Regularization theory for Ill-posed problems: selected topics
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin ; Boston, Mass.
<<De>> Gruyter
2013
|
| Schriftenreihe: | Inverse and ill-posed problems series
58 |
| Schlagwörter: | |
| Links: | http://deposit.dnb.de/cgi-bin/dokserv?id=4276989&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=027179826&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIV, 287 S. graph. Darst. |
| ISBN: | 3110286467 9783110286465 9783110286502 |
Internformat
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| 245 | 1 | 0 | |a Regularization theory for Ill-posed problems |b selected topics |c Shuai Lu ; Sergei V. Pereverzev |
| 264 | 1 | |a Berlin ; Boston, Mass. |b <<De>> Gruyter |c 2013 | |
| 300 | |a XIV, 287 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Inverse and ill-posed problems series |v 58 | |
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| 650 | 0 | 7 | |a Inkorrekt gestelltes Problem |0 (DE-588)4186951-5 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1819256362915332096 |
|---|---|
| adam_text | IMAGE 1
CONTENTS
PREFACE VII
1 AN INTRODUCTION USING CLASSICAL EXAMPLES 1
1.1 NUMERICAL DIFFERENTIATION. FIRST LOOK AT THE PROBLEM O F
REGULARIZATION. THE BALANCING PRINCIPLE 1
1.1.1 FINITE-DIFFERENCE FORMULAE 1
1.1.2 FINITE-DIFFERENCE FORMULAE FOR NONEXACT DATA. A PRIORI CHOICE O F
THE STEPSIZE 3
1.1.3 A POSTERIORI CHOICE O F THE STEPSIZE 6
1.1.4 NUMERICAL ILLUSTRATION 9
1.1.5 THE BALANCING PRINCIPLE IN A GENERAL FRAMEWORK 10
1.2 STABLE SUMMATION O F ORTHOGONAL SERIES WITH NOISY COEFFICIENTS.
DETERMINISTIC AND STOCHASTIC NOISE MODELS. DESCRIPTION O F SMOOTHNESS
PROPERTIES 12
1.2.1 SUMMATION METHODS 13
1.2.2 DETERMINISTIC NOISE MODEL 14
1.2.3 STOCHASTIC NOISE MODEL 15
1.2.4 SMOOTHNESS ASSOCIATED WITH A BASIS 18
1.2.5 APPROXIMATION AND STABILITY PROPERTIES O F A-METHODS 19
1.2.6 ERROR BOUNDS 21
1.3 THE ELLIPTIC CAUCHY PROBLEM AND REGULARIZATION BY DISCRETIZATION . .
. . 25 1.3.1 NATURAL LINEARIZATION O F THE ELLIPTIC CAUCHY PROBLEM 27
1.3.2 REGULARIZATION BY DISCRETIZATION 36
1.3.3 APPLICATION IN DETECTING CORROSION 39
2 BASICS O F SINGLE PARAMETER REGULARIZATION SCHEMES 47
2.1 SIMPLE EXAMPLE FOR MOTIVATION 47
2.2 ESSENTIALLY ILL-POSED LINEAR OPERATOR EQUATIONS. LEAST-SQUARES
SOLUTION. GENERAL VIEW ON REGULARIZATION 49
2.3 SMOOTHNESS IN THE CONTEXT O F THE PROBLEM. BENCHMARK ACCURACY LEVELS
FOR DETERMINISTIC AND STOCHASTIC DATA NOISE MODELS 65
2.3.1 THE BEST POSSIBLE ACCURACY FOR THE DETERMINISTIC NOISE MODEL 68
HTTP://D-NB.INFO/1032207221
IMAGE 2
XLL
CONTENTS
2.3.2 THE BEST POSSIBLE ACCURACY FOR THE GAUSSIAN WHITE
NOISE MODEL 73
2.4 OPTIMAL ORDER AND THE SATURATION O F REGULARIZATION METHODS IN
HILBERT SPACES 80
2.5 CHANGING THE PENALTY TERM FOR VARIANCE REDUCTION. REGULARIZATION IN
HILBERT SCALES 90
2.6 ESTIMATION O F LINEAR FUNCTIONALS FROM INDIRECT NOISY OBSERVATIONS .
. . . 101
2.7 REGULARIZATION BY FINITE-DIMENSIONAL APPROXIMATION 113
2.8 MODEL SELECTION BASED ON INDIRECT OBSERVATION IN GAUSSIAN WHITE
NOISE 124 2.8.1 LINEAR MODELS GIVEN BY LEAST-SQUARES METHODS 127
2.8.2 OPERATOR MONOTONE FUNCTIONS 131
2.8.3 THE PROBLEM O F MODEL SELECTION (CONTINUATION) 137
2.9 A WARNING EXAMPLE: AN OPERATOR EQUATION FORMULATION IS NOT ALWAYS
ADEQUATE (NUMERICAL DIFFERENTIATION REVISITED) 141
2.9.1 NUMERICAL DIFFERENTIATION IN VARIABLE HILBERT SCALES ASSOCIATED
WITH DESIGNS 143
2.9.2 ERROR BOUNDS IN L2 147
2.9.3 ADAPTATION TO THE UNKNOWN BOUND O F THE APPROXIMATION ERROR 150
2.9.4 NUMERICAL DIFFERENTIATION IN THE SPACE O F CONTINUOUS FUNCTIONS
151
2.9.5 RELATION TO THE SAVITZKY-GOLAY METHOD. NUMERICAL EXAMPLES 155
3 MULTIPARAMETER REGULARIZATION 163
3.1 WHEN DO WE REALLY NEED MULTIPARAMETER REGULARIZATION? 163
3.2 MULTIPARAMETER DISCREPANCY PRINCIPLE 165
3.2.1 MODEL FUNCTION BASED ON THE MULTIPARAMETER DISCREPANCY PRINCIPLE
168
3.2.2 A USE O F THE MODEL FUNCTION TO APPROXIMATE ONE SET OF PARAMETERS
SATISFYING THE DISCREPANCY PRINCIPLE 170
3.2.3 PROPERTIES O F THE MODEL FUNCTION APPROXIMATION 172
3.2.4 DISCREPANCY CURVE AND THE CONVERGENCE ANALYSIS 173
3.2.5 HEURISTIC ALGORITHM FOR THE MODEL FUNCTION APPROXIMATION O F THE
MULTIPARAMETER DISCREPANCY PRINCIPLE 174
3.2.6 GENERALIZATION IN THE CASE O F MORE THAN TWO REGULARIZATION
PARAMETERS 175
3.3 NUMERICAL REALIZATION AND TESTING 177
3.3.1 NUMERICAL EXAMPLES AND COMPARISON 177
IMAGE 3
CONTENTS
XLLL
3.3.2 TWO-PARAMETER DISCREPANCY CURVE 182
3.3.3 A NUMERICAL CHECK O F PROPOSITION 3.1 AND USE O F A DISCREPANCY
CURVE 184
3.3.4 EXPERIMENTS WITH THREE-PARAMETER REGULARIZATION 187
3.4 TWO-PARAMETER REGULARIZATION WITH ONE NEGATIVE PARAMETER FOR
PROBLEMS WITH NOISY OPERATORS AND RIGHT-HAND SIDE 189
3.4.1 COMPUTATIONAL ASPECTS FOR REGULARIZED TOTAL LEAST SQUARES 191
3.4.2 COMPUTATIONAL ASPECTS FOR DUAL REGULARIZED TOTAL LEAST SQUARES .
192 3.4.3 ERROR BOUNDS IN THE CASE B = I 193
3.4.4 ERROR BOUNDS FOR B ^ I 195
3.4.5 NUMERICAL ILLUSTRATIONS. MODEL FUNCTION APPROXIMATION IN DUAL
REGULARIZED TOTAL LEAST SQUARES 197
4 REGULARIZATION ALGORITHMS IN LEARNING THEORY 203
4.1 SUPERVISED LEARNING PROBLEM AS AN OPERATOR EQUATION IN A REPRODUCING
KERNEL HILBERT SPACE (RKHS) 203
4.1.1 REPRODUCING KERNEL HILBERT SPACES AND RELATED OPERATORS 205 4.1.2
A PRIORI ASSUMPTION ON THE PROBLEM: GENERAL SOURCE CONDITIONS 207
4.2 KERNEL INDEPENDENT LEARNING RATES 209
4.2.1 REGULARIZATION FOR BINARY CLASSIFICATION: RISK BOUNDS AND BAYES
CONSISTENCY 217
4.3 ADAPTIVE KERNEL METHODS USING THE BALANCING PRINCIPLE 218
4.3.1 ADAPTIVE LEARNING WHEN THE ERROR MEASURE IS KNOWN 220
4.3.2 ADAPTIVE LEARNING WHEN THE ERROR MEASURE IS UNKNOWN 223 4.3.3
PROOFS O F PROPOSITIONS 4.6 AND 4.7 225
4.3.4 NUMERICAL EXPERIMENTS. QUASIBALANCING PRINCIPLE 231
4.4 KERNEL ADAPTIVE REGULARIZATION WITH APPLICATION TO BLOOD GLUCOSE
READING 235
4.4.1 READING THE BLOOD GLUCOSE LEVEL FROM SUBCUTANEOUS ELECTRIC CURRENT
MEASUREMENTS 242
4.5 MULTIPARAMETER REGULARIZATION IN LEARNING THEORY 249
5 META-LEARNING APPROACH TO REGULARIZATION - CASE STUDY: BLOOD GLUCOSE
PREDICTION 255
5.1 A BRIEF INTRODUCTION TO META-LEARNING AND BLOOD GLUCOSE PREDICTION .
. . 255
5.2 A TRADITIONAL LEARNING THEORY APPROACH: ISSUES AND CONCERNS 259
5.3 META-LEARNING APPROACH TO CHOOSING A KERNEL AND A REGULARIZATION
PARAMETER 261
5.3.1 OPTIMIZATION OPERATION 263
IMAGE 4
X I V CONTENTS
5.3.2 HEURISTIC OPERATION 267
5.3.3 LEARNING AT METALEVEL 267
5.4 CASE-STUDY: BLOOD GLUCOSE PREDICTION 269
BIBLIOGRAPHY 277
INDEX 289
|
| any_adam_object | 1 |
| author | Lu, Shuai 1979- Pereverzev, Sergej V. |
| author_GND | (DE-588)138671664 |
| author_facet | Lu, Shuai 1979- Pereverzev, Sergej V. |
| author_role | aut aut |
| author_sort | Lu, Shuai 1979- |
| author_variant | s l sl s v p sv svp |
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| dewey-full | 518.1 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.1 |
| dewey-search | 518.1 |
| dewey-sort | 3518.1 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041733082 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T16:53:47Z |
| institution | BVB |
| isbn | 3110286467 9783110286465 9783110286502 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027179826 |
| oclc_num | 846519091 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XIV, 287 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | <<De>> Gruyter |
| record_format | marc |
| series2 | Inverse and ill-posed problems series |
| spellingShingle | Lu, Shuai 1979- Pereverzev, Sergej V. Regularization theory for Ill-posed problems selected topics Regularisierungsverfahren (DE-588)4846428-4 gnd Inverses Problem (DE-588)4125161-1 gnd Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd |
| subject_GND | (DE-588)4846428-4 (DE-588)4125161-1 (DE-588)4186951-5 |
| title | Regularization theory for Ill-posed problems selected topics |
| title_auth | Regularization theory for Ill-posed problems selected topics |
| title_exact_search | Regularization theory for Ill-posed problems selected topics |
| title_full | Regularization theory for Ill-posed problems selected topics Shuai Lu ; Sergei V. Pereverzev |
| title_fullStr | Regularization theory for Ill-posed problems selected topics Shuai Lu ; Sergei V. Pereverzev |
| title_full_unstemmed | Regularization theory for Ill-posed problems selected topics Shuai Lu ; Sergei V. Pereverzev |
| title_short | Regularization theory for Ill-posed problems |
| title_sort | regularization theory for ill posed problems selected topics |
| title_sub | selected topics |
| topic | Regularisierungsverfahren (DE-588)4846428-4 gnd Inverses Problem (DE-588)4125161-1 gnd Inkorrekt gestelltes Problem (DE-588)4186951-5 gnd |
| topic_facet | Regularisierungsverfahren Inverses Problem Inkorrekt gestelltes Problem |
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