Topics in Industrial Mathematics

by Neunzert, Helmut; Siddiqi, Abul Hasan

• ISBN13: 9780792364177
• ISBN10: 0792364171
• Format: Hardcover
• Copyright: 2000-07-01
• Publisher: Kluwer Academic Pub

Summary

This book is devoted to some analytical and numerical methods for analyzing industrial problems related to emerging technologies such as digital image processing, material sciences and financial derivatives affecting banking and financial institutions. Case studies are based on industrial projects given by reputable industrial organizations of Europe to the Institute of Industrial and Business Mathematics, Kaiserslautern, Germany. Mathematical methods presented in the book which are most reliable for understanding current industrial problems include Iterative Optimization Algorithms, Galerkin's Method, Finite Element Method, Boundary Element Method, Quasi-Monte Carlo Method, Wavelet Analysis, and Fractal Analysis. The Black-Scholes model of Option Pricing, which was awarded the 1997 Nobel Prize in Economics, is presented in the book. In addition, basic concepts related to modeling are incorporated in the book. Audience: The book is appropriate for a course in Industrial Mathematics for upper-level undergraduate or beginning graduate-level students of mathematics or any branch of engineering.

Table of Contents

Preface	xi
Case Studies at Kaiserslautern	1	(52)
Molecular alignment	2	(5)
The problem	2	(1)
The model	2	(4)
The evaluation	6	(1)
The interpretation of the results	6	(1)
Acoustic identification of vehicles	7	(10)
The problem	7	(1)
The model	7	(2)
The evaluation	9	(7)
The interpretation of the results	16	(1)
The Airbag-sensor	17	(10)
The objective	17	(1)
The modelling project	17	(1)
The algorithmic project	18	(1)
The modelling of Safing sensor	18	(4)
The advanced model	22	(5)
How to judge the quality of a nonwoven fabric	27	(15)
The problem	27	(1)
The models: A first approach	28	(7)
Evaluation of our first model	35	(1)
The second model	36	(6)
Fatigue lifetime	42	(11)
Introduction	42	(2)
Physical situation, modelling, rate independence, and rainflow counting	44	(6)
Damage estimation using rainflow counted data	50	(3)
Algorithms for Optimization	53	(26)
Introduction	53	(1)
General results about optimization	54	(6)
Special classes of optimization problem	60	(2)
Programming problem	60	(1)
Calculus of variation	60	(1)
Minimum norm problem and projection	61	(1)
Optimal control problem for a system represented by differential equations	61	(1)
Newton algorithm and its generalization	62	(9)
Conjugate gradient method	71	(3)
Variable metric methods (DFP and BFGS methods)	74	(2)
Problems	76	(3)
Maxwell's Equations and Numerical Methods	79	(74)
Maxwell's equations	80	(16)
Brief historical note and physical laws	80	(6)
Maxwell's equations and their consequences	86	(6)
Variational formulation of Maxwell's equations	92	(2)
Variational formulation (weak formulation) of magnetostatics of a surface current	94	(2)
Finite element method	96	(24)
Introduction to numerical methods	96	(8)
Finite element method	104	(2)
Abstract finite element method	106	(7)
Finite element method in concrete cases	113	(7)
Boundary element method	120	(18)
Basic mathematical results for the boundary element method	121	(5)
Formulation of boundary value problems in terms of integral equations over boundary of the given domain	126	(4)
Main ingredients of the boundary element method	130	(6)
Coupling of boundary element and finite element methods	136	(2)
Problems	138	(15)
Monte Carlo Methods	153	(28)
Monte Carlo method	154	(5)
Motivation	154	(3)
Monte Carlo method in m-dimensional Euclidean space	157	(2)
Quasi-Monte Carlo methods	159	(10)
Basic results	159	(5)
Properties of discrepancy and star discrepancy	164	(5)
The particle methods	169	(3)
Introduction	169	(1)
Particle approximations - measure-theoretic approach	170	(1)
Functional analytic approach	171	(1)
A current study of the particle method	172	(6)
Introduction	172	(2)
Derivation of distance	174	(3)
Computational results	177	(1)
Spatially homogeneous Boltzmann equation	178	(1)
Problems	178	(3)
Image Processing	181	(84)
Image model and methods of image processing	182	(19)
Image model	182	(1)
Image enhancement	183	(2)
Image smoothing	185	(3)
Image restoration	188	(3)
Image analysis	191	(7)
Variational methods in image processing	198	(3)
Introduction to Fourier analysis	201	(18)
Amplitude, frequency and phase	201	(1)
Basic results	202	(4)
Continuous and discrete Fourier transforms	206	(6)
The fast Fourier transforms	212	(3)
Fourier analysis via computer	215	(4)
Wavelets with applications	219	(34)
Introduction	219	(1)
Wavelets and multi-resolution analysis	220	(10)
Special features of wavelets	230	(9)
Performance of Fourier, fractal and wavelet methods in image compression	239	(10)
Differential equation and wavelets	249	(4)
Fractal image compression	253	(8)
Introduction	253	(1)
IFS theory	254	(7)
Problems	261	(4)
Models of Hysteresis and Applications	265	(22)
Introduction to hysteresis	265	(3)
Hysteresis operators	268	(6)
Rainflow counting method	274	(7)
Energy dissipation	281	(3)
Hysteresis in the wave equation	284	(3)
Appendix	287	(58)
Introduction to mathematical models	287	(8)
Fractal image compression	295	(9)
Some basic results	304	(11)
Results from Sobolev spaces	315	(11)
Numerical solutions of linear systems	326	(7)
Black-Scholes world of option pricing	333	(12)
Bibliography	345	(22)
Symbols	367	(2)
Index	369