Multiscale methods: averaging and homogenization
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
[2008]
|
| Series: | Texts in applied mathematics
53 |
| Subjects: | |
| Online Access: | Inhaltstext Publisher description Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVIII, 307 S. graph. Darst. |
| ISBN: | 9780387738284 0387738282 |
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MARC
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| 100 | 1 | |a Pavliotis, Grigorios A. |e Verfasser |0 (DE-588)1274035414 |4 aut | |
| 245 | 1 | 0 | |a Multiscale methods |b averaging and homogenization |c Grigorios A. Pavliotis ; Andrew M. Stuart |
| 264 | 1 | |a New York, NY |b Springer |c [2008] | |
| 300 | |a XVIII, 307 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Texts in applied mathematics |v 53 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Équations aux dérivées partielles | |
| 650 | 4 | |a Équations différentielles | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Differential equations, Partial | |
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| 650 | 0 | 7 | |a Mehrskalenanalyse |0 (DE-588)4416235-2 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Stuart, Andrew |e Verfasser |0 (DE-588)1078856400 |4 aut | |
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Record in the Search Index
| _version_ | 1825638375126728704 |
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| adam_text |
Contents
Series
Preface
.
V
Preface
.
IX
1
Introduction
. 1
1.1
Overview
. 1
1.2
Motivating Examples
. 1
1.2.1
Example I: Steady Heat Conduction in a Composite Material
2
1.2.2
Example II: Homogenization for Advection-Diffusion
Equations
. 3
1.2.3
Example III: Averaging, Homogenization, and Dynamics
. 4
1.2.4
Example IV: Dimension Reduction in Dynamical Systems
. 5
1.3
Averaging Versus Homogenization
. 6
1.3.1
Averaging for Systems of Linear Equations
. 7
1.3.2
Homogenization for Systems of Linear Equations
. 8
1.4
Discussion and Bibliography
. 9
Parti Background
2
Analysis
. 13
2.1
Setup
. 13
2.2
Notation
. 14
2.3
Banach and Hubert Spaces
. 16
2.3.1
Banach Spaces
. 16
2.3.2
Hubert Spaces
. 18
2.4
Function Spaces
. 18
2.4.1
Spaces of Continuous Functions
. 18
2.4.2
IP Spaces
. 19
2.4.3
Sobolev Spaces
. 21
2.4.4
Banach
Space-Vaîued
Spaces
. 22
2.4.5
Sobolev Spaces of Periodic Functions
. 23
XIV Contents
2.5
Two-Scale Convergence
. 25
2.5.1
Two-Scale Convergence for Steady Problems
. 25
2.5.2
Two-Scale Convergence for Time-Dependent Problems
----- 28
2.6
Equations in Hubert Spaces
. 29
2.6.1
Lax-Milgram Theory
. 30
2.6.2
The
Fredholm
Alternative
. 31
2.7
Discussion and Bibliography
. 32
2.8
Exercises
. 34
3
Probability Theory and Stochastic Processes
. 37
3.1
Setup
. 37
3.2
Probability, Expectation, and Conditional Expectation
. 37
3.3
Stochastic Processes
. 40
3.4
Martingales and Stochastic Integrals
. 46
3.4.1
Martingales
. 46
3.4.2
The
Ito
Stochastic Integral
. 47
3.4.3
The Stratonovich Stochastic Integral
. 49
3.5
Weak Convergence of Probability Measures
. 49
3.6
Discussion and Bibliography
. 54
3.7
Exercises
. 56
4
Ordinary Differential Equations
. 59
4.1
Setup
. 59
4.2
Existence and Uniqueness
. 59
4.3
The Generator
. 62
4.4
Ergodicity
. 66
4.5
Discussion and Bibliography
. 72
4.6
Exercises
. 72
5
Markov Chains
. 73
5.1
Setup
. 73
5.2
Discrete-Time Markov Chains
. 73
5.3
Continuous-Time Markov Chains
. 75
5.4
The Generator
. 76
5.5
Existence and Uniqueness
. 80
5.6
Ergodicity
. 81
5.7
Discussion and Bibliography
. 83
5.8
Exercises
. 84
6
Stochastic Differential Equations
. 85
6.1
Setup
. 85
6.2
Existence and Uniqueness
. 86
6.3
The Generator
. 88
6.4
Ergodicity
. 94
6.5
Discussion and Bibliography
. 99
6.6
Exercises
. 100
Contents
XV
7
Partial
Differential
Equations
. 103
7.1 Setup. 103
7.2
Elliptic PDEs
. 104
7.2.1 The Dirichlet Problem. 105
7.2.2
The Periodic
Problem . 107
7.2.3 The Fredholm Alternative. 107
7.2.4 The Maximum
Principle
. 113
7.3
Parabolic PDEs
. 114
7.3.1
Bounded Domains
. 114
7.3.2
The Maximum Principle
. 115
7.3.3
Unbounded Domains: The Cauchy Problem
. 117
7.4
Transport PDEs
. 118
7.5
Semigroups
. 121
7.6
Discussion and Bibliography
. 122
7.7
Exercises
. 123
Part II Perturbation Expansions
8
Invariant Manifolds for ODEs
. 127
8.1
Introduction
. 127
8.2
Full Equations
. 127
8.3
Simplified Equations
. 128
8.4
Derivation
. 129
8.5
Applications
. 130
8.5.1
Linear Fast Dynamics
. 130
8.5.2
Large Time Dynamics
. 131
8.5.3
Center Manifold
. 131
8.6
Discussion and Bibliography
. 133
8.7
Exercises
. 134
9
Averaging for Markov Chains
. 137
9.1
Introduction
. 137
9.2
Full Equations
. 137
9.3
Simplified Equations
. 139
9.4
Derivation
. 140
9.5
Application
. 141
9.6
Discussion and Bibliography
. 142
9.7
Exercises
. 142
10
Averaging for ODEs and SDEs
. 145
10.1
Introduction
. 145
10.2
Full Equations
. 145
10.3
Simplified Equations
. 146
10.4
Derivation
. 147
XVI Contents
10.5
Deterministic Problems
. 148
10.6
Applications
. 149
10.6.1
A Skew-Product SDE
. 149
10.6.2
Hamiltonian Mechanics
. 150
10.7
Discussion and Bibliography
. 153
10.8
Exercises
. 155
11
Homogenization for ODEs and SDEs
. 157
11.1
Introduction
. 157
11.2
Full Equations
. 158
11.3
Simplified Equations
. 159
11.4
Derivation
. 160
11.5
Properties of the Simplified Equations
. 162
11.6
Deterministic Problems
. 163
11.7
Applications
. 166
11.7.1
Fast Ornstein-Uhlenbeck Noise
. 166
11.7.2
Fast Chaotic Noise
. 169
11.7.3
Stratonovich Corrections
. 170
11.7.4
Stokes' Law
. 171
11.7.5
Green-Kubo Formula
. 173
11.7.6
Neither
Ito
nor Stratonovich
. 174
11.7.7
The Levy Area Correction
. 176
11.8
Discussion and Bibliography
. 177
11.9
Exercises
. 180
12
Homogenization for Elliptic PDEs
. 183
12.1
Introduction
. 183
12.2
Full Equations
. 183
12.3
Simplified Equations
. 184
12.4
Derivation
. 185
12.5
Properties of the Simplified Equations
. 188
12.6
Applications
. 191
12.6.1
The One-Dimensional Case
. 191
12.6.2
Layered Materials
. 193
12.7
Discussion and Bibliography
. 195
12.8
Exercises
. 199
13
Homogenization for Parabolic PDEs
. 203
13.1
Introduction
. 203
13.2
Full Equations
. 203
13.3
Simplified Equations
. 205
13.4
Derivation
. 206
13.5
Properties of the Simplified Equations
. 207
13.6
Applications
. 209
13.6.1
Gradient Vector Fields
. 209
13.6.2
Divergence-Free Fields
. 214
Contents XVII
13.7
The Connection to SDEs
. 221
13.8
Discussion and Bibliography
. 222
13.9
Exercises
. 224
14
Averaging for Linear Transport and Parabolic PDEs
. 227
14.1
Introduction
. 227
14.2
Full Equations
. 227
14.3
Simplified Equations
. 228
14.4
Derivation
. 229
14.5
Transport Equations:
D
= 0. 230
14.5.1
The One-Dimensional Case
. 231
14.5.2
Divergence-Free Velocity Fields
. 232
14.6
The Connection to ODEs and SDEs
. 233
14.7
Discussion and Bibliography
. 235
14.8
Exercises
. 236
Partili
Theory
15
Invariant Manifolds for ODEs: The Convergence Theorem
. 239
15.1
Introduction
. 239
15.2
The Theorem
. 239
15.3
The Proof
. 241
15.4
Discussion and Bibliography
. 242
15.5
Exercises
. 243
16
Averaging for Markov Chains: The Convergence Theorem
. 245
16.1
Introduction
. 245
16.2
The Theorem
. 245
16.3
The Proof
. 246
16.4
Discussion and Bibliography
. 247
16.5
Exercises
. 248
17
Averaging for SDEs: The Convergence Theorem
. 249
17.1
Introduction
. 249
17.2
The Theorem
. 249
17.3
The Proof
. 250
17.4
Discussion and Bibliography
. 253
17.5
Exercises
. 253
18
Homogenization for SDEs: The Convergence Theorem
. 255
18.1
Introduction
. 255
18.2
The Theorem
. 255
18.3
The Proof
. 25?
18.4
Discussion and Bibliography
. 261
18.5
Exercises
.
261
XVIII
Contents
19
Homogenization
for Elliptic PDEs: The Convergence Theorem
. 263
19.1
Introduction
. 263
19.2
The Theorems
. 263
19.3
The Proof: Strong Convergence in L2
. 264
19.4
The Proof: Strong Convergence in H1
. 268
19.5
Discussion and Bibliography
. 270
19.6
Exercises
. 271
20
Homogenization for Parabolic PDEs: The Convergence Theorem
_ 273
20.1
Introduction
. 273
20.2
The Theorem
. 273
20.3
The Proof
. 274
20.4
Discussion and Bibliography
. 277
20.5
Exercises
. 278
21
Averaging for Linear Transport and Parabolic PDEs: The
Convergence Theorem
. 279
21.1
Introduction
. 279
21.2
The Theorems
. 279
21.3
The Proof:
D
> 0. 281
21.4
The Proof:
D
= 0. 282
21.5
Discussion and Bibliography
. 284
21.6
Exercises
. 285
References
. 287
Index
. 303 |
| adam_txt |
Contents
Series
Preface
.
V
Preface
.
IX
1
Introduction
. 1
1.1
Overview
. 1
1.2
Motivating Examples
. 1
1.2.1
Example I: Steady Heat Conduction in a Composite Material
2
1.2.2
Example II: Homogenization for Advection-Diffusion
Equations
. 3
1.2.3
Example III: Averaging, Homogenization, and Dynamics
. 4
1.2.4
Example IV: Dimension Reduction in Dynamical Systems
. 5
1.3
Averaging Versus Homogenization
. 6
1.3.1
Averaging for Systems of Linear Equations
. 7
1.3.2
Homogenization for Systems of Linear Equations
. 8
1.4
Discussion and Bibliography
. 9
Parti Background
2
Analysis
. 13
2.1
Setup
. 13
2.2
Notation
. 14
2.3
Banach and Hubert Spaces
. 16
2.3.1
Banach Spaces
. 16
2.3.2
Hubert Spaces
. 18
2.4
Function Spaces
. 18
2.4.1
Spaces of Continuous Functions
. 18
2.4.2
IP Spaces
. 19
2.4.3
Sobolev Spaces
. 21
2.4.4
Banach
Space-Vaîued
Spaces
. 22
2.4.5
Sobolev Spaces of Periodic Functions
. 23
XIV Contents
2.5
Two-Scale Convergence
. 25
2.5.1
Two-Scale Convergence for Steady Problems
. 25
2.5.2
Two-Scale Convergence for Time-Dependent Problems
----- 28
2.6
Equations in Hubert Spaces
. 29
2.6.1
Lax-Milgram Theory
. 30
2.6.2
The
Fredholm
Alternative
. 31
2.7
Discussion and Bibliography
. 32
2.8
Exercises
. 34
3
Probability Theory and Stochastic Processes
. 37
3.1
Setup
. 37
3.2
Probability, Expectation, and Conditional Expectation
. 37
3.3
Stochastic Processes
. 40
3.4
Martingales and Stochastic Integrals
. 46
3.4.1
Martingales
. 46
3.4.2
The
Ito
Stochastic Integral
. 47
3.4.3
The Stratonovich Stochastic Integral
. 49
3.5
Weak Convergence of Probability Measures
. 49
3.6
Discussion and Bibliography
. 54
3.7
Exercises
. 56
4
Ordinary Differential Equations
. 59
4.1
Setup
. 59
4.2
Existence and Uniqueness
. 59
4.3
The Generator
. 62
4.4
Ergodicity
. 66
4.5
Discussion and Bibliography
. 72
4.6
Exercises
. 72
5
Markov Chains
. 73
5.1
Setup
. 73
5.2
Discrete-Time Markov Chains
. 73
5.3
Continuous-Time Markov Chains
. 75
5.4
The Generator
. 76
5.5
Existence and Uniqueness
. 80
5.6
Ergodicity
. 81
5.7
Discussion and Bibliography
. 83
5.8
Exercises
. 84
6
Stochastic Differential Equations
. 85
6.1
Setup
. 85
6.2
Existence and Uniqueness
. 86
6.3
The Generator
. 88
6.4
Ergodicity
. 94
6.5
Discussion and Bibliography
. 99
6.6
Exercises
. 100
Contents
XV
7
Partial
Differential
Equations
. 103
7.1 Setup. 103
7.2
Elliptic PDEs
. 104
7.2.1 The Dirichlet Problem. 105
7.2.2
The Periodic
Problem . 107
7.2.3 The Fredholm Alternative. 107
7.2.4 The Maximum
Principle
. 113
7.3
Parabolic PDEs
. 114
7.3.1
Bounded Domains
. 114
7.3.2
The Maximum Principle
. 115
7.3.3
Unbounded Domains: The Cauchy Problem
. 117
7.4
Transport PDEs
. 118
7.5
Semigroups
. 121
7.6
Discussion and Bibliography
. 122
7.7
Exercises
. 123
Part II Perturbation Expansions
8
Invariant Manifolds for ODEs
. 127
8.1
Introduction
. 127
8.2
Full Equations
. 127
8.3
Simplified Equations
. 128
8.4
Derivation
. 129
8.5
Applications
. 130
8.5.1
Linear Fast Dynamics
. 130
8.5.2
Large Time Dynamics
. 131
8.5.3
Center Manifold
. 131
8.6
Discussion and Bibliography
. 133
8.7
Exercises
. 134
9
Averaging for Markov Chains
. 137
9.1
Introduction
. 137
9.2
Full Equations
. 137
9.3
Simplified Equations
. 139
9.4
Derivation
. 140
9.5
Application
. 141
9.6
Discussion and Bibliography
. 142
9.7
Exercises
. 142
10
Averaging for ODEs and SDEs
. 145
10.1
Introduction
. 145
10.2
Full Equations
. 145
10.3
Simplified Equations
. 146
10.4
Derivation
. 147
XVI Contents
10.5
Deterministic Problems
. 148
10.6
Applications
. 149
10.6.1
A Skew-Product SDE
. 149
10.6.2
Hamiltonian Mechanics
. 150
10.7
Discussion and Bibliography
. 153
10.8
Exercises
. 155
11
Homogenization for ODEs and SDEs
. 157
11.1
Introduction
. 157
11.2
Full Equations
. 158
11.3
Simplified Equations
. 159
11.4
Derivation
. 160
11.5
Properties of the Simplified Equations
. 162
11.6
Deterministic Problems
. 163
11.7
Applications
. 166
11.7.1
Fast Ornstein-Uhlenbeck Noise
. 166
11.7.2
Fast Chaotic Noise
. 169
11.7.3
Stratonovich Corrections
. 170
11.7.4
Stokes' Law
. 171
11.7.5
Green-Kubo Formula
. 173
11.7.6
Neither
Ito
nor Stratonovich
. 174
11.7.7
The Levy Area Correction
. 176
11.8
Discussion and Bibliography
. 177
11.9
Exercises
. 180
12
Homogenization for Elliptic PDEs
. 183
12.1
Introduction
. 183
12.2
Full Equations
. 183
12.3
Simplified Equations
. 184
12.4
Derivation
. 185
12.5
Properties of the Simplified Equations
. 188
12.6
Applications
. 191
12.6.1
The One-Dimensional Case
. 191
12.6.2
Layered Materials
. 193
12.7
Discussion and Bibliography
. 195
12.8
Exercises
. 199
13
Homogenization for Parabolic PDEs
. 203
13.1
Introduction
. 203
13.2
Full Equations
. 203
13.3
Simplified Equations
. 205
13.4
Derivation
. 206
13.5
Properties of the Simplified Equations
. 207
13.6
Applications
. 209
13.6.1
Gradient Vector Fields
. 209
13.6.2
Divergence-Free Fields
. 214
Contents XVII
13.7
The Connection to SDEs
. 221
13.8
Discussion and Bibliography
. 222
13.9
Exercises
. 224
14
Averaging for Linear Transport and Parabolic PDEs
. 227
14.1
Introduction
. 227
14.2
Full Equations
. 227
14.3
Simplified Equations
. 228
14.4
Derivation
. 229
14.5
Transport Equations:
D
= 0. 230
14.5.1
The One-Dimensional Case
. 231
14.5.2
Divergence-Free Velocity Fields
. 232
14.6
The Connection to ODEs and SDEs
. 233
14.7
Discussion and Bibliography
. 235
14.8
Exercises
. 236
Partili
Theory
15
Invariant Manifolds for ODEs: The Convergence Theorem
. 239
15.1
Introduction
. 239
15.2
The Theorem
. 239
15.3
The Proof
. 241
15.4
Discussion and Bibliography
. 242
15.5
Exercises
. 243
16
Averaging for Markov Chains: The Convergence Theorem
. 245
16.1
Introduction
. 245
16.2
The Theorem
. 245
16.3
The Proof
. 246
16.4
Discussion and Bibliography
. 247
16.5
Exercises
. 248
17
Averaging for SDEs: The Convergence Theorem
. 249
17.1
Introduction
. 249
17.2
The Theorem
. 249
17.3
The Proof
. 250
17.4
Discussion and Bibliography
. 253
17.5
Exercises
. 253
18
Homogenization for SDEs: The Convergence Theorem
. 255
18.1
Introduction
. 255
18.2
The Theorem
. 255
18.3
The Proof
. 25?
18.4
Discussion and Bibliography
. 261
18.5
Exercises
.
261
XVIII
Contents
19
Homogenization
for Elliptic PDEs: The Convergence Theorem
. 263
19.1
Introduction
. 263
19.2
The Theorems
. 263
19.3
The Proof: Strong Convergence in L2
. 264
19.4
The Proof: Strong Convergence in H1
. 268
19.5
Discussion and Bibliography
. 270
19.6
Exercises
. 271
20
Homogenization for Parabolic PDEs: The Convergence Theorem
_ 273
20.1
Introduction
. 273
20.2
The Theorem
. 273
20.3
The Proof
. 274
20.4
Discussion and Bibliography
. 277
20.5
Exercises
. 278
21
Averaging for Linear Transport and Parabolic PDEs: The
Convergence Theorem
. 279
21.1
Introduction
. 279
21.2
The Theorems
. 279
21.3
The Proof:
D
> 0. 281
21.4
The Proof:
D
= 0. 282
21.5
Discussion and Bibliography
. 284
21.6
Exercises
. 285
References
. 287
Index
. 303 |
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| author | Pavliotis, Grigorios A. Stuart, Andrew |
| author_GND | (DE-588)1274035414 (DE-588)1078856400 |
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| author_role | aut aut |
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| callnumber-first | Q - Science |
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| dewey-raw | 515.35 |
| dewey-search | 515.35 |
| dewey-sort | 3515.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| illustrated | Illustrated |
| index_date | 2024-07-02T21:57:08Z |
| indexdate | 2025-03-04T05:00:29Z |
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| isbn | 9780387738284 0387738282 |
| language | English |
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| physical | XVIII, 307 S. graph. Darst. |
| publishDate | 2008 |
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| series | Texts in applied mathematics |
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| spelling | Pavliotis, Grigorios A. Verfasser (DE-588)1274035414 aut Multiscale methods averaging and homogenization Grigorios A. Pavliotis ; Andrew M. Stuart New York, NY Springer [2008] XVIII, 307 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 53 Includes bibliographical references and index Équations aux dérivées partielles Équations différentielles Differential equations Differential equations, Partial Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Mehrskalenanalyse (DE-588)4416235-2 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Mehrskalenanalyse (DE-588)4416235-2 s Differentialgleichung (DE-588)4012249-9 s DE-604 Stuart, Andrew Verfasser (DE-588)1078856400 aut Erscheint auch als Online-Ausgabe 978-0-387-73829-1 0-387-73829-0 Texts in applied mathematics 53 (DE-604)BV002476038 53 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2990945&prov=M&dok_var=1&dok_ext=htm Inhaltstext http://www.loc.gov/catdir/enhancements/fy0829/2007941385-d.html Publisher description Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016721106&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Pavliotis, Grigorios A. Stuart, Andrew Multiscale methods averaging and homogenization Texts in applied mathematics Équations aux dérivées partielles Équations différentielles Differential equations Differential equations, Partial Differentialgleichung (DE-588)4012249-9 gnd Mehrskalenanalyse (DE-588)4416235-2 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4416235-2 (DE-588)4123623-3 |
| title | Multiscale methods averaging and homogenization |
| title_auth | Multiscale methods averaging and homogenization |
| title_exact_search | Multiscale methods averaging and homogenization |
| title_exact_search_txtP | Multiscale methods averaging and homogenization |
| title_full | Multiscale methods averaging and homogenization Grigorios A. Pavliotis ; Andrew M. Stuart |
| title_fullStr | Multiscale methods averaging and homogenization Grigorios A. Pavliotis ; Andrew M. Stuart |
| title_full_unstemmed | Multiscale methods averaging and homogenization Grigorios A. Pavliotis ; Andrew M. Stuart |
| title_short | Multiscale methods |
| title_sort | multiscale methods averaging and homogenization |
| title_sub | averaging and homogenization |
| topic | Équations aux dérivées partielles Équations différentielles Differential equations Differential equations, Partial Differentialgleichung (DE-588)4012249-9 gnd Mehrskalenanalyse (DE-588)4416235-2 gnd |
| topic_facet | Équations aux dérivées partielles Équations différentielles Differential equations Differential equations, Partial Differentialgleichung Mehrskalenanalyse Lehrbuch |
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| volume_link | (DE-604)BV002476038 |
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