Gespeichert in:
| Titel: | Tractability of multivariate problems |
|---|---|
| Unterteilung: | Linear information Volume I |
| Von: |
Erich Novak ; Henryk Woźniakowski
|
| Person: |
Novak, Erich
1953- Verfasser aut Woźniakowski, Henryk 1946- |
| Hauptverfassende: | , |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Zürich
European Mathematical Society
[2008]
|
| Schriftenreihe: | EMS tracts in mathematics
6 |
| Online-Zugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019324884&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | XI, 384 Seiten |
| ISBN: | 9783037190265 |
Internformat
MARC
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| 245 | 1 | 0 | |a Tractability of multivariate problems |n Volume I |p Linear information |c Erich Novak ; Henryk Woźniakowski |
| 264 | 1 | |a Zürich |b European Mathematical Society |c [2008] | |
| 300 | |a XI, 384 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a EMS tracts in mathematics |v 6 | |
| 490 | 0 | |a EMS tracts in mathematics | |
| 700 | 1 | |a Woźniakowski, Henryk |d 1946- |e Verfasser |0 (DE-588)135875994 |4 aut | |
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Datensatz im Suchindex
| _version_ | 1819321886809522176 |
|---|---|
| adam_text | Contents
Preface vii
1
Overview
1
2 Motivation
for Tractability
Studies
3
2.1
Notes and Remarks
........................... 13
3
Twelve Examples
14
3.1
Tractability in the Worst Case Setting
.................. 14
3.1.1
Example
1 :
Integration of Lipschitz Functions
.......... 14
3.1.2
Example
2:
Integration of Trigonometric Polynomials
...... 16
3.1.3
Example
3:
Integration of Smooth Periodic Functions
...... 19
3.1.4
Example
4:
Approximation of
С
°° Functions
.......... 23
3.1.5
Example
5:
Discrepancy
..................... 34
3.1.6
Example
6:
Diagonal Problems for Weighted Spaces
...... 40
3.2
Tractability in Other Settings
...................... 48
3.2.1
Average Case Setting
....................... 50
3.2.2
Example
7: Isotropie
Wiener Measure
.............. 50
3.2.3
Example
8:
Folded Wiener Sheet Measure
............ 58
3.2.4
Probabilistic Setting
....................... 63
3.2.5
Example
9:
Absolute and Normalized Errors
.......... 63
3.2.6
Example
10:
Relative Error
.................... 65
3.2.7
Randomized Setting
....................... 67
3.2.8
Example
11:
Montecarlo
Algorithm
............... 68
3.2.9
Example
12:
Class A 11
...................... 73
3.3
Open Problems
.............................. 75
3.4
Notes and Remarks
........................... 85
4
Basic Concepts and Survey of IBC Results
89
4.1
Complexity in the Worst Case Setting
.................. 89
4.1.1
Types of Information
....................... 92
4.1.2
Algorithms and Their Cost
.................... 97
4.1.3
Why Do We Use the Real Number Model?
........... 99
4.1.4
Errors and Complexity
...................... 105
4.1.5
Information Complexity and Total Complexity
......... 107
4.2
Basic Results for Linear Problems in the Worst Case Setting
...... 108
4.2.1
On the Power of Adaption
.................... 109
4.2.2
Linear and Nonlinear Algorithms for Linear Problems
..... 115
4.2.3
Linear Problems on
Hubert
Spaces for
Ла11
........... 118
4.2.4
Sobolev Embeddings
....................... 119
χ
Contents
4.3
Some Results for Different Settings
................... 127
4.3.1
Average Case Setting
....................... 127
4.3.2
Probabilistic Setting
....................... 137
4.3.3
Randomized Algorithms
..................... 140
4.4
Multivariate Problems and Tractability
................. 148
4.4.1
Polynomial Tractability
...................... 149
4.4.2
Weak Tractability
......................... 150
4.4.3
Generalized Tractability
..................... 151
4.5
Notes and Remarks
........................... 152
5
Worst Case Setting
155
5.1
Linear Problems Defined over Hubert Spaces
.............. 156
5.2
Linear Tensor Product Problems
..................... 173
5.3
Linear Weighted Tensor Product Problems
............... 188
5.3.1
Weighted Hubert Spaces
..................... 189
5.3.2
Types of Weights
......................... 194
5.3.3
Weighted Operators
........................ 199
5.3.4
Tractability of Weighted Problems
................ 200
5.4
Other Ways of Obtaining Linear Weighted Problems
.......... 220
5.4.1
Weighted Sobolev Space of Smooth Functions
......... 221
5.4.2
General Linear Weighted Problems
............... 229
5.5
Notes and Remarks
........................... 238
6
Average Case Setting
241
6.1
Linear Problems
............................. 242
6.2
Linear Tensor Product Problems
..................... 255
6.3
Linear Weighted Tensor Product Problems
............... 265
6.4
Notes and Remarks
........................... 282
7
Randomized Setting
284
7.1
Tractability of Linear Problems for
Λ*11
................ 284
8
Generalized Tractability
286
8.1
Motivation of Generalized Tractability
................. 289
8.2
Linear Tensor Product Problems
..................... 295
8.3
Restricted Tractability Domain
..................... 296
8.3.1
Restricted Tractability in
ε
.................... 296
8.3.2
Restricted Tractability in
d
.................... 299
8.3.3
Restricted Tractability with
d
* > 1
ande0
< 1......... 314
8.4
Unrestricted Tractability Domain
.................... 319
8.4.1
Finitely Many Eigenvalues
.................... 320
8.4.2
Exponential Decay of Eigenvalues
................ 327
8.4.3
Polynomial Decay of Eigenvalues
................ 331
8.5
Comparison
............................... 338
Contents xi
8.6 Notes
and Remarks
...........................340
Appendices
341
A Reproducing Kernel
Hubert
Spaces of Sobolev Type
341
A.I Korobov Spaces
............................. 341
АЛЛ
Weighted Korobov Spaces
.................... 347
A.2 Sobolev Spaces
.............................
35О
A.2.1 The First Weighted Sobolev Space
................
35О
A.2.2 The Second Weighted Sobolev Space
.............. 353
A.2.3 The Third Weighted Sobolev Space
............... 355
В
Gaussian Measures
357
B.I Gaussian Measures on
В
anach Spaces
.................357
B.2 Gaussian Measures and Reproducing Kernel Hubert Spaces
......361
С
List of Open Problems
З63
Bibliography
355
Index
38 !
|
| any_adam_object | 1 |
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| author_sort | Novak, Erich 1953- |
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| bvnumber | BV025720552 |
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| id | DE-604.BV025720552 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T14:12:52Z |
| institution | BVB |
| isbn | 9783037190265 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-019324884 |
| oclc_num | 315927014 |
| open_access_boolean | |
| owner | DE-11 DE-188 DE-739 DE-83 DE-384 DE-703 DE-945 |
| owner_facet | DE-11 DE-188 DE-739 DE-83 DE-384 DE-703 DE-945 |
| physical | XI, 384 Seiten |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | European Mathematical Society |
| record_format | marc |
| series | EMS tracts in mathematics |
| series2 | EMS tracts in mathematics |
| spellingShingle | Novak, Erich 1953- Woźniakowski, Henryk 1946- Tractability of multivariate problems EMS tracts in mathematics |
| title | Tractability of multivariate problems |
| title_auth | Tractability of multivariate problems |
| title_exact_search | Tractability of multivariate problems |
| title_full | Tractability of multivariate problems Volume I Linear information Erich Novak ; Henryk Woźniakowski |
| title_fullStr | Tractability of multivariate problems Volume I Linear information Erich Novak ; Henryk Woźniakowski |
| title_full_unstemmed | Tractability of multivariate problems Volume I Linear information Erich Novak ; Henryk Woźniakowski |
| title_short | Tractability of multivariate problems |
| title_sort | tractability of multivariate problems linear information |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019324884&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV025521833 (DE-604)BV022480257 |
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