Iterative methods for ill-posed problems: an introduction
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Berlin [u.a.]
De Gruyter
2011
|
| Series: | Inverse and ill-posed problems series
54 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XI, 136 S. graph. Darst. 25 cm, 399 g |
| ISBN: | 9783110250640 9783110250657 |
Staff View
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| 100 | 1 | |a Bakušinskij, Anatolij B. |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Iterativnye metody resenija nekorrektnych zadač |
| 245 | 1 | 0 | |a Iterative methods for ill-posed problems |b an introduction |c Anatoly B. Bakushinsky ; Mihail Yu. Kokurin ; Alexandra Smirnova |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2011 | |
| 300 | |a XI, 136 S. |b graph. Darst. |c 25 cm, 399 g | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Record in the Search Index
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IMAGE 1
CONTENTS
PREFACE V
1 THE REGULARITY CONDITION. NEWTON'S METHOD 1
1.1 PRELIMINARY RESULTS 1
1.2 LINEARIZATION PROCEDURE 2
1.3 ERROR ANALYSIS 4
PROBLEMS 6
2 THE GAUSS-NEWTON METHOD 10
2.1 MOTIVATION 10
2.2 CONVERGENCE RATES 12
PROBLEMS 14
3 THE GRADIENT METHOD 16
3.1 THE GRADIENT METHOD FOR REGULAR PROBLEMS 16
3.2 ILL-POSED CASE 18
PROBLEMS 20
4 TIKHONOV'S SCHEME 23
4.1 THE TIKHONOV FUNCTIONAL 23
4.2 PROPERTIES OF A MINIMIZING SEQUENCE 24
4.3 OTHER TYPES OF CONVERGENCE 27
4.4 EQUATIONS WITH NOISY DATA 29
PROBLEMS 30
5 TIKHONOV'S SCHEME FOR LINEAR EQUATIONS 32
5.1 THE MAIN CONVERGENCE RESULT 32
5.2 ELEMENTS OF SPECTRAL THEORY 34
5.3 MINIMIZING SEQUENCES FOR LINEAR EQUATIONS 35
5.4 A PRIORI AGREEMENT BETWEEN THE REGULARIZATION PARAMETER AND THE
ERROR FOR EQUATIONS WITH PERTURBED RIGHT-HAND SIDES 37
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1009312847
DIGITALISIERT DURCH
IMAGE 2
X CONTENTS
5.5 THE DISCREPANCY PRINCIPLE 40
5.6 APPROXIMATION OF A QUASI-SOLUTION 43
PROBLEMS 43
6 THE GRADIENT SCHEME FOR LINEAR EQUATIONS 45
6.1 THE TECHNIQUE OF SPECTRAL ANALYSIS 45
6.2 A PRIORI STOPPING RULE 48
6.3 A POSTERIORI STOPPING RULE 49
PROBLEMS 53
7 CONVERGENCE RATES FOR THE APPROXIMATION METHODS IN THE CASE OF LINEAR
IRREGULAR EQUATIONS 54
7.1 THE SOURCE-TYPE CONDITION (STC) 54
7.2 STC FOR THE GRADIENT METHOD 57
7.3 THE SATURATION PHENOMENA 59
7.4 APPROXIMATIONS IN CASE OF A PERTURBED STC 61
7.5 ACCURACY OF THE ESTIMATES 62
PROBLEMS 63
8 EQUATIONS WITH A CONVEX DISCREPANCY FUNCTIONAL BY TIKHONOV'S METHOD 64
8.1 SOME DIFFICULTIES ASSOCIATED WITH TIKHONOV'S METHOD IN CASE OF A
CONVEX DISCREPANCY FUNCTIONAL 64
8.2 AN ILLUSTRATIVE EXAMPLE 65
PROBLEMS 67
9 ITERATIVE REGULARIZARON PRINCIPLE 69
9.1 THE IDEA OF ITERATIVE REGULARIZATION 69
9.2 THE ITERATIVELY REGULARIZED GRADIENT METHOD 70
PROBLEMS 74
10 THE ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD 76
10.1 CONVERGENCE ANALYSIS 76
10.2 FURTHER PROPERTIES OF IRGN ITERATIONS 79
10.3 A UNIFIED APPROACH TO THE CONSTRUCTION OF ITERATIVE METHODS FOR
IRREGULAR EQUATIONS 83
10.4 THE REVERSE CONNECTION CONTROL 84
PROBLEMS 88
11 THE STABLE GRADIENT METHOD FOR IRREGULAR NONLINEAR EQUATIONS 90 11.1
SOLVING AN AUXILIARY FINITE DIMENSIONAL PROBLEM BY THE GRADIENT DESCENT
METHOD 90
11.2 INVESTIGATION OF A DIFFERENCE INEQUALITY 94
11.3 THE CASE OF NOISY DATA 95
PROBLEMS 97
IMAGE 3
CONTENTS XI
12 RELATIVE COMPUTATIONAL EFFICIENCY OF ITERATIVELY REGULARIZED METHODS
. 98 12.1 GENERALIZED GAUSS-NEWTON METHODS 98
12.2 A MORE RESTRICTIVE SOURCE CONDITION 100
12.3 COMPARISON TO ITERATIVELY REGULARIZED GRADIENT SCHEME 101 PROBLEMS
102
13 NUMERICAL INVESTIGATION OF TWO-DIMENSIONAL INVERSE GRAVIMETRY PROBLEM
103
13.1 PROBLEM FORMULATION 103
13.2 THE ALGORITHM 104
13.3 SIMULATIONS 105
PROBLEMS 109
14 ITERATIVELY REGULARIZED METHODS FOR INVERSE PROBLEM IN OPTICAL
TOMOGRAPHY I LL
14.1 STATEMENT OF THE PROBLEM I LL
14.2 SIMPLE EXAMPLE 112
14.3 FORWARD SIMULATION 114
14.4 THE INVERSE PROBLEM 116
14.5 NUMERICAL RESULTS 119
PROBLEMS 121
15 FEIGENBAUM'S UNIVERSALITY EQUATION 123
15.1 THE UNIVERSAL CONSTANTS 123
15.2 IU-POSEDNESS 125
15.3 NUMERICAL ALGORITHM FOR 2 Z 12 125
15.4 REGULARIZED METHOD FOR Z 13 127
PROBLEMS 128
16 CONCLUSION 130
REFERENCES 132
INDEX 137 |
| any_adam_object | 1 |
| author | Bakušinskij, Anatolij B. Kokurin, Michail Ju Smirnova, Aleksandra B. |
| author_facet | Bakušinskij, Anatolij B. Kokurin, Michail Ju Smirnova, Aleksandra B. |
| author_role | aut aut aut |
| author_sort | Bakušinskij, Anatolij B. |
| author_variant | a b b ab abb m j k mj mjk a b s ab abs |
| building | Verbundindex |
| bvnumber | BV037266888 |
| classification_rvk | SK 620 SK 910 SK 920 |
| ctrlnum | (OCoLC)711846836 (DE-599)DNB1009312847 |
| dewey-full | 515.357 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.357 |
| dewey-search | 515.357 |
| dewey-sort | 3515.357 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV037266888 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:00:49Z |
| institution | BVB |
| isbn | 9783110250640 9783110250657 |
| language | English Russian |
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| owner | DE-20 DE-19 DE-BY-UBM |
| owner_facet | DE-20 DE-19 DE-BY-UBM |
| physical | XI, 136 S. graph. Darst. 25 cm, 399 g |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | De Gruyter |
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| series | Inverse and ill-posed problems series |
| series2 | Inverse and ill-posed problems series |
| spelling | Bakušinskij, Anatolij B. Verfasser aut Iterativnye metody resenija nekorrektnych zadač Iterative methods for ill-posed problems an introduction Anatoly B. Bakushinsky ; Mihail Yu. Kokurin ; Alexandra Smirnova Berlin [u.a.] De Gruyter 2011 XI, 136 S. graph. Darst. 25 cm, 399 g txt rdacontent n rdamedia nc rdacarrier Inverse and ill-posed problems series 54 Literaturangaben Iteration (DE-588)4123457-1 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 gnd rswk-swf Operatorgleichung (DE-588)4043601-9 gnd rswk-swf Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd rswk-swf Inkorrekt gestelltes Problem (DE-588)4186951-5 s Iteration (DE-588)4123457-1 s Operatorgleichung (DE-588)4043601-9 s Hilbert-Raum (DE-588)4159850-7 s DE-604 Kokurin, Michail Ju. Verfasser aut Smirnova, Aleksandra B. Verfasser aut Inverse and ill-posed problems series 54 (DE-604)BV037286556 54 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3645007&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=021179915&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bakušinskij, Anatolij B. Kokurin, Michail Ju Smirnova, Aleksandra B. Iterative methods for ill-posed problems an introduction Inverse and ill-posed problems series Iteration (DE-588)4123457-1 gnd Hilbert-Raum (DE-588)4159850-7 gnd Operatorgleichung (DE-588)4043601-9 gnd Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd |
| subject_GND | (DE-588)4123457-1 (DE-588)4159850-7 (DE-588)4043601-9 (DE-588)4186951-5 |
| title | Iterative methods for ill-posed problems an introduction |
| title_alt | Iterativnye metody resenija nekorrektnych zadač |
| title_auth | Iterative methods for ill-posed problems an introduction |
| title_exact_search | Iterative methods for ill-posed problems an introduction |
| title_full | Iterative methods for ill-posed problems an introduction Anatoly B. Bakushinsky ; Mihail Yu. Kokurin ; Alexandra Smirnova |
| title_fullStr | Iterative methods for ill-posed problems an introduction Anatoly B. Bakushinsky ; Mihail Yu. Kokurin ; Alexandra Smirnova |
| title_full_unstemmed | Iterative methods for ill-posed problems an introduction Anatoly B. Bakushinsky ; Mihail Yu. Kokurin ; Alexandra Smirnova |
| title_short | Iterative methods for ill-posed problems |
| title_sort | iterative methods for ill posed problems an introduction |
| title_sub | an introduction |
| topic | Iteration (DE-588)4123457-1 gnd Hilbert-Raum (DE-588)4159850-7 gnd Operatorgleichung (DE-588)4043601-9 gnd Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd |
| topic_facet | Iteration Hilbert-Raum Operatorgleichung Inkorrekt gestelltes Problem |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3645007&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=021179915&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV037286556 |
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