Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs:
Saved in:
| Main Author: | |
|---|---|
| Format: | Thesis Book |
| Language: | English |
| Published: |
Göttingen
Universitätsverlag Göttingen
2019
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | 222 Seiten Diagramme 24 cm |
| ISBN: | 9783863954116 3863954114 |
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MARC
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| 100 | 1 | |a Weidling, Frederic |e Verfasser |0 (DE-588)1194586449 |4 aut | |
| 245 | 1 | 0 | |a Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs |c Frederic Weidling |
| 264 | 1 | |a Göttingen |b Universitätsverlag Göttingen |c 2019 | |
| 300 | |a 222 Seiten |b Diagramme |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 502 | |b Dissertation |c Universität Göttingen |d 2019 | ||
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| 653 | |a Variationelle Quellbedingungen | ||
| 653 | |a Verifizierung | ||
| 653 | |a 1620: Hardcover, Softcover / Mathematik | ||
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Record in the Search Index
| _version_ | 1804180746924982272 |
|---|---|
| adam_text | C
ONTENTS
INTRODUCTION
5
CONTENTS
11
1
TIKHONOV
REGULARIZATION
15
1.1
ILL-POSED
PROBLEMS
AND
THEIR
REGULARIZATION
.........................................
15
1.2
GENERALIZED
TIKHONOV
FUNCTIONALS
......................................................
16
1.2.1
NOISE
MODEL
...............................................................................
17
1.2.2
REGULARIZING
PROPERTY
..............................................................
19
1.2.3
NUMERICAL
MINIMIZATION
OF
TIKHONOV
FUNCTIONALS
...................
22
2
CONVERGENCE
RATES
THEORY
27
2.1
SPECTRAL
SOURCE
CONDITIONS
....................................................................
28
2.1.1
CONVERGENCE
RATES
....................................................................
30
2.1.2
LOWER
BOUNDS
ON
RATES
..............................................................
32
2.1.3
PARAMETER
CHOICE
RULES
..............................................................
34
2.1.4
NONLINEAR
OPERATORS
.................................................................
37
2.2
RANGE
CONDITIONS
.................................................................................
38
2.3
RATES
BY
STABILITY
ESTIMATES
.................................................................
40
2.4
VARIATIONAL
SOURCE
CONDITIONS
..............................................................
43
2.4.1
CONVERGENCE
RATES
....................................................................
44
12
CONTENTS
2.4.2
A
META-THEOREM
TO
PROVE
VARIATIONAL
SOURCE
CONDITIONS
WITH
BREGMAN
LOSS
............................................................................
46
2.4.3
RELATION
TO
PREVIOUS
CONVERGENCE
RATE
RESULTS
...........................
55
2.4.4
FURTHER
RESULTS
ON
VARIATIONAL
SOURCE
CONDITIONS
......................
60
3
EQUIVALENCE
RESULTS
IN
HILBERT
SPACES
67
3.1
SPECTRAL
TAIL
CONDITION
.........................................................................
69
3.2
APPROXIMATION
PROPERTY
OF
REGULARIZATION
METHODS
...........................
71
3.3
CONVERGENCE
RATES
RESULTS
....................................................................
75
3.3.1
CONVERGENCE
RATES
FOR
AN
ORACLE
PARAMETER
CHOICE
...................
75
3.3.2
CONVERGENCE
RATES
FOR
QUASIOPTIMAL
PARAMETER
CHOICE
RULES
.
.
79
3.4
WHITE
NOISE
ERROR
MODEL
.......................................................................
83
3.5
INTERPRETATION
OF
MAXISETS
....................................................................
87
3.6
EXAMPLES
...............................................................................................
90
4
OPTIMAL
CONVERGENCE
RATES
IN
BESOV
SPACES
ON
THE
TORUS
97
4.1
LOWER
BOUNDS
ON
CONVERGENCE
RATES
.................................................
99
4.2
SUBGRADIENT
SMOOTHNESS
.........................................................................
101
4.2.1
SMOOTHNESS
OF
BESOV
NORM
SUBGRADIENTS
...................................
101
4.2.2
JACKSON-TYPE
INEQUALITY
..............................................................
105
4.3
ABSTRACT
CONVERGENCE
RATES
FOR
GAUSSIAN
WHITE
NOISE
........................
107
4.4
APPLICATION
TO
FINITELY
SMOOTHING
OPERATORS
........................................
112
4.4.1
VERIFICATION
OF
VARIATIONAL
SOURCE
CONDITIONS
AND
CONVERGENCE
RATES
............................................................................................
112
4.4.2
LOWER
BOUNDS
..............................................................................
117
4.4.3
NUMERICAL
VALIDATION
...................................................................
119
4.5
APPLICATION
TO
BACKWARD
HEAT
EQUATION
.................................................
120
5
CONVERGENCE
RATES
FOR
THE
REGULARIZED
SCHRODINGER
EQUATION
125
5.1
PROBLEM
DESCRIPTION
..............................................................................
126
5.2
REGULARIZATION
APPROACH
......................................................................
127
5.3
ILL-POSEDNESS
ESTIMATE
...........................................................................
130
5.3.1
COMPLEX
GEOMETRIC
OPTICS
SOLUTIONS
...........................................
131
5.3.2
ENERGY
DEPENDENT
ESTIMATES
OF
LOW
FOURIER
FREQUENCIES
FOR
THE
DIFFERENCE
OF
TWO
POTENTIALS
........................................................
138
5.4
CONVERGENCE
RATES
.................................................................................
143
5.4.1
EXTENSION
TO
FAR
FIELD
DATA
...........................................................
143
5.4.2
PROOF
OF
THE
THEOREMS
.................................................................
144
CONTENTS
13
5.4.3
DISCUSSION
OF
THE
RESULT
..............................................................
151
6
ELECTRICAL
IMPEDANCE
TOMOGRAPHY
155
6.1
THE
ELECTRICAL
IMPEDANCE
TOMOGRAPHY
PROBLEM
...................................
156
6.1.1
PROBLEM
SETUP
..............................................................................
157
6.1.2
REGULARIZATION
OF
ELECTRICAL
IMPEDANCE
TOMOGRAPHY
..................
159
6.2
STABILITY
ESTIMATES
FOR
ELECTRICAL
IMPEDANCE
TOMOGRAPHY
.....................
162
6.2.1
CONNECTION
TO
SCHRODINGER
EQUATION
...........................................
163
6.2.2
COMPLEX
GEOMETRIC
OPTICS
SOLUTIONS
FOR
LESS
REGULAR
POTENTIALS
.
167
6.2.3
STABILITY
ESTIMATES
......................................................................
174
6.3
CONVERGENCE
RATES
.................................................................................
180
6.3.1
EXCURSION:
FROM
STABILITY
TO
VARIATIONAL
SOURCE
CONDITIONS
....
181
6.3.2
VARIATIONAL
SOURCE
CONDITIONS
FOR ELECTRICAL
IMPEDANCE
TOMOGRA
PHY
..............................................................................................
184
CONCLUSION
AND
OUTLOOK
187
A
CONVEX
ANALYSIS
189
B
FUNCTION
SPACES
193
B.L
GEOMETRY
OF
BANACH
SPACES
...................................................................
193
B.1.1
SMOOTHNESS
AND
CONVEXITY
........................................................
193
B.1.2
DUALITY
MAPPINGS
......................................................................
195
B.2
REAL
INTERPOLATION
...................................................................................
196
B.3
BESOV
SPACES
...........................................................................................
198
B.3.1
BESOV
SPACES
ON
THE
EUCLIDEAN
SPACE
...........................................
199
B.3.2
PERIODIC
BESOV
SPACES
................................................................
203
BIBLIOGRAPHY
205
ACKNOWLEDGEMENT
221
|
| any_adam_object | 1 |
| author | Weidling, Frederic |
| author_GND | (DE-588)1194586449 |
| author_facet | Weidling, Frederic |
| author_role | aut |
| author_sort | Weidling, Frederic |
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| building | Verbundindex |
| bvnumber | BV046288372 |
| ctrlnum | (OCoLC)1112883982 (DE-599)DNB1193019354 |
| discipline | Mathematik |
| format | Thesis Book |
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| id | DE-604.BV046288372 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:40:40Z |
| institution | BVB |
| institution_GND | (DE-588)6146838-1 |
| isbn | 9783863954116 3863954114 |
| language | English |
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| physical | 222 Seiten Diagramme 24 cm |
| publishDate | 2019 |
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| publisher | Universitätsverlag Göttingen |
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| spelling | Weidling, Frederic Verfasser (DE-588)1194586449 aut Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs Frederic Weidling Göttingen Universitätsverlag Göttingen 2019 222 Seiten Diagramme 24 cm txt rdacontent n rdamedia nc rdacarrier Dissertation Universität Göttingen 2019 Inverses Problem (DE-588)4125161-1 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Paperback / softback )Unsewn / adhesive bound Glattheit Variationelle Quellbedingungen Verifizierung 1620: Hardcover, Softcover / Mathematik (DE-588)4113937-9 Hochschulschrift gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s Inverses Problem (DE-588)4125161-1 s DE-604 Universitätsverlag Göttingen (DE-588)6146838-1 pbl B:DE-101 application/pdf https://d-nb.info/1193019354/04 Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031665905&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Weidling, Frederic Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs Inverses Problem (DE-588)4125161-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4125161-1 (DE-588)4044779-0 (DE-588)4113937-9 |
| title | Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs |
| title_auth | Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs |
| title_exact_search | Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs |
| title_full | Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs Frederic Weidling |
| title_fullStr | Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs Frederic Weidling |
| title_full_unstemmed | Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs Frederic Weidling |
| title_short | Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs |
| title_sort | variational source conditions and conditional stability estimates for inverse problems in pdes |
| topic | Inverses Problem (DE-588)4125161-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Inverses Problem Partielle Differentialgleichung Hochschulschrift |
| url | https://d-nb.info/1193019354/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031665905&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT weidlingfrederic variationalsourceconditionsandconditionalstabilityestimatesforinverseproblemsinpdes AT universitatsverlaggottingen variationalsourceconditionsandconditionalstabilityestimatesforinverseproblemsinpdes |
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