Regularization methods in Banach spaces:
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
De Gruyter
2012
|
| Series: | Radon series on computational and applied mathematics
10 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literatur-Verz.: S. 265 - 279 |
| Physical Description: | XI, 283 S. graph. Darst. 25 cm |
| ISBN: | 3110255243 9783110255249 |
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MARC
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| 245 | 1 | 0 | |a Regularization methods in Banach spaces |c Thomas Schuster ... |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2012 | |
| 300 | |a XI, 283 S. |b graph. Darst. |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Radon series on computational and applied mathematics |v 10 | |
| 500 | |a Literatur-Verz.: S. 265 - 279 | ||
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| 650 | 0 | 7 | |a Regularisierung |0 (DE-588)4124043-1 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1807953726786240512 |
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| adam_text |
IMAGE 1
CONTENTS
PREFACE VII
I WHY TO USE BANACH SPACES IN REGULARIZATION THEORY?
1 APPLICATIONS WITH A BANACH SPACE SETTING 4
1.1 X-RAY DIFFRACTOMETRY 4
1.2 TWO PHASE RETRIEVAL PROBLEMS 6
1.3 A PARAMETER IDENTIFICATION PROBLEM FOR AN ELLIPTIC PARTIAL
DIFFERENTIAL EQUATION 9
1.4 AN INVERSE PROBLEM FROM FINANCE 13
1.5 SPARSITY CONSTRAINTS 18
II GEOMETRY AND MATHEMATICAL TOOLS OF BANACH SPACES
2 PRELIMINARIES AND BASIC DEFINITIONS 28
2.1 BASIC MATHEMATICAL TOOLS 28
2.2 CONVEX ANALYSIS 31
2.2.1 THE SUBGRADIENT O F CONVEX FUNCTIONALS 31
2.2.2 DUALITY MAPPINGS 34
2.3 GEOMETRY O F BANACH SPACE NORMS 36
2.3.1 CONVEXITY AND SMOOTHNESS 37
2.3.2 BREGMAN DISTANCE 44
3 ILL-POSED OPERATOR EQUATIONS AND REGULARIZATION 49
3.1 OPERATOR EQUATIONS AND THE ILL-POSEDNESS PHENOMENON 4 9
3.1.1 .LINEAR PROBLEMS 50
3.1.2 NONLINEAR PROBLEMS 52
3.1.3 CONDITIONAL WELL-POSEDNESS 55
3.2 MATHEMATICAL TOOLS IN REGULARIZATION THEORY 56
3.2.1 REGULARIZATION APPROACHES 57
3.2.2 SOURCE CONDITIONS AND DISTANCE FUNCTIONS 63
3.2.3 VARIATIONAL INEQUALITIES 67
3.2.4 DIFFERENCES BETWEEN THE LINEAR AND THE NONLINEAR CASE . . . . 69
HTTP://D-NB.INFO/1020937068
IMAGE 2
CONTENTS
III TIKHONOV-TYPE REGULARIZATION
4 TIKHONOV REGULARIZATION IN BANACH SPACES WITH GENERAL CONVEX PENALTIES
81
4.1 BASIC PROPERTIES O F REGULARIZED SOLUTIONS 81
4.1.1 EXISTENCE AND STABILITY O F REGULARIZED SOLUTIONS 81
4.1.2 CONVERGENCE O F REGULARIZED SOLUTIONS 84
4.2 ERROR ESTIMATES AND CONVERGENCE RATES 89
4.2.1 ERROR ESTIMATES UNDER VARIATIONAL INEQUALITIES 90
4.2.2 CONVERGENCE RATES FOR THE BREGMAN DISTANCE 95
4.2.3 TIKHONOV REGULARIZATION UNDER CONVEX CONSTRAINTS 99
4.2.4 HIGHER RATES BRIEFLY VISITED 101
4.2.5 RATE RESULTS UNDER CONDITIONAL STABILITY ESTIMATES 103
4.2.6 A GLIMPSE O F RATE RESULTS UNDER SPARSITY CONSTRAINTS 105
5 TIKHONOV REGULARIZATION O F LINEAR OPERATORS WITH POWER-TYPE PENALTIES
108
5.1 SOURCE CONDITIONS 108
5.2 CHOICE O F THE REGULARIZATION PARAMETER 113
5.2.1 A PRIORI PARAMETER CHOICE 113
5.2.2 MOROZOV'S DISCREPANCY PRINCIPLE 115
5.2.3 MODIFIED DISCREPANCY PRINCIPLE 116
5.3 MINIMIZATION O F THE TIKHONOV FUNCTIONALS 122
5.3.1 PRIMAL METHOD 123
5.3.2 DUAL METHOD 135
IV ITERATIVE REGULARIZATION
6 LINEAR OPERATOR EQUATIONS 144
6.1 THE LANDWEBER ITERATION 146
6.1.1 NOISE-FREE CASE 146
6.1.2 REGULARIZATION PROPERTIES 152
6.2 SEQUENTIAL SUBSPACE OPTIMIZATION METHODS 157
6.2.1 BREGMAN PROJECTIONS 158
6.2.2 THE METHOD FOR EXACT DATA (SESOP) 163
6.2.3 THE REGULARIZATION METHOD FOR NOISY DATA (RESESOP). . . . 165
6.3 ITERATIVE SOLUTION O F SPLIT FEASIBILITY PROBLEMS (SFP) 177
6.3.1 CONTINUITY O F BREGMAN AND METRIC PROJECTIONS 179
6.3.2 A REGULARIZATION METHOD FOR THE SOLUTION O F SFPS 183
IMAGE 3
CONTENTS XI
7 NONLINEAR OPERATOR EQUATIONS 193
7.1 PRELIMINARIES 193
7.1.1 CONDITIONS ON THE SPACES 193
7.1.2 VARIATIONAL INEQUALITIES 194
7.1.3 CONDITIONS ON THE FORWARD OPERATOR 195
7.2 GRADIENT TYPE METHODS 199
7.2.1 CONVERGENCE O F THE LANDWEBER ITERATION WITH THE DISCREPANCY
PRINCIPLE 199
7.2.2 CONVERGENCE RATES FOR THE ITERATIVELY REGULARIZED LANDWEBER
ITERATION WITH A PRIORI STOPPING RULE 203
7.3 THE ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD 212
7.3.1 CONVERGENCE WITH A PRIORI PARAMETER CHOICE 215
7.3.2 CONVERGENCE WITH A POSTERIORI PARAMETER CHOICE 225
7.3.3 NUMERICAL ILLUSTRATION 230
V THE METHOD OF APPROXIMATE INVERSE
8 SETTING O F THE METHOD 236
9 CONVERGENCE ANALYSIS IN L P ( Q ) AND ~C(K) 239
9.1 THE CASE X - L P ( Q ) 239
9.2 THE CASE X - ~ C ( K ) 244
9.3 AN APPLICATION TO X-RAY DIFFRACTOMETRY 248
10 A GLIMPSE O F SEMI-DISCRETE OPERATOR EQUATIONS 253
BIBLIOGRAPHY 265
INDEX 280 |
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| dewey-raw | 515/.732 |
| dewey-search | 515/.732 |
| dewey-sort | 3515 3732 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV040424773 |
| illustrated | Illustrated |
| indexdate | 2024-08-21T00:10:30Z |
| institution | BVB |
| isbn | 3110255243 9783110255249 |
| language | English |
| lccn | 2012013065 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025277399 |
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| owner | DE-83 DE-188 DE-19 DE-BY-UBM DE-20 DE-384 |
| owner_facet | DE-83 DE-188 DE-19 DE-BY-UBM DE-20 DE-384 |
| physical | XI, 283 S. graph. Darst. 25 cm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | De Gruyter |
| record_format | marc |
| series | Radon series on computational and applied mathematics |
| series2 | Radon series on computational and applied mathematics |
| spelling | Regularization methods in Banach spaces Thomas Schuster ... Berlin [u.a.] De Gruyter 2012 XI, 283 S. graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Radon series on computational and applied mathematics 10 Literatur-Verz.: S. 265 - 279 Banach-Raum (DE-588)4004402-6 gnd rswk-swf Regularisierung (DE-588)4124043-1 gnd rswk-swf Regularisierung (DE-588)4124043-1 s Banach-Raum (DE-588)4004402-6 s DE-604 Schuster, Thomas 1971- Sonstige (DE-588)137817053 oth Erscheint auch als Online-Ausgabe 978-3-11-025572-0 Radon series on computational and applied mathematics 10 (DE-604)BV023335470 10 application/pdf http://d-nb.info/1020937068/04 Inhaltsverzeichnis text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3996621&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025277399&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Regularization methods in Banach spaces Radon series on computational and applied mathematics Banach-Raum (DE-588)4004402-6 gnd Regularisierung (DE-588)4124043-1 gnd |
| subject_GND | (DE-588)4004402-6 (DE-588)4124043-1 |
| title | Regularization methods in Banach spaces |
| title_auth | Regularization methods in Banach spaces |
| title_exact_search | Regularization methods in Banach spaces |
| title_full | Regularization methods in Banach spaces Thomas Schuster ... |
| title_fullStr | Regularization methods in Banach spaces Thomas Schuster ... |
| title_full_unstemmed | Regularization methods in Banach spaces Thomas Schuster ... |
| title_short | Regularization methods in Banach spaces |
| title_sort | regularization methods in banach spaces |
| topic | Banach-Raum (DE-588)4004402-6 gnd Regularisierung (DE-588)4124043-1 gnd |
| topic_facet | Banach-Raum Regularisierung |
| url | http://d-nb.info/1020937068/04 http://deposit.dnb.de/cgi-bin/dokserv?id=3996621&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025277399&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023335470 |
| work_keys_str_mv | AT schusterthomas regularizationmethodsinbanachspaces |
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