Linear operator equations: approximation and regularization
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2009
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIII, 249 S. |
| ISBN: | 9789812835642 9812835644 |
Staff View
MARC
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| 020 | |a 9789812835642 |c hard cover : alk. paper |9 978-981-283-564-2 | ||
| 020 | |a 9812835644 |c hard cover : alk. paper |9 981-283-564-4 | ||
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| 035 | |a (DE-599)BVBBV035755189 | ||
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| 100 | 1 | |a Nair, M. Thamban |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Linear operator equations |b approximation and regularization |c M. Thamban Nair |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2009 | |
| 300 | |a XIII, 249 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Linear operators | |
| 650 | 4 | |a Operator equations | |
| 650 | 0 | 7 | |a Lineare Operatorgleichung |0 (DE-588)4123658-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Lineare Operatorgleichung |0 (DE-588)4123658-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018615170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-018615170 | ||
Record in the Search Index
| _version_ | 1804140675721068544 |
|---|---|
| adam_text | Contents
Preface
УІІ
1.
Introduction
1
1.1
General Introduction
..................... 1
1.2
Well-Posedness and Ill-Posedness
.............. 2
1.2.1
Examples of well-posed equations
.........
З
1.2.2
Examples of ill-posed equations
.......... 4
1.3
What Do We Do in This Book?
............... 4
2.
Basic Results from Functional Analysis
7
2.1
Spaces and Operators
.................... 7
2.1.1
Spaces
........................ 7
2.1.2
Bounded operators
................. 14
2.1.3
Compact operators
................. 21
2.2
Some Important Theorems
................. 29
2.2.1
Uniform boundedness principle
........... 29
2.2.2
Closed graph theorem
................ 34
2.2.3
Hahn-Banach theorem
............... 36
2.2.4
Projection and Riesz representation theorems
. . 38
2.3
Spectral Results for Operators
............... 42
2.3.1
Invertibility of operators
.............. 42
2.3.2
Spectral notions
................... 44
2.3.3
Spectrum of a compact operator
.......... 48
2.3.4
Spectral Mapping Theorem
............. 49
2.3.5
Spectral representation for compact self adjoint
operators
....................... 50
xii
Linear Operator Equations: Approximation and Regularization
2.3.6
Singular value representation
............ 51
2.3.7
Spectral theorem for self adjoint operators
.... 53
Problems
............................ 54
3.
Well-Posed Equations and Their Approximations
57
3.1
Introduction
.......................... 57
3.2
Convergence and Error Estimates
.............. 60
3.3
Conditions for Stability
................... 65
3.3.1
Norm approximation
................ 66
3.3.2
Norm-square approximation
............ 70
3.4
Projection Based Approximation
.............. 71
3.4.1
Interpolatory
projections
.............. 71
3.4.2
A projection based approximation
......... 74
3.5
Quadrature Based Approximation
............. 76
3.5.1
Quadrature rule
................... 77
3.5.2
Interpolatory
quadrature rule
........... 78
3.5.3
Nyström
approximation
............... 79
3.5.4
Collectively compact approximation
........ 82
3.6
Norm-square Approximation Revisited
........... 84
3.7
Second Kind Equations
................... 88
3.7.1
Iterated versions of approximations
........ 89
3.8
Methods for Second Kind Equations
............ 90
3.8.1
Projection methods
................. 91
3.8.2
Computational aspects
............... 105
3.8.3
Error estimates under smoothness assumptions
. . 108
3.8.4
Quadrature methods for integral equations
.... 118
3.8.5
Accelerated methods
................ 122
3.9
Qualitative
Properties of Convergence
........... 124
3.9.1
Uniform and arbitrarily slow convergence
..... 124
3.9.2
Modification of ASC-methods
........... 127
Problems
............................ 130
4.
Ill-Posed Equations and Their Regularizations
135
4.1
Ill-Posedness of Operator Equations
............ 135
4.1.1
Compact operator equations
............ 136
4.1.2
A backward heat conduction problem
....... 137
4.2
LRN Solution and Generalized Inverse
........... 140
4.2.1
LRN solution
..................... 140
Contents xiii
4.2.2
Generalized inverse
................. 145
4.2.3
Normal equation
................... 148
4.2.4
Picard
criterion
................... 150
4.3
Regularization Procedure
.................. 151
4.3.1
Regularization family
................ 151
4.3.2
Regularization algorithm
.............. 153
4.4
Tikhonov Regularization
................... 155
4.4.1
Source conditions and order of convergence
.... 162
4.4.2
Error estimates with inexact data
......... 166
4.4.3
An illustration of the source condition
....... 171
4.4.4
Discrepancy principles
............... 172
4.4.5
Remarks on general regularization
......... 183
4.5
Best Possible Worst Case Error
............... 185
4.5.1
Estimates for
ω(Μ,δ)
................ 188
4.5.2
Illustration with differentiation problem
...... 190
4.5.3
Illustration with backward heat equation
..... 192
4.6
General Source Conditions
.................. 193
4.6.1
Why do we need a general source condition?
. . . 193
4.6.2
Error estimates for Tikhonov regularization
.... 195
4.6.3
Parameter choice strategies
............. 196
4.6.4
Estimate for
ω(Μψ>ρ, δ)...............
201
Problems
............................ 203
5.
Regularized Approximation Methods
207
5.1
Introduction
.......................... 207
5.2
Using an Approximation (Tn) of
Γ
............. 209
5.2.1
Convergence and general error estimates
..... 209
5.2.2
Error estimates under source conditions
...... 213
5.2.3
Error estimates for
Ritz
method
.......... 214
5.3
Using an Approximation (An) of T*T
........... 217
5.3.1
Results under norm convergence
.......... 218
5.4
Methods for Integral Equations
............... 220
5.4.1
A degenerate kernel method
............ 222
5.4.2
Regularized
Nyström
method
............ 230
Problems
............................ 238
Bibliography
241
Index
247
|
| any_adam_object | 1 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV035755189 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:03:45Z |
| institution | BVB |
| isbn | 9789812835642 9812835644 |
| language | English |
| lccn | 2009007531 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018615170 |
| oclc_num | 244765483 |
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| owner | DE-355 DE-BY-UBR DE-11 DE-83 DE-703 DE-824 |
| owner_facet | DE-355 DE-BY-UBR DE-11 DE-83 DE-703 DE-824 |
| physical | XIII, 249 S. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Nair, M. Thamban Verfasser aut Linear operator equations approximation and regularization M. Thamban Nair Singapore [u.a.] World Scientific 2009 XIII, 249 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Linear operators Operator equations Lineare Operatorgleichung (DE-588)4123658-0 gnd rswk-swf Lineare Operatorgleichung (DE-588)4123658-0 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018615170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nair, M. Thamban Linear operator equations approximation and regularization Linear operators Operator equations Lineare Operatorgleichung (DE-588)4123658-0 gnd |
| subject_GND | (DE-588)4123658-0 |
| title | Linear operator equations approximation and regularization |
| title_auth | Linear operator equations approximation and regularization |
| title_exact_search | Linear operator equations approximation and regularization |
| title_full | Linear operator equations approximation and regularization M. Thamban Nair |
| title_fullStr | Linear operator equations approximation and regularization M. Thamban Nair |
| title_full_unstemmed | Linear operator equations approximation and regularization M. Thamban Nair |
| title_short | Linear operator equations |
| title_sort | linear operator equations approximation and regularization |
| title_sub | approximation and regularization |
| topic | Linear operators Operator equations Lineare Operatorgleichung (DE-588)4123658-0 gnd |
| topic_facet | Linear operators Operator equations Lineare Operatorgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018615170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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