Iterative methods for fixed point problems in Hilbert spaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2012
|
| Series: | Lecture notes in mathematics
2057 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 275 - 289 |
| Physical Description: | XVI, 298 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642309007 |
Staff View
MARC
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| 245 | 1 | 0 | |a Iterative methods for fixed point problems in Hilbert spaces |c Andrzej Cegielski |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2012 | |
| 300 | |a XVI, 298 S. |b graph. Darst. |c 235 mm x 155 mm | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 2057 | |
| 500 | |a Literaturverz. S. 275 - 289 | ||
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Record in the Search Index
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IMAGE 1
CONTENTS
1 INTRODUCTION 1
1.1 BACKGROUND 1
1.1.1 HILBERT SPACE 1
1.1.2 NOTATIONS AND BASIC FACTS 5
1.2 METRIC PROJECTION 16
1.2.1 EXISTENCE AND UNIQUENESS O F THE METRIC PROJECTION 17
1.2.2 CHARACTERIZATION O F THE METRIC PROJECTION 19
1.2.3 FIRST APPLICATIONS O F THE CHARACTERIZATION THEOREM 20
1.3 CONVEX OPTIMIZATION PROBLEMS 23
1.3.1 CONVEX MINIMIZATION PROBLEMS 24
1.3.2 VARIATIONAL INEQUALITY 27
1.3.3 COMMON FIXED POINT PROBLEM 27
1.3.4 CONVEX FEASIBILITY PROBLEM 27
1.3.5 LINEAR FEASIBILITY PROBLEM 30
1.3.6 GENERAL CONVEX FEASIBILITY PROBLEM 33
1.3.7 SPLIT FEASIBILITY PROBLEM 34
1.3.8 LINEAR SPLIT FEASIBILITY PROBLEM 35
1.3.9 MULTIPLE-SETS SPLIT FEASIBILITY PROBLEM 35
1.4 EXERCISES 36
2 ALGORITHMIC OPERATORS 39
2.1 BASIC DEFINITIONS AND PROPERTIES 4 0
2.1.L NONEXPANSIVE OPERATORS 41
2.1.2 QUASI-NONEXPANSIVE OPERATORS 45
2.1.3 CUTTERS AND STRONGLY QUASI-NONEXPANSIVE OPERATORS 53
2.2 FIRMLY NONEXPANSIVE OPERATORS 65
2.2.1 BASIC PROPERTIES O F FIRMLY NONEXPANSIVE OPERATORS 66
2.2.2 RELATIONSHIPS BETWEEN FIRMLY NONEXPANSIVE AND NONEXPANSIVE
OPERATORS 70
2.2.3 FURTHER PROPERTIES O F THE METRIC PROJECTION 76
XIII
HTTP://D-NB.INFO/1022400045
IMAGE 2
XIV CONTENTS
2.2.4 METRIC PROJECTION ONTO A CLOSED SUBSPACE 80
2.2.5 METRIC PROJECTION ONTO A CLOSED AFFINE SUBSPACE 82
2.2.6 PROPERTIES O F RELAXED FIRMLY NONEXPANSIVE OPERATORS 84 2.2.7
FIXED POINTS O F FIRMLY NONEXPANSIVE OPERATORS 90
2.3 STRONGLY NONEXPANSIVE OPERATORS 91
2.4 GENERALIZED RELAXATIONS OF ALGORITHMIC OPERATORS 96
2.5 EXERCISES 102
3 CONVERGENCE O F ITERATIVE METHODS 105
3.1 ITERATIVE METHODS 105
3.2 PROPERTIES O F THE WEAK CONVERGENCE 106
3.3 PROPERTIES O F FEJER MONOTONE SEQUENCES 108
3.4 ASYMPTOTICALLY REGULAR OPERATORS I L L
3.5 OPIAL'S THEOREM AND ITS CONSEQUENCES 114
3.6 GENERALIZATION O F OPIAL'S THEOREM 116
3.7 OPIAL-TYPE THEOREMS FOR CUTTERS 118
3.8 STRONG CONVERGENCE O F FEJER MONOTONE SEQUENCES 123
3.9 RELATIONSHIPS AMONG ALGORITHMIC OPERATORS 126
3.10 EXERCISES 127
4 ALGORITHMIC PROJECTION OPERATORS 1 29
4.1 EXAMPLES O F METRIC PROJECTIONS 129
4.1.1 METRIC PROJECTION ONTO A HYPERPLANE 129
4.1.2 METRIC PROJECTION ONTO A FINITE DIMENSIONAL AFFINE SUBSPACE 132
4.1.3 METRIC PROJECTION ONTO A HALF-SPACE 133
4.1.4 METRIC PROJECTION ONTO A BAND 133
4.1.5 METRIC PROJECTION ONTO THE ORTHANT 134
4.1.6 METRIC PROJECTION ONTO BOX CONSTRAINTS 135
4.1.7 METRIC PROJECTION ONTO A BALL 137
4.1.8 METRIC PROJECTION ONTO AN ELLIPSOID 137
4.1.9 METRIC PROJECTION ONTO AN ICE CREAM CONE 140
4.2 CUTTERS 142
4.2.1 CHARACTERIZATION O F CUTTERS 142
4.2.2 CUTTERS WITH SUBSETS O F FIXED POINTS BEING AFFINE SUBSPACES 143
4.2.3 SUBGRADIENT PROJECTION 144
4.3 ALTERNATING PROJECTION 147
4.3.1 BASIC PROPERTIES 147
4.3.2 FIXED POINTS O F THE ALTERNATING PROJECTION 148
4.3.3 ALTERNATING PROJECTION FOR A CLOSED AFFINE SUBSPACE 151 4.3.4
GENERALIZED RELAXATION O F THE ALTERNATING PROJECTION 152 4.3.5 AVERAGED
ALTERNATING REFLECTION 160
4.4 SIMULTANEOUS PROJECTION 162
4.4.1 SIMULTANEOUS PROJECTION AS AN ALTERNATING PROJECTION IN A PRODUCT
SPACE 163
IMAGE 3
CONTENTS XV
4.4.2 PROPERTIES O F THE SIMULTANEOUS PROJECTION 165
4.4.3 SIMULTANEOUS PROJECTION FOR A SYSTEM O F LINEAR EQUATIONS 168
4.4.4 SIMULTANEOUS PROJECTION FOR THE LINEAR FEASIBILITY PROBLEM 169
4.5 CYCLIC PROJECTION 171
4.5.1 CYCLIC RELAXED PROJECTION 172
4.5.2 CYCLIC-SIMULTANEOUS PROJECTION 173
4.5.3 PROJECTIONS WITH REFLECTION ONTO AN OBTUSE CONE 174
4.5.4 CYCLIC CUTTER 176
4.6 LANDWEBER OPERATOR 176
4.6.1 MAIN PROPERTIES 177
4.6.2 LANDWEBER OPERATOR FOR LINEAR SYSTEMS 178
4.6.3 EXTRAPOLATED LANDWEBER OPERATOR FOR A SYSTEM O F LINEAR EQUATIONS
181
4.7 PROJECTED LANDWEBER OPERATOR 184
4.8 SIMULTANEOUS CUTTER 185
4.9 EXTRAPOLATED SIMULTANEOUS CUTTER 187
4.9.1 PROPERTIES OF THE EXTRAPOLATED SIMULTANEOUS CUTTER 187
4.9.2 EXTRAPOLATED SIMULTANEOUS PROJECTION 189
4.9.3 EXTRAPOLATED SIMULTANEOUS PROJECTION FOR LFP 190
4.9.4 SURROGATE PROJECTION 191
4.9.5 SURROGATE PROJECTION WITH RESIDUAL SELECTION 196
4.9.6 EXTRAPOLATED SIMULTANEOUS SUBGRADIENT PROJECTION 198 4.10
EXTRAPOLATED CYCLIC CUTTER 199
4.10.1 USEFUL INEQUALITIES 200
4.10.2 PROPERTIES O F THE EXTRAPOLATED CYCLIC CUTTER 201
4.11 EXERCISES 202
5 PROJECTION METHODS 203
5.1 ALTERNATING PROJECTION METHODS 204
5.1.1 GENERAL CASE 204
5.1.2 ALTERNATING PROJECTION METHOD FOR CLOSED LINEAR SUBSPACES 206
5.2 EXTRAPOLATED ALTERNATING PROJECTION METHODS 208
5.2.1 ACCELERATION TECHNIQUES FOR CONSISTENT PROBLEMS 209
5.2.2 ACCELERATION TECHNIQUES FOR INCONSISTENT PROBLEMS 210 5.2.3
DOUGLAS-RACHFORD ALGORITHM 212
5.3 PROJECTED GRADIENT METHOD 213
5.4 SIMULTANEOUS PROJECTION METHOD 215
5.4.1 CONVERGENCE O F THE SPM 215
5.4.2 PROJECTED SIMULTANEOUS PROJECTION METHODS 217
IMAGE 4
XVI C O N T E N T S
5.5 CYCLIC PROJECTION METHODS 218
5.5.1 CONVERGENCE 219
5.5.2 PROJECTION-REFLECTION METHOD 220
5.6 SUCCESSIVE PROJECTION METHODS 222
5.6.1 CONVERGENCE 222
5.6.2 CONTROL SEQUENCES 223
5.6.3 EXAMPLES 227
5.7 LANDWEBER METHOD AND PROJECTED LANDWEBER METHOD 228
5.8 SIMULTANEOUS CUTTER METHODS 230
5.8.1 ASSUMPTIONS ON WEIGHT FUNCTIONS 231
5.8.2 CONVERGENCE THEOREM 242
5.8.3 EXAMPLES 245
5.8.4 BLOCK ITERATIVE PROJECTION METHODS 249
5.9 SEQUENTIAL CUTTER METHODS 250
5.9.1 CONVERGENCE THEOREM 251
5.9.2 CONTROL SEQUENCES FOR SEQUENTIAL CUTTER METHODS 251
5.9.3 EXAMPLES 252
5.10 EXTRAPOLATED SIMULTANEOUS CUTTER METHODS 253
5.10.1 ASSUMPTIONS ON STEP SIZES 254
5.10.2 CONVERGENCE THEOREM 255
5.10.3 EXTRAPOLATED SIMULTANEOUS SUBGRADIENT PROJECTION METHOD 258
5.11 EXTRAPOLATED CYCLIC CUTTER METHOD 259
5.11.1 CONVERGENCE 260
5.11.2 ACCELERATED KACZMARZ METHOD FOR A SYSTEM O F LINEAR EQUATIONS 262
5.12 SURROGATE CONSTRAINTS METHODS 263
5.12.1 PROPER CONTROL 264
5.12.2 CONVERGENCE THEOREM 265
5.12.3 EXAMPLES O F PROPER CONTROL 266
5.13 SCM WITH RESIDUAL SELECTION 268
5.13.1 GENERAL PROPERTIES 268
5.13.2 DESCRIPTION OF THE METHOD 271
5.13.3 OBTUSE CONE SELECTION 272
5.13.4 REGULAR OBTUSE CONE SELECTION 273
5.14 EXERCISES 274
REFERENCES 275
INDEX 295 |
| any_adam_object | 1 |
| author | Cegielski, Andrzej |
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| author_facet | Cegielski, Andrzej |
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| bvnumber | BV040494135 |
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| ctrlnum | (OCoLC)813209987 (DE-599)DNB1022400045 |
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| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.26 |
| dewey-search | 518.26 |
| dewey-sort | 3518.26 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-08-21T00:15:38Z |
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| isbn | 9783642309007 |
| language | English |
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| physical | XVI, 298 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2012 |
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| spelling | Cegielski, Andrzej Verfasser (DE-588)1027021344 aut Iterative methods for fixed point problems in Hilbert spaces Andrzej Cegielski Berlin [u.a.] Springer 2012 XVI, 298 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 2057 Literaturverz. S. 275 - 289 Iteration (DE-588)4123457-1 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 gnd rswk-swf Fixpunkt (DE-588)4154496-1 gnd rswk-swf Fixpunkt (DE-588)4154496-1 s Iteration (DE-588)4123457-1 s Hilbert-Raum (DE-588)4159850-7 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-30901-4 Lecture notes in mathematics 2057 (DE-604)BV000676446 2057 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4040291&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=025341058&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cegielski, Andrzej Iterative methods for fixed point problems in Hilbert spaces Lecture notes in mathematics Iteration (DE-588)4123457-1 gnd Hilbert-Raum (DE-588)4159850-7 gnd Fixpunkt (DE-588)4154496-1 gnd |
| subject_GND | (DE-588)4123457-1 (DE-588)4159850-7 (DE-588)4154496-1 |
| title | Iterative methods for fixed point problems in Hilbert spaces |
| title_auth | Iterative methods for fixed point problems in Hilbert spaces |
| title_exact_search | Iterative methods for fixed point problems in Hilbert spaces |
| title_full | Iterative methods for fixed point problems in Hilbert spaces Andrzej Cegielski |
| title_fullStr | Iterative methods for fixed point problems in Hilbert spaces Andrzej Cegielski |
| title_full_unstemmed | Iterative methods for fixed point problems in Hilbert spaces Andrzej Cegielski |
| title_short | Iterative methods for fixed point problems in Hilbert spaces |
| title_sort | iterative methods for fixed point problems in hilbert spaces |
| topic | Iteration (DE-588)4123457-1 gnd Hilbert-Raum (DE-588)4159850-7 gnd Fixpunkt (DE-588)4154496-1 gnd |
| topic_facet | Iteration Hilbert-Raum Fixpunkt |
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| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT cegielskiandrzej iterativemethodsforfixedpointproblemsinhilbertspaces |