysis of approximate solutions in various norms, and under various regularity assumptions on the exact solution.
Dealing with Uncertainties : A Guide to Error Analysis
Dealing with Uncertainties proposes and explains a new approach for the analysis of uncertainties. Firstly, it is shown that uncertainties are the consequence of modern science rather than of measurements. Secondly, it stresses the importance of the deductive approach to uncertainties.
Continuous-Time Sigma-Delta A/D Conversion : Fundamentals, Performance Limits and Robust Implementations
This comprehensive book deals with all relevant aspects arising during the analysis, design and simulation of the now widespread continuous-time implementations of sigma-delta modulators. The results of several years of research by the authors in the field of CT sigma-delta modulators are covered, including the analysis and modeling of different CT modulator architectures, CT/DT loop filter synthesis, a detailed error analysis of all components, and possible compensation/correction schemes for the non-ideal behavior in CT sigma-delta modulators. Guidance for obtaining low-power consumption and several practical implementations are also presented. It is shown that all the proposed new theories, architectures and possible correction techniques have been confirmed by measurements on discrete or integrated circuits. Quantitative results are also provided, thus enabling prediction of the resulting accuracy.
Computational Methods in Transport : Verification and Validation
The focus of this book deals with a cross cutting issue affecting all particle transport algorithms and applications; verification and validation (V&V). In other words, are the equations being solved correctly and are the correct equations being solved? Verification and validation assures a scientist, engineer or mathematician that a simulation code is a mirror of reality and not just an expensive computer game. In this book, we will learn what the astrophysicist, atmospheric scientist, mathematician or nuclear engineer do to assess the accuracy of their code. What convergence studies, what error analysis, what problems do each field use to benchmark and ascertain the accuracy of their transport simulations.
CMOS Cascade Sigma-Delta Modulators for Sensors and Telecom : Error Analysis and Practical Design
CMOS Cascade Sigma-Delta Modulators for Sensors and Telecom: Error Analysis and Practical Design starts with a tutorial presentation of the fundamentals of low-pass sigma-delta modulators, their applications, and their most common architectures. It then presents an exhaustive analysis of SC circuit errors with a twofold outcome. On the one hand, compact expressions are derived to support design plans and quick top-down design. On the other, detailed behavioral models are presented to support accurate verification. This set of models allows the designer to determine the required specifications for the different modulator building blocks and form the basis of a systematic design approach. The book is completed in subsequent chapters with the detailed presentation of three high-performance modulator ICs: the first two are intended for DSL-like applications, whereas the third one is intended for automotive sensors.
Iterative Approximation of Fixed Points
The aim of this monograph is to give a unified introductory treatment of the most important iterative methods for constructing fixed points of nonlinear contractive type mappings. It summarizes the most significant contributions in the area by presenting, for each iterative method considered (Picard iteration, Krasnoselskij iteration, Mann iteration, Ishikawa iteration etc.), some of the most relevant, interesting, representative and actual convergence theorems. Applications to the solution of nonlinear operator equations as well as the appropriate error analysis of the main iterative methods, are also presented.
Approximation of Additive Convolution-Like Operators : Real C*-Algebra Approach
Various aspects of numerical analysis for equations arising in boundary integral equation methods have been the subject of several books published in the last 15 years [95, 102, 183, 196, 198]. Prominent examples include various classes of o- dimensional singular integral equations or equations related to single and double layer potentials. Usually, a mathematically rigorous foundation and error analysis for the approximate solution of such equations is by no means an easy task. One reason is the fact that boundary integral operators generally are neither integral operatorsof the formidentity plus compact operatornor identity plus an operator with a small norm. Consequently, existing standard theories for the numerical analysis of Fredholm integral equations of the second kind are not applicable. In the last 15 years it became clear that the Banach algebra technique is a powerful tool to analyze the stability problem for relevant approximation methods [102, 103, 183, 189]. The starting point for this approach is the observation that the ? stability problem is an invertibility problem in a certain BanachorC -algebra. As a rule, this algebra is very complicated – and one has to ?nd relevant subalgebras to use such tools as local principles and representation theory.
A Posteriori Error Analysis Via Duality Theory : With Applications in Modeling and Numerical Approximations
This volume provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear variational problems. The author avoids giving the results in the most general, abstract form so that it is easier for the reader to understand more clearly the essential ideas involved. Many examples are included to show the usefulness of the derived error estimates.
