Multiscale methods for Fredholm integral equations:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2015
|
| Series: | Cambridge monographs on applied and computational mathematics
28 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xiii, 536 Seiten |
| ISBN: | 9781107103474 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
Preface page ix
List of symbols xi
Introduction 1
1 A review of the Fredholm approach 5
1.1 Introduction 5
l .2 Second-kind matrix Fredholm equations 7
1.3 Fredholm functions 11
1.4 Resolvent kernels 17
1.5 Fredholm determinants 20
1.6 Eigenvalue estimates and a trace formula 24
1.7 Bibliographical remarks 31
2 Fredholm equations and projection theory 32
2.1 Fredholm integral equations 32
2.2 General theory of projection methods 53
2.3 Bibliographical remarks 78
3 Conventional numerical methods 80
3.1 Degenerate kernel methods 80
3.2 Quadrature methods 86
3.3 Galerkin methods 94
3.4 Collocation methods 105
3.5 Petrov-Galerkin methods 112
3.6 Bibliographical remarks 142
4 Multiscale basis functions 144
4.1 Multiscale functions on the unit interval 145
4.2 Multiscale partitions 153
v
Contents
vi
4.3 Multiscale orthogonal bases 166
4.4 Refinable sets and set wavelets 169
4.5 Multiscale interpolating bases 184
4.6 Bibliographical remarks 197
5 Multiscale Galerkin methods 199
5.1 The multiscale Galerkin method 200
5.2 The fast multiscale Galerkin method 205
5.3 Theoretical analysis 209
5.4 Bibliographical remarks 221
6 Multiscale Petrov-Galerkin methods 223
6.1 Fast multiscale Petrov-Galerkin methods 223
6.2 Discrete multiscale Petrov—Galerkin methods 231
6.3 Bibliographical remarks 263
7 Multiscale collocation methods 265
7.1 Multiscale basis functions and collocation functionals 266
7.2 Multiscale collocation methods 281
7.3 Analysis of the truncation scheme 288
7.4 Bibliographical remarks 298
8 Numerical integrations and error control 300
8.1 Discrete systems of the multiscale collocation method 300
8.2 Quadrature rules with polynomial order of accuracy 302
8.3 Quadrature rules with exponential order of accuracy 314
8.4 Numerical experiments 318
8.5 Bibliographical remarks 321
9 Fast solvers for discrete systems 322
9.1 Multilevel augmentation methods 322
9.2 Multilevel iteration methods 347
9.3 Bibliographical remarks 354
10 Multiscale methods for nonlinear integral equations 356
10.1 Critical issues in solving nonlinear equations 356
10.2 Multiscale methods for the Hammerstein equation 359
10.3 Multiscale methods for nonlinear boundary
integral equations 377
10.4 Numerical experiments 402
10.5 Bibliographical remarks 413
Contents vii
11 Muitfseale methods for ill-posed integral equations 416
n.i Numerical solutions of regularization problems 416
11.2 Multiscale Galerkin methods via the Lavrentiev
regularization 420
11.3 Multiscale collocation methods via the Tikhonov
regularization 438
11.4 Numerical experiments 456
11.5 Bibliographical remarks 463
12 Eigen-problems of weakly singular integral operators 465
12.1 Introduction 465
12.2 An abstract framework 466
12.3 A multiscale collocation method 474
12.4 Analysis of the fast algorithm 478
12.5 A power iteration algorithm 483
12.6 A numerical example 484
12.7 Bibliographical remarks 487
Appendix Basic results from functional analysis 488
A.l Metric spaces 488
A.2 Linear operator theory 494
A.3 Invariant sets 502
References 519
Index 534
Cambridge Monographs on Applied and Computational Mathematic
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, it is well known that discretization of Fredholm integral equations leads to dense matrices which present computational difficulties in developing their efficient solvers. This book gives an account of the state of the art in the study of fast multiscale methods for solving the 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 summarized in the appendix.
The book is an essential reference for practitioners wishing to use the new techniques presented here, 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.
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.
Charles A. Micchelll 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. 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 the author or co-author of over 300 research publications and an owner of several patents.
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. He has published over 150 research papers. His research interests include approximation theory and wavelet analysis, the numerical solution of integral equations and PDEs, and image and signal processing.
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| illustrated | Not Illustrated |
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| institution | BVB |
| isbn | 9781107103474 |
| language | English |
| lccn | 2014050239 |
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| physical | xiii, 536 Seiten |
| publishDate | 2015 |
| publishDateSearch | 2015 |
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| series | Cambridge monographs on applied and computational mathematics |
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| spelling | Chen, Zhongying Verfasser aut Multiscale methods for Fredholm integral equations Zhongying Chen, Sun Yat-Sen University, Guangzhou, China, Charles A. Micchelli, State University of New York, Albany, Yuesheng Xu, Sun Yat-Sen University, Guangzhou, China Cambridge Cambridge University Press 2015 xiii, 536 Seiten txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on applied and computational mathematics 28 Includes bibliographical references and index Fredholm-Integralgleichung (DE-588)4155261-1 gnd rswk-swf Mehrskalenmodell (DE-588)7600619-0 gnd rswk-swf Fredholm-Integralgleichung (DE-588)4155261-1 s Mehrskalenmodell (DE-588)7600619-0 s DE-604 Micchelli, Charles A. Verfasser aut Xu, Yuesheng Verfasser aut Cambridge monographs on applied and computational mathematics 28 (DE-604)BV011073737 28 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028335896&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028335896&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Chen, Zhongying Micchelli, Charles A. Xu, Yuesheng Multiscale methods for Fredholm integral equations Cambridge monographs on applied and computational mathematics Fredholm-Integralgleichung (DE-588)4155261-1 gnd Mehrskalenmodell (DE-588)7600619-0 gnd |
| subject_GND | (DE-588)4155261-1 (DE-588)7600619-0 |
| title | Multiscale methods for Fredholm integral equations |
| title_auth | Multiscale methods for Fredholm integral equations |
| title_exact_search | Multiscale methods for Fredholm integral equations |
| title_full | Multiscale methods for Fredholm integral equations Zhongying Chen, Sun Yat-Sen University, Guangzhou, China, Charles A. Micchelli, State University of New York, Albany, Yuesheng Xu, Sun Yat-Sen University, Guangzhou, China |
| title_fullStr | Multiscale methods for Fredholm integral equations Zhongying Chen, Sun Yat-Sen University, Guangzhou, China, Charles A. Micchelli, State University of New York, Albany, Yuesheng Xu, Sun Yat-Sen University, Guangzhou, China |
| title_full_unstemmed | Multiscale methods for Fredholm integral equations Zhongying Chen, Sun Yat-Sen University, Guangzhou, China, Charles A. Micchelli, State University of New York, Albany, Yuesheng Xu, Sun Yat-Sen University, Guangzhou, China |
| title_short | Multiscale methods for Fredholm integral equations |
| title_sort | multiscale methods for fredholm integral equations |
| topic | Fredholm-Integralgleichung (DE-588)4155261-1 gnd Mehrskalenmodell (DE-588)7600619-0 gnd |
| topic_facet | Fredholm-Integralgleichung Mehrskalenmodell |
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