Wavelet methods for elliptic partial differential equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2009
|
| Edition: | 1. publ. |
| Series: | Numerical mathematics and scientific computation
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVII, 480 S. Ill., graph. Darst. |
| ISBN: | 9780198526056 |
Staff View
MARC
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| 100 | 1 | |a Urban, Karsten |d 1966- |e Verfasser |0 (DE-588)141064684 |4 aut | |
| 245 | 1 | 0 | |a Wavelet methods for elliptic partial differential equations |c Karsten Urban |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2009 | |
| 300 | |a XXVII, 480 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Numerical mathematics and scientific computation | |
| 650 | 4 | |a Differential equations, Elliptic | |
| 650 | 4 | |a Wavelets (Mathematics) | |
| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Wavelet |0 (DE-588)4215427-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |D s |
| 689 | 0 | 1 | |a Wavelet |0 (DE-588)4215427-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016411133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016411133 | ||
Record in the Search Index
| _version_ | 1804137514906157056 |
|---|---|
| adam_text | CONTENTS
List of Algorithms
xii
Preface
xiii
Acknowledgements
xv
List of Figures
xvii
List of Tables
xxvi
1
Introduction
1
1.1
Some aspects of the history of wavelets
1
1.2
The scope of this book
4
1.3
Outline
6
2
Multiscale approximation and multiresolution
9
2.1
The
Haar
system
9
2.1.1
Projection by interpolation
9
2.1.2
Orthogonal projection
12
2.2
Piecewise linear systems
17
2.3
Similar properties
24
2.3.1
Stability
24
2.3.2
Refinement relation
25
2.3.3
Multiresolution
26
2.3.4
Locality
27
2.4
Multiresolution analysis on the real line
28
2.4.1
The scaling function
29
2.4.2
When does a mask define a refinable function?
30
2.4.3
Consequences of the refinability
31
2.5
Daubechies
orthonormal
scaling functions
35
2.6
B-splines
37
2.6.1
Centralized B-splines
38
2.7
Dual scaling functions associated to B-splines
39
2.8
Multilevel projectors
43
2.9
Approximation properties
46
2.9.1
A general framework
46
2.9.2
Stability properties
48
2.9.3
Error estimates
49
2.10
Plotting scaling functions
50
2.10.1
Subdivision
51
2.10.2
Cascade algorithm
55
2.11
Periodization
57
2.12
Exercises and programs
58
vu
CONTENTS
Elliptic boundary value problems
63
3.1
A model problem
63
3.1.1
Variational formulation
64
3.1.2
Existence and uniqueness
68
3.2
Variational formulation
70
3.2.1
Operators associated by the bilinear form
74
3.2.2
Reduction to homogeneous boundary
conditions
75
3.2.3
Stability
75
3.3
Regularity theory
76
3.4
Galerkin methods
77
3.4.1
Discretization
77
3.4.2
Stability
78
3.4.3
Error estimates
78
3.4.4
^-estimates
82
3.4.5
Numerical solution
83
3.5
Exercises and programs
85
Multiresolution Galerkin methods
87
4.1
Multiscale discretization
87
4.2
Multiresolution multiscale discretization
89
4.2.1
Piecewise linear multiresolution
89
4.2.2
Periodic boundary value problems
90
4.2.3
Common properties
94
4.3
Error estimates
95
4.4
Some numerical examples
96
4.5
Setup of the algebraic system
97
4.5.1
Refinable integrals
98
4.5.2
The right-hand side
101
4.5.3
Quadrature
101
4.6
The BPX preconditioner
102
4.7
MultiGrid
104
4.8
Numerical examples for the model problem
108
4.9
Exercises and programs
115
Wavelets
120
5.1
Detail spaces
120
5.1.1
Updating
120
5.1.2
The
Haar
system again
121
5.2
Orthogonal wavelets
124
5.2.1
Multilevel decomposition
124
5.2.2
The construction of wavelets
126
5.2.3
Wavelet projectors
129
5.3 Biorthogonal
wavelets
131
5.3.1 Biorthogonal
complement spaces
132
CONTENTS ix
5.3.2 Biorthogonal
projectors
133
5.3.3 Biorthogonal B-spline
wavelets
134
5.4 Fast
Wavelet Transform (FWT)
134
5.4.1
Decomposition
141
5.4.2
Reconstruction
143
5.4.3
Efficiency
145
5.4.4
A generał
framework
145
5.5
Vanishing moments and compression
147
5.6
Norm equivalences
150
5.6.1
Jackson inequality
151
5.6.2
Bernstein inequality
152
5.6.3
A characterization theorem
152
5.7
Other kinds of wavelets
156
5.7.1
Interpolatory
wavelets
157
5.7.2 Semiorthogonal
wavelets
161
5.7.3
Noncompactly supported wavelets
163
5.7.4
Multiwavelets
164
5.7.5
Frames
166
5.7.6
Curvelets
166
5.8
Exercises and programs
167
Wavelet-Galerkin methods
170
6.1
Wavelet preconditioning
170
6.2
The role of the FWT
175
6.3
Numerical examples for the model problem
177
6.3.1
Rate of convergence
177
6.3.2
Compression
181
6.4
Exercises and programs
184
Adaptive wavelet methods
186
7.1
Adaptive approximation of functions
186
7.1.1
Best TV-term approximation
187
7.1.2
The size and decay of the wavelet coefficients
190
7.2
A posteriori error estimates and adaptivity
191
7.2.1
A posteriori error estimates
192
7.2.2
Ad hoc refinement strategies
194
7.3
Infinite-dimensional iterations
200
7.4
An equivalent
£2
problem: Using wavelets
203
7.5
Compressible matrices
205
7.5.1
Numerical realization of APPLY
215
7.5.2
Numerical experiments for APPLY
216
CONTENTS
7.6
Approximate iterations
221
7.6.1
Adaptive Wavelet-Richardson method
221
7.6.2
Adaptive scheme with inner iteration
224
7.6.3
Optimality
226
7.7
Quantitative efficiency
232
7.7.1
Quantitative aspects of the efficiency
232
7.7.2
An efficient modified scheme: Ad hoc strategy revisited
234
7.8
Nonlinear problems
238
7.8.1
Nonlinear variational problems
239
7.8.2
The DSX algorithm
241
7.8.3
Prediction
243
7.8.4
Reconstruction
246
7.8.5
Quasi-interpolation
249
7.8.6
Decomposition
252
7.9
Exercises and programs
254
Wavelets on general domains
257
8.1
Multiresolution on the interval
258
8.1.1
Refinement matrices
260
8.1.2
Boundary scaling functions
262
8.1.3 Biorthogonal
multiresolution
271
8.1.4
Refinement matrices
284
8.1.5
Boundary conditions
287
8.1.6
Symmetry
291
8.2
Wavelets on the interval
292
8.2.1
Stable completion
292
8.2.2
Spline-wavelets on the interval
297
8.2.3
Further examples
304
8.2.4
Dirichlet boundary conditions
307
8.2.5
Quantitative aspects
323
8.2.6
Other constructions on the interval
325
8.2.7
Software for wavelets on the interval
327
8.2.8
Numerical experiments
328
8.3
Tensor product wavelets
335
8.4
The Wavelet Element Method
(WEM) 342
8.4.1
Matching in ID
344
8.4.2
The setting in arbitrary dimension
357
8.4.3
The
WEM in
the two-dimensional case
372
8.4.4
Trivariate matched wavelets
386
8.4.5
Software for the
WEM 387
8.5
Embedding methods
390
8.6
Exercises and programs
391
CONTENTS xi
9
Some applications
394
9.1
Elliptic problems on bounded domains
394
9.1.1
Numerical realization of the
WEM 394
9.1.2
Model problem on the L-shaped domain
395
9.2
More complicated domains
401
9.2.1
Influence of the mapping
-
A non-rectangular domain.
401
9.2.2
Influence of the matching
403
9.2.3
Comparison of the different adaptive methods
406
9.3
Saddle point problems
407
9.3.1
The standard Galerkin discretization:
The LBB condition
409
9.3.2
An equivalent £2 problem
410
9.3.3
The adaptive wavelet method: Convergence
without LBB
412
9.4
The Stokes problem
417
9.4.1
Formulation
417
9.4.2
Discretization
418
9.4.3
B-spline wavelets and the exact application of the
divergence
419
9.4.4
Bounded domains
423
9.4.5
The divergence operator
425
9.4.6
Compressibility of A and BT
427
9.4.7
Numerical experiments
428
9.4.8
Rate of convergence
429
9.5
Exercises and programs
434
A Sobolev spaces and variational formulations
437
A.I Weak derivatives and sobolev spaces with integer order
437
A.2 Sobolev spaces with fractional order
443
A.3 Sobolev spaces with negative order
445
A.4 Variational formulations
446
A.5 Regularity theory
452
B Besov
spaces
455
B.I Sobolev and
Besov
embedding
457
B.2 Convergence of approximation schemes
459
С
Basic iterations
462
References
465
Index
477
|
| adam_txt |
CONTENTS
List of Algorithms
xii
Preface
xiii
Acknowledgements
xv
List of Figures
xvii
List of Tables
xxvi
1
Introduction
1
1.1
Some aspects of the history of wavelets
1
1.2
The scope of this book
4
1.3
Outline
6
2
Multiscale approximation and multiresolution
9
2.1
The
Haar
system
9
2.1.1
Projection by interpolation
9
2.1.2
Orthogonal projection
12
2.2
Piecewise linear systems
17
2.3
Similar properties
24
2.3.1
Stability
24
2.3.2
Refinement relation
25
2.3.3
Multiresolution
26
2.3.4
Locality
27
2.4
Multiresolution analysis on the real line
28
2.4.1
The scaling function
29
2.4.2
When does a mask define a refinable function?
30
2.4.3
Consequences of the refinability
31
2.5
Daubechies
orthonormal
scaling functions
35
2.6
B-splines
37
2.6.1
Centralized B-splines
38
2.7
Dual scaling functions associated to B-splines
39
2.8
Multilevel projectors
43
2.9
Approximation properties
46
2.9.1
A general framework
46
2.9.2
Stability properties
48
2.9.3
Error estimates
49
2.10
Plotting scaling functions
50
2.10.1
Subdivision
51
2.10.2
Cascade algorithm
55
2.11
Periodization
57
2.12
Exercises and programs
58
vu
CONTENTS
Elliptic boundary value problems
63
3.1
A model problem
63
3.1.1
Variational formulation
64
3.1.2
Existence and uniqueness
68
3.2
Variational formulation
70
3.2.1
Operators associated by the bilinear form
74
3.2.2
Reduction to homogeneous boundary
conditions
75
3.2.3
Stability
75
3.3
Regularity theory
76
3.4
Galerkin methods
77
3.4.1
Discretization
77
3.4.2
Stability
78
3.4.3
Error estimates
78
3.4.4
^-estimates
82
3.4.5
Numerical solution
83
3.5
Exercises and programs
85
Multiresolution Galerkin methods
87
4.1
Multiscale discretization
87
4.2
Multiresolution multiscale discretization
89
4.2.1
Piecewise linear multiresolution
89
4.2.2
Periodic boundary value problems
90
4.2.3
Common properties
94
4.3
Error estimates
95
4.4
Some numerical examples
96
4.5
Setup of the algebraic system
97
4.5.1
Refinable integrals
98
4.5.2
The right-hand side
101
4.5.3
Quadrature
101
4.6
The BPX preconditioner
102
4.7
MultiGrid
104
4.8
Numerical examples for the model problem
108
4.9
Exercises and programs
115
Wavelets
120
5.1
Detail spaces
120
5.1.1
Updating
120
5.1.2
The
Haar
system again
121
5.2
Orthogonal wavelets
124
5.2.1
Multilevel decomposition
124
5.2.2
The construction of wavelets
126
5.2.3
Wavelet projectors
129
5.3 Biorthogonal
wavelets
131
5.3.1 Biorthogonal
complement spaces
132
CONTENTS ix
5.3.2 Biorthogonal
projectors
133
5.3.3 Biorthogonal B-spline
wavelets
134
5.4 Fast
Wavelet Transform (FWT)
134
5.4.1
Decomposition
141
5.4.2
Reconstruction
143
5.4.3
Efficiency
145
5.4.4
A generał
framework
145
5.5
Vanishing moments and compression
147
5.6
Norm equivalences
150
5.6.1
Jackson inequality
151
5.6.2
Bernstein inequality
152
5.6.3
A characterization theorem
152
5.7
Other kinds of wavelets
156
5.7.1
Interpolatory
wavelets
157
5.7.2 Semiorthogonal
wavelets
161
5.7.3
Noncompactly supported wavelets
163
5.7.4
Multiwavelets
164
5.7.5
Frames
166
5.7.6
Curvelets
166
5.8
Exercises and programs
167
Wavelet-Galerkin methods
170
6.1
Wavelet preconditioning
170
6.2
The role of the FWT
175
6.3
Numerical examples for the model problem
177
6.3.1
Rate of convergence
177
6.3.2
Compression
181
6.4
Exercises and programs
184
Adaptive wavelet methods
186
7.1
Adaptive approximation of functions
186
7.1.1
Best TV-term approximation
187
7.1.2
The size and decay of the wavelet coefficients
190
7.2
A posteriori error estimates and adaptivity
191
7.2.1
A posteriori error estimates
192
7.2.2
Ad hoc refinement strategies
194
7.3
Infinite-dimensional iterations
200
7.4
An equivalent
£2
problem: Using wavelets
203
7.5
Compressible matrices
205
7.5.1
Numerical realization of APPLY
215
7.5.2
Numerical experiments for APPLY
216
CONTENTS
7.6
Approximate iterations
221
7.6.1
Adaptive Wavelet-Richardson method
221
7.6.2
Adaptive scheme with inner iteration
224
7.6.3
Optimality
226
7.7
Quantitative efficiency
232
7.7.1
Quantitative aspects of the efficiency
232
7.7.2
An efficient modified scheme: Ad hoc strategy revisited
234
7.8
Nonlinear problems
238
7.8.1
Nonlinear variational problems
239
7.8.2
The DSX algorithm
241
7.8.3
Prediction
243
7.8.4
Reconstruction
246
7.8.5
Quasi-interpolation
249
7.8.6
Decomposition
252
7.9
Exercises and programs
254
Wavelets on general domains
257
8.1
Multiresolution on the interval
258
8.1.1
Refinement matrices
260
8.1.2
Boundary scaling functions
262
8.1.3 Biorthogonal
multiresolution
271
8.1.4
Refinement matrices
284
8.1.5
Boundary conditions
287
8.1.6
Symmetry
291
8.2
Wavelets on the interval
292
8.2.1
Stable completion
292
8.2.2
Spline-wavelets on the interval
297
8.2.3
Further examples
304
8.2.4
Dirichlet boundary conditions
307
8.2.5
Quantitative aspects
323
8.2.6
Other constructions on the interval
325
8.2.7
Software for wavelets on the interval
327
8.2.8
Numerical experiments
328
8.3
Tensor product wavelets
335
8.4
The Wavelet Element Method
(WEM) 342
8.4.1
Matching in ID
344
8.4.2
The setting in arbitrary dimension
357
8.4.3
The
WEM in
the two-dimensional case
372
8.4.4
Trivariate matched wavelets
386
8.4.5
Software for the
WEM 387
8.5
Embedding methods
390
8.6
Exercises and programs
391
CONTENTS xi
9
Some applications
394
9.1
Elliptic problems on bounded domains
394
9.1.1
Numerical realization of the
WEM 394
9.1.2
Model problem on the L-shaped domain
395
9.2
More complicated domains
401
9.2.1
Influence of the mapping
-
A non-rectangular domain.
401
9.2.2
Influence of the matching
403
9.2.3
Comparison of the different adaptive methods
406
9.3
Saddle point problems
407
9.3.1
The standard Galerkin discretization:
The LBB condition
409
9.3.2
An equivalent £2 problem
410
9.3.3
The adaptive wavelet method: Convergence
without LBB
412
9.4
The Stokes problem
417
9.4.1
Formulation
417
9.4.2
Discretization
418
9.4.3
B-spline wavelets and the exact application of the
divergence
419
9.4.4
Bounded domains
423
9.4.5
The divergence operator
425
9.4.6
Compressibility of A and BT
427
9.4.7
Numerical experiments
428
9.4.8
Rate of convergence
429
9.5
Exercises and programs
434
A Sobolev spaces and variational formulations
437
A.I Weak derivatives and sobolev spaces with integer order
437
A.2 Sobolev spaces with fractional order
443
A.3 Sobolev spaces with negative order
445
A.4 Variational formulations
446
A.5 Regularity theory
452
B Besov
spaces
455
B.I Sobolev and
Besov
embedding
457
B.2 Convergence of approximation schemes
459
С
Basic iterations
462
References
465
Index
477 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Urban, Karsten 1966- |
| author_GND | (DE-588)141064684 |
| author_facet | Urban, Karsten 1966- |
| author_role | aut |
| author_sort | Urban, Karsten 1966- |
| author_variant | k u ku |
| building | Verbundindex |
| bvnumber | BV023225298 |
| callnumber-first | Q - Science |
| callnumber-label | QA403 |
| callnumber-raw | QA403.3 |
| callnumber-search | QA403.3 |
| callnumber-sort | QA 3403.3 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 560 |
| ctrlnum | (OCoLC)604423252 (DE-599)BVBBV023225298 |
| dewey-full | 515.3533 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.3533 |
| dewey-search | 515.3533 |
| dewey-sort | 3515.3533 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV023225298 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:17:43Z |
| indexdate | 2024-07-09T21:13:31Z |
| institution | BVB |
| isbn | 9780198526056 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016411133 |
| oclc_num | 604423252 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-29T DE-83 DE-11 DE-824 DE-20 DE-384 DE-634 |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-29T DE-83 DE-11 DE-824 DE-20 DE-384 DE-634 |
| physical | XXVII, 480 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| series2 | Numerical mathematics and scientific computation |
| spelling | Urban, Karsten 1966- Verfasser (DE-588)141064684 aut Wavelet methods for elliptic partial differential equations Karsten Urban 1. publ. Oxford [u.a.] Oxford Univ. Press 2009 XXVII, 480 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Numerical mathematics and scientific computation Differential equations, Elliptic Wavelets (Mathematics) Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Wavelet (DE-588)4215427-3 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Wavelet (DE-588)4215427-3 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016411133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Urban, Karsten 1966- Wavelet methods for elliptic partial differential equations Differential equations, Elliptic Wavelets (Mathematics) Elliptische Differentialgleichung (DE-588)4014485-9 gnd Wavelet (DE-588)4215427-3 gnd |
| subject_GND | (DE-588)4014485-9 (DE-588)4215427-3 |
| title | Wavelet methods for elliptic partial differential equations |
| title_auth | Wavelet methods for elliptic partial differential equations |
| title_exact_search | Wavelet methods for elliptic partial differential equations |
| title_exact_search_txtP | Wavelet methods for elliptic partial differential equations |
| title_full | Wavelet methods for elliptic partial differential equations Karsten Urban |
| title_fullStr | Wavelet methods for elliptic partial differential equations Karsten Urban |
| title_full_unstemmed | Wavelet methods for elliptic partial differential equations Karsten Urban |
| title_short | Wavelet methods for elliptic partial differential equations |
| title_sort | wavelet methods for elliptic partial differential equations |
| topic | Differential equations, Elliptic Wavelets (Mathematics) Elliptische Differentialgleichung (DE-588)4014485-9 gnd Wavelet (DE-588)4215427-3 gnd |
| topic_facet | Differential equations, Elliptic Wavelets (Mathematics) Elliptische Differentialgleichung Wavelet |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016411133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT urbankarsten waveletmethodsforellipticpartialdifferentialequations |