Accuracy of mathematical models: dimension reduction, homogenization, and simplification
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin
European Mathematical Society
[2020]
|
| Schriftenreihe: | EMS tracts in mathematics
33 |
| Schlagwörter: | |
| Links: | https://d-nb.info/1213478901/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032329818&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xvi, 317 Seiten 25 cm |
| ISBN: | 9783037192061 3037192062 |
Internformat
MARC
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|---|---|---|---|
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| 007 | t| | ||
| 008 | 200930s2020 gw |||| 00||| eng d | ||
| 016 | 7 | |a 1213478901 |2 DE-101 | |
| 020 | |a 9783037192061 |c Festeinband |9 978-3-03719-206-1 | ||
| 020 | |a 3037192062 |9 3-03719-206-2 | ||
| 035 | |a (OCoLC)1183717845 | ||
| 035 | |a (DE-599)DNB1213478901 | ||
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| 100 | 1 | |a Repin, Sergej Igorevič |d 1953- |e Verfasser |0 (DE-588)1089800177 |4 aut | |
| 245 | 1 | 0 | |a Accuracy of mathematical models |b dimension reduction, homogenization, and simplification |c Sergey I. Repin, Stefan A. Sauter |
| 264 | 1 | |a Berlin |b European Mathematical Society |c [2020] | |
| 264 | 4 | |c © 2020 | |
| 300 | |a xvi, 317 Seiten |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a EMS tracts in mathematics |v 33 | |
| 650 | 0 | 7 | |a Mathematisches Modell |0 (DE-588)4114528-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtlineare elliptische Differentialgleichung |0 (DE-588)4310554-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Homogenität |0 (DE-588)4300430-1 |2 gnd |9 rswk-swf |
| 653 | |a conversion of models | ||
| 653 | |a modelling error | ||
| 653 | |a homogenization | ||
| 653 | |a a posteriori error majorant | ||
| 653 | |a model simplification | ||
| 653 | |a dimension reduction | ||
| 689 | 0 | 0 | |a Mathematisches Modell |0 (DE-588)4114528-8 |D s |
| 689 | 0 | 1 | |a Homogenität |0 (DE-588)4300430-1 |D s |
| 689 | 0 | 2 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | 3 | |a Nichtlineare elliptische Differentialgleichung |0 (DE-588)4310554-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Sauter, Stefan |d 1964- |e Verfasser |0 (DE-588)129230006 |4 aut | |
| 710 | 2 | |a European Mathematical Society Publishing House ETH-Zentrum SEW A27 |0 (DE-588)1066118477 |4 pbl | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-03719-706-6 |
| 830 | 0 | |a EMS tracts in mathematics |v 33 |w (DE-604)BV022480257 |9 33 | |
| 856 | 4 | 2 | |m B:DE-101 |q application/pdf |u https://d-nb.info/1213478901/04 |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032329818&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-032329818 | |
Datensatz im Suchindex
| _version_ | 1819328514841640960 |
|---|---|
| adam_text | CONTENTS
PREFACE
...................................................................................................................
XI
1
INTRODUCTION
...................................................................................................
1
1.1
BASIC
NOTATION
.......................................................................................
1
1.1.1
DOMAINS
AND
OPERATORS
..............................................................
1
1.1.2
SPACES
OF
FUNCTIONS
....................................................................
2
1.1.3
CONVEX
FUNCTIONALS
....................................................................
4
1.2
FUNCTIONAL
INEQUALITIES
..........................................................................
7
1.2.1
HOLDER
TYPE
INEQUALITIES
...........................................................
7
1.2.2
FRIEDRICHS
AND
POINCARE
INEQUALITIES
........................................
7
1.2.3
INEQUALITIES
FOR
FUNCTIONS
WITH
ZERO
MEAN
TRACES
ON
THE
BOUNDARY
..........................................................................
10
1.2.4
KORN
*
S
INEQUALITIES
....................................................................
11
1.2.5
INF-SUP
CONDITION
....................................................................
12
1.3
COMPUTABLE
BOUNDS
OF
CONSTANTS
IN
FUNCTIONAL
INEQUALITIES
.......................................................................
15
1.3.1
CONSTANT
IN
THE
FRIEDRICHS
INEQUALITY
........................................
16
1.3.2
CONSTANTS
IN
POINCARE-TYPE
INEQUALITIES
..................................
17
1.3.3
CONSTANTS
IN
TRACE-TYPE
INEQUALITIES
...............................................
20
1.3.4
ESTIMATES
OF
CONSTANTS
BASED
ON
DOMAIN
DECOMPOSITION
....
20
2
DISTANCE
TO
EXACT
SOLUTIONS
.................................................................................
25
2.1
A
CLASS
OF
BOUNDARY
VALUE
PROBLEMS
.........................................................
25
2.2
THE
MAIN
ERROR
IDENTITY
..............................................................................
29
2.2.1
ERROR
MEASURE
.................................................................................
29
2.2.2
DECOMPOSITION
OF
THE
ERROR
MEASURE
............................................
31
2.2.3
PROBLEMS
WITH
LINEAR
J
7
..................................................................
33
2.2.4
ERROR
IDENTITIES
IN
VECTOR
FORM
.....................................................
42
2.2.5
DIFFERENCE
BETWEEN
THE
EXACT
SOLUTIONS
OF
TWO
PROBLEMS
....
43
2.3
LINEAR
PROBLEMS
........................................................................................44
2.3.1
ERROR
RELATIONS
IN
THE
GENERAL
FORM
...............................................
44
2.3.2
SPECIAL
CASE
....................................................................................47
2.3.3
PRIMAL-DUAL
NORMS
OF
ERRORS
IN
V
X
Y
*
......................................
50
2.3.4
ERRORS
IN
THE
FULL
PRIMAL-DUAL
NORM
...............................................
52
2.3.5
MAJORANT
AS
A
SOURCE
OF
NEW
MODELS
............................................
54
2.3.6
NON-HOMOGENEOUS
BOUNDARY
CONDITIONS
......................................
55
2.4
APPLICATIONS
TO
PARTICULAR
MATHEMATICAL
MODELS
.......................................
57
2.4.1
DIFFUSION
TYPE
MODELS
..................................................................
57
2.4.2
MIXED
BOUNDARY
CONDITIONS
............................................................
59
2.4.3
PROBLEMS
WITH
PERIODIC
BOUNDARY
CONDITIONS
................................
60
2.4.4
ADVANCED
ESTIMATES
BASED
ON
DOMAIN
DECOMPOSITION
............
61
VIII
CONTENTS
2.4.5
ELASTICITY
........................................................................................
64
2.4.6
VARIATIONAL
FUNCTIONALS
WITH
POWER
GROWTH
....................................
70
2.4.7
STOKES
PROBLEM
...............................................................................
75
2.4.8
BINGHAM
PROBLEM
............................................................................
79
2.4.9
ANOTHER
ERROR
ESTIMATION
METHOD
...................................................
81
2.5
VALIDATION
OF
MATHEMATICAL
MODELS
............................................................
86
2.6
ERRORS
OF
NUMERICAL
APPROXIMATIONS
.........................................................
89
2.6.1
TWO-SIDED
ESTIMATES
OF
APPROXIMATION
ERRORS
.............................90
2.6.2
REDUCTION
OF
THE
SET
Q*
*
............................................................
91
2.6.3
TRANSFORMATION
OF
..................................................
92
2.6.4
USING
EXTRA
REGULARITY
OF
THE
EXACT
SOLUTION
................................
93
2.6.5
USING
AN
AUXILIARY
FINITE-DIMENSIONAL
PROBLEM
.............................94
2.6.6
APPLICATIONS
TO
LEAST
SQUARES
TYPE
METHODS
...................................
99
2.6.7
NONCONFORMING
APPROXIMATIONS
.................................................
101
3
DIMENSION
REDUCTION
MODELS
..........................................................................
103
3.1
DIMENSION
REDUCTION
................................................................................
103
3.2
SECOND-ORDER
ELLIPTIC
PROBLEMS
.................................................................
107
3.2.1
BASIC
PROBLEM
................................................................................
107
3.2.2
REDUCED
PROBLEM
..........................................................................
108
3.2.3
ERROR
GENERATED
BY
DIMENSION
REDUCTION
.....................................
110
3.2.4
PARTICULAR
CASES
.............................................................................
115
3.2.5
EXAMPLES
......................................................................................
117
3.3
DIMENSION
REDUCTION
IN
LINEAR
ELASTICITY
.................................................
123
3.3.1
THE
PLANE
STRESS
PROBLEM
.............................................................
123
3.3.2
THE
FUNCTION
/
.............................................................................
130
3.3.3
BEHAVIOR
OF
THE
MODELLING
ERROR
AS
T
*
0
.................................
132
3.3.4
EXAMPLE
.........................................................................................
134
3.4
BENDING
OF
ELASTIC
PLATES
..........................................................................
136
3.4.1
STATEMENT
OF
THE
PROBLEM
.............................................................
136
3.4.2
THE
KIRCHHOFF-LOVE
PLATE
MODEL
.................................................
137
3.4.3
RECONSTRUCTION
OF
3D
DISPLACEMENTS
...........................................
140
3.4.4
RECONSTRUCTION
OF
3D
STRESSES
....................................................
140
3.4.5
ERROR
ESTIMATES
FOR
PLATE-TYPE
DOMAINS
........................................
141
3.4.6
ACCURACY
OF
THE
KL
PLATE
MODEL
.................................................
149
3.4.7
ESTIMATES
OF
THE
MODELLING
ERROR
.................................................
150
3.4.8
ASYMPTOTIC
BEHAVIOUR
OF
THE
ERROR
MAJORANT
..............................
152
4
MODEL
SIMPLIFICATION
.........................................................................................
159
4.1
MODEL
SIMPLIFICATION
BASED
ON
THE
CONCEPT
OF
ENERGY
............................
159
4.2
SIMPLIFICATION
OF
COEFFICIENTS
....................................................................
164
4.2.1
SECOND-ORDER
ELLIPTIC
PROBLEMS
....................................................
165
4.2.2
GENERAL
ELLIPTIC
PROBLEM
.............................................................
168
4.2.3
USING
EXTRA
REGULARITY
OF
U
..........................................................
170
CONTENTS
IX
4.2.4
ASYMPTOTIC
RATE
OF
CONVERGENCE
OF
THE
ERROR
ESTIMATOR
(V)
IN
TERMS
OF
THE
MEASURE
OF
THE
NON-RESOLVED
GEOMETRY
............
172
4.3
GEOMETRICAL
SIMPLIFICATION
......................................................................
173
4.3.1
SIMPLIFICATION
OF
THE
DIRICHLET
BOUNDARY
....................................
173
4.3.2
GENERALIZATIONS
.............................................................................
177
4.3.3
SIMPLIFICATION
OF
THE
NEUMANN
BOUNDARY
.................................
179
4.4
COMMENTS
...............................................................................................
182
4.4.1
PROBLEMS
WITH
*
ROUGH
*
COEFFICIENTS
...........................................
182
4.4.2
MODELLING-DISCRETIZATION
ADAPTATION
STRATEGIES
...........................
184
4.4.3
PROBLEMS
WITH
UNCERTAIN
DATA
.......................................................
185
5
ELLIPTIC
HOMOGENIZATION
...................................................................................
187
5.1
SETTING
......................................................................................................
187
5.2
MATHEMATICAL
HOMOGENIZATION
VIA
ASYMPTOTIC
EXPANSIONS
.....................
190
5.3
PROPERTIES
OF
THE
HOMOGENIZED
PROBLEM
.................................................
195
5.3.1
WELL-POSEDNESS
OF
THE
HOMOGENIZED
EQUATION
...........................
196
5.3.2
REGULARITY
ESTIMATES
FOR
THE
HOMOGENIZED
EQUATION
...............
201
5.3.3
REGULARITY
ESTIMATES
FOR
THE
CELL
PROBLEM
....................................209
5.3.4
CONVERGENCE
OF
THE
FIRST-ORDER
APPROXIMATION
...........................
213
5.4
DISCRETIZATION
............................................................................................
218
5.5
ERROR ESTIMATION
......................................................................................
221
5.5.1
GENERAL
COMMENTS
......................................................................
221
5.5.2
ESTIMATES
OF
THE
MODELLING
ERROR
................................................
225
5.5.3
ERROR
OF
THE
FULLY
DISCRETE
FIRST-ORDER
APPROXIMATION
...............230
5.6
COMMENTS
...............................................................................................
245
5.6.1
REGULARITY
AND
EMBEDDING
CONSTANTS
..........................................
245
5.6.2
MODELING-DISCRETIZATION
STRATEGIES
.............................................
247
5.6.3
MULTISCALE
PROBLEMS
...................................................................
248
6
CONVERSION
OF
MODELS
......................................................................................
251
6.1
REGULARIZATION
OF
MODELS
.........................................................................
251
6.1.1
ADDING
A
REGULARIZING
TERM
..........................................................
251
6.1.2
SMOOTHING
...................................................................................
253
6.1.3
PROX-TYPE
REGULARIZATION
.............................................................
259
6.2
ERRORS
OF
PENALTY-TYPE
MODELS
................................................................262
6.2.1
GENERAL
APPROACH
.........................................................................
262
6.2.2
VARIATIONAL
PROBLEMS
DEFINED
IN
SUBSPACES
.................................
265
6.3
FICTITIOUS
DOMAIN
METHODS
......................................................................
269
6.4
LINEARIZATION
............................................................................................
275
6.5
ERRORS
OF
TIME-INCREMENTAL
MODELS
..........................................................
278
X
CONTENTS
A
W
1,P
-REGULARITY
CONSTANT
FOR
SECOND-ORDER
ELLIPTIC
PROBLEMS
WITH
NONSMOOTH
COEFFICIENTS
......................................................................
283
BIBLIOGRAPHY
............................................................................................................
293
LIST
OF
NOTATIONS
......................................................................................................
311
INDEX
........................................................................................................................
315
|
| any_adam_object | 1 |
| author | Repin, Sergej Igorevič 1953- Sauter, Stefan 1964- |
| author_GND | (DE-588)1089800177 (DE-588)129230006 |
| author_facet | Repin, Sergej Igorevič 1953- Sauter, Stefan 1964- |
| author_role | aut aut |
| author_sort | Repin, Sergej Igorevič 1953- |
| author_variant | s i r si sir s s ss |
| building | Verbundindex |
| bvnumber | BV046920600 |
| classification_rvk | SK 560 SK 660 |
| ctrlnum | (OCoLC)1183717845 (DE-599)DNB1213478901 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV046920600 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T19:04:41Z |
| institution | BVB |
| institution_GND | (DE-588)1066118477 |
| isbn | 9783037192061 3037192062 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032329818 |
| oclc_num | 1183717845 |
| open_access_boolean | |
| owner | DE-634 DE-11 DE-706 |
| owner_facet | DE-634 DE-11 DE-706 |
| physical | xvi, 317 Seiten 25 cm |
| publishDate | 2020 |
| publishDateSearch | 2020 |
| publishDateSort | 2020 |
| publisher | European Mathematical Society |
| record_format | marc |
| series | EMS tracts in mathematics |
| series2 | EMS tracts in mathematics |
| spellingShingle | Repin, Sergej Igorevič 1953- Sauter, Stefan 1964- Accuracy of mathematical models dimension reduction, homogenization, and simplification EMS tracts in mathematics Mathematisches Modell (DE-588)4114528-8 gnd Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Homogenität (DE-588)4300430-1 gnd |
| subject_GND | (DE-588)4114528-8 (DE-588)4310554-3 (DE-588)4044779-0 (DE-588)4300430-1 |
| title | Accuracy of mathematical models dimension reduction, homogenization, and simplification |
| title_auth | Accuracy of mathematical models dimension reduction, homogenization, and simplification |
| title_exact_search | Accuracy of mathematical models dimension reduction, homogenization, and simplification |
| title_full | Accuracy of mathematical models dimension reduction, homogenization, and simplification Sergey I. Repin, Stefan A. Sauter |
| title_fullStr | Accuracy of mathematical models dimension reduction, homogenization, and simplification Sergey I. Repin, Stefan A. Sauter |
| title_full_unstemmed | Accuracy of mathematical models dimension reduction, homogenization, and simplification Sergey I. Repin, Stefan A. Sauter |
| title_short | Accuracy of mathematical models |
| title_sort | accuracy of mathematical models dimension reduction homogenization and simplification |
| title_sub | dimension reduction, homogenization, and simplification |
| topic | Mathematisches Modell (DE-588)4114528-8 gnd Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Homogenität (DE-588)4300430-1 gnd |
| topic_facet | Mathematisches Modell Nichtlineare elliptische Differentialgleichung Partielle Differentialgleichung Homogenität |
| url | https://d-nb.info/1213478901/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032329818&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV022480257 |
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