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Multiscale Methods for Fredholm Integral Equations [Kõva köide]

, (State University of New York, Albany),
  • Formaat: Hardback, 552 pages, kõrgus x laius x paksus: 237x160x40 mm, kaal: 970 g, Worked examples or Exercises; 25 Tables, black and white; 5 Halftones, unspecified; 20 Line drawings, unspecified
  • Sari: Cambridge Monographs on Applied and Computational Mathematics
  • Ilmumisaeg: 16-Jul-2015
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107103479
  • ISBN-13: 9781107103474
Teised raamatud teemal:
Multiscale Methods for Fredholm Integral EquationsSuurem pilt
  • Formaat: Hardback, 552 pages, kõrgus x laius x paksus: 237x160x40 mm, kaal: 970 g, Worked examples or Exercises; 25 Tables, black and white; 5 Halftones, unspecified; 20 Line drawings, unspecified
  • Sari: Cambridge Monographs on Applied and Computational Mathematics
  • Ilmumisaeg: 16-Jul-2015
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107103479
  • ISBN-13: 9781107103474
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781107103474.html
Märksõnad:
The authors present the state of the art in fast multiscale methods, from traditional numerical methods to the recently developed wavelet-based approach. Theorems of functional analysis used throughout the book are summarised in an appendix. Selected chapters are suitable for a one-semester course for advanced undergraduates or beginning graduates.

The recent appearance of wavelets as a new computational tool in applied mathematics has given a new impetus to the field of numerical analysis of Fredholm integral equations. This book gives an account of the state of the art in the study of fast multiscale methods for solving these equations based on wavelets. The authors begin by introducing essential concepts and describing conventional numerical methods. They then develop fast algorithms and apply these to solving linear, nonlinear Fredholm integral equations of the second kind, ill-posed integral equations of the first kind and eigen-problems of compact integral operators. Theorems of functional analysis used throughout the book are summarised in the appendix. The book is an essential reference for practitioners wishing to use the new techniques. It may also be used as a text, with the first five chapters forming the basis of a one-semester course for advanced undergraduates or beginning graduates.

Muu info

Presents the state of the art in the study of fast multiscale methods for solving these equations based on wavelets.
Preface ix
List of symbols
xi
Introduction 1(4)
1 A review of the Fredholm approach
5(27)
1.1 Introduction
5(2)
1.2 Second-kind matrix Fredholm equations
7(4)
1.3 Fredholm functions
11(6)
1.4 Resolvent kernels
17(3)
1.5 Fredholm determinants
20(4)
1.6 Eigenvalue estimates and a trace formula
24(7)
1.7 Bibliographical remarks
31(1)
2 Fredholm equations and projection theory
32(48)
2.1 Fredholm integral equations
32(21)
2.2 General theory of projection methods
53(25)
2.3 Bibliographical remarks
78(2)
3 Conventional numerical methods
80(64)
3.1 Degenerate kernel methods
80(6)
3.2 Quadrature methods
86(8)
3.3 Galerkin methods
94(11)
3.4 Collocation methods
105(7)
3.5 Petrov--Galerkin methods
112(30)
3.6 Bibliographical remarks
142(2)
4 Multiscale basis functions
144(55)
4.1 Multiscale functions on the unit interval
145(8)
4.2 Multiscale partitions
153(13)
4.3 Multiscale orthogonal bases
166(3)
4.4 Refinable sets and set wavelets
169(15)
4.5 Multiscale interpolating bases
184(13)
4.6 Bibliographical remarks
197(2)
5 Multiscale Galerkin methods
199(24)
5.1 The multiscale Galerkin method
200(5)
5.2 The fast multiscale Galerkin method
205(4)
5.3 Theoretical analysis
209(12)
5.4 Bibliographical remarks
221(2)
6 Multiscale Petrov-Galerkin methods
223(42)
6.1 Fast multiscale Petrov--Galerkin methods
223(8)
6.2 Discrete multiscale Petrov--Galerkin methods
231(32)
6.3 Bibliographical remarks
263(2)
7 Multiscale collocation methods
265(35)
7.1 Multiscale basis functions and collocation functionals
266(15)
7.2 Multiscale collocation methods
281(7)
7.3 Analysis of the truncation scheme
288(10)
7.4 Bibliographical remarks
298(2)
8 Numerical integrations and error control
300(22)
8.1 Discrete systems of the multiscale collocation method
300(2)
8.2 Quadrature rules with polynomial order of accuracy
302(12)
8.3 Quadrature rules with exponential order of accuracy
314(4)
8.4 Numerical experiments
318(3)
8.5 Bibliographical remarks
321(1)
9 Fast solvers for discrete systems
322(34)
9.1 Multilevel augmentation methods
322(25)
9.2 Multilevel iteration methods
347(7)
9.3 Bibliographical remarks
354(2)
10 Multiscale methods for nonlinear integral equations
356(60)
10.1 Critical issues in solving nonlinear equations
356(3)
10.2 Multiscale methods for the Hammerstein equation
359(18)
10.3 Multiscale methods for nonlinear boundary integral equations
377(25)
10.4 Numerical experiments
402(11)
10.5 Bibliographical remarks
413(3)
11 Multiscale methods for ill-posed integral equations
416(49)
11.1 Numerical solutions of regularization problems
416(4)
11.2 Multiscale Galerkin methods via the Lavrentiev regularization
420(18)
11.3 Multiscale collocation methods via the Tikhonov regularization
438(18)
11.4 Numerical experiments
456(7)
11.5 Bibliographical remarks
463(2)
12 Eigen-problems of weakly singular integral operators
465(23)
12.1 Introduction
465(1)
12.2 An abstract framework
466(8)
12.3 A multiscale collocation method
474(4)
12.4 Analysis of the fast algorithm
478(5)
12.5 A power iteration algorithm
483(1)
12.6 A numerical example
484(3)
12.7 Bibliographical remarks
487(1)
Appendix Basic results from functional analysis
488(31)
A.1 Metric spaces
488(6)
A.2 Linear operator theory
494(8)
A.3 Invariant sets
502(17)
References 519(15)
Index 534
Zhongying Chen is a professor of computational mathematics at Sun Yat-Sen University, China. He is the author or co-author of more than 70 professional publications, including the books Generalized Difference Methods for Differential Equations and Approximate Solutions of Operator Equations. He has served on the editorial board of four journals including Advances in Computational Mathematics, and two book series including the Series in Information and Computational Science, China. Charles A. Micchelli is a distinguished research professor of mathematics at SUNY Albany and an international leader of approximation theory and wavelet analysis. His research interests cover approximation theory, wavelet analysis, computer aided geometric design, multiscale methods for Fredholm integral equations, speech recognition and mathematical learning theory. Prior to his current position, he was employed by IBM T. J. Watson Research Center as a senior research staff member for over 30 years. Honors he has received include the Senior Alexander von Humboldt Prize and an invitation to speak at the International Congress of Mathematicians (1983). Micchelli is an editor of several leading journals in approximation theory and computational mathematics and also the founding editor of Advances in Computational Mathematics, for which he served as editor-in-chief from 1994 to 2011. He is the author or co-author of over 300 research publications and an owner of several patents. His research has been supported by the US National Science Foundation and the Office of Science of the US Air Force. Yuesheng Xu is a professor emeritus of mathematics at Syracuse University, USA and Guohua Chair Professor at Sun Yat-Sen University, China. He is a National Scholar of China, the director of Guangdong Province Key Laboratory of Computational Science, and the president of the Guangdong Province Association of Computational Mathematics. Xu is also a member of the executive committee of the Association of Computational Mathematics of China and an Adjunct Professor of radiology at SUNY Upstate Medical University, USA. His research interests include approximation theory and wavelet analysis, the numerical solution of integral equations and PDEs, and image and signal processing. He currently serves on the editorial board of seven academic journals, including Advances in Computational Mathematics, where he was the managing editor from 1999 to 2011, and the Journal of Integral Equations and Applications. He has published over 150 research papers. His research has been supported by the US National Science Foundation, the Office of Science of the US Air Force, NASA and the National Natural Science Foundation of China, Guangdong Province Government of China.