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E-raamat: Uncertainty Quantification in Variational Inequalities: Theory, Numerics, and Applications

, (University of Catania, Italy), ,
  • Formaat: 404 pages
  • Ilmumisaeg: 24-Dec-2021
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351857666
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Uncertainty Quantification in Variational Inequalities: Theory, Numerics, and ApplicationsSuurem pilt
  • Formaat: 404 pages
  • Ilmumisaeg: 24-Dec-2021
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351857666
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Uncertainty Quantification (UQ) is an emerging and extremely active research discipline which aims to quantitatively treat any uncertainty in applied models. The primary objective of Uncertainty Quantification in Variational Inequalities: Theory, Numerics, and Applications is to present a comprehensive treatment of UQ in variational inequalities and some of its generalizations emerging from various network, economic, and engineering models. Some of the developed techniques also apply to machine learning, neural networks, and related fields.

Features











First book on UQ in variational inequalities emerging from various network, economic, and engineering models





Completely self-contained and lucid in style





Aimed for a diverse audience including applied mathematicians, engineers, economists, and professionals from academia





Includes the most recent developments on the subject which so far have only been available in the research literature
List of Figures
xi
List of Tables
xiii
Symbol Description xv
Preface xvii
I Variational Inequalities
1(172)
1 Preliminaries
3(24)
1.1 Elements of Functional Analysis
3(6)
1.2 Fundamentals of Measure Theory and Integration
9(6)
1.3 Essentials of Operator Theory
15(5)
1.4 An Overview of Convex Analysis and Optimization
20(5)
1.5 Comments and Bibliographical Notes
25(2)
2 Probability
27(12)
2.1 Probability Measure
27(2)
2.2 Conditional Probability and Independence
29(1)
2.3 Random Variables and Expectation
30(3)
2.4 Correlation, Independence, and Conditional Expectation
33(3)
2.5 Modes of Convergence of Random Variables
36(1)
2.6 Comments and Bibliographical Notes
37(2)
3 Projections on Convex Sets
39(18)
3.1 Projections on Convex Sets in Hilbert Spaces
40(12)
3.2 Projections on Convex Sets in Banach Spaces
52(3)
3.2.1 Metric Projection
52(1)
3.2.2 Generalized Projection
53(2)
3.3 Comments and Bibliographical Notes
55(2)
4 Variational and Quasi-Variational Inequalities
57(50)
4.1 Illustrative Examples
58(2)
4.2 Linear Variational Inequalities
60(12)
4.2.1 Some Basic Results
60(6)
4.2.2 Regularization for Linear Variational Inequalities
66(3)
4.2.3 Recession Approach for Linear Variational Inequalities
69(3)
4.3 Nonlinear Variational Inequalities
72(27)
4.3.1 Existence Results and Stability of Solutions
72(6)
4.3.2 Variational Inequalities with Monotone-Type Maps
78(6)
4.3.3 Some Geometric and Hypercircle Estimates
84(4)
4.3.4 Regularization and Convergence Rates
88(11)
4.4 Quasi-Variational Inequalities
99(6)
4.5 Comments and Bibliographical Notes
105(2)
5 Numerical Methods for Variational and Quasi-Variational Inequalities
107(66)
5.1 Projection Methods
108(18)
5.1.1 Projection Methods with a Constant Step-Length
108(3)
5.1.2 Projection Methods with a Variable Step-Length
111(11)
5.1.3 A Projection Method with Hypercircle Error Bounds
122(4)
5.2 Extragradient Methods
126(13)
5.2.1 The Extragradient Method of Korpelevich and Its Variants
126(5)
5.2.2 Hyperplane Extragradient Methods
131(1)
5.2.3 Subgradient Extragradient Methods
132(2)
5.2.4 Projected Reflected Gradient Method
134(5)
5.3 Gap Functions and Descent Methods
139(12)
5.3.1 Auchmuty's Gap Functions
139(3)
5.3.2 The Regularized Gap Function
142(6)
5.3.3 The D-Gap Function
148(3)
5.4 The Auxiliary Problem Principle
151(4)
5.5 Relaxation Method for Variational Inequalities
155(5)
5.6 Projection Methods for Quasi-Variational Inequalities
160(7)
5.7 Convergence of Recursive Sequences
167(4)
5.8 Comments and Bibliographical Notes
171(2)
II Uncertainty Quantification
173(104)
Prologue on Uncertainty Quantification
175(2)
6 An Lp-Approach for Variational Inequalities with Uncertain Data
177(36)
6.1 Linear Variational Inequalities with Random Data
177(17)
6.1.1 A Probabilistic Approximation Scheme
183(8)
6.1.2 An Illustrative Example
191(3)
6.2 Nonlinear Variational Inequalities with Random Data
194(5)
6.2.1 A Probabilistic Approximation Scheme
197(2)
6.3 Regularization of Variational Inequalities with Random Data
199(4)
6.4 Variational Inequalities with Mean-Value Constraints
203(6)
6.4.1 A Probabilistic Approximation Scheme
205(4)
6.5 Comments and Bibliographical Notes
209(4)
7 Expected Residual Minimization (ERM)
213(20)
7.1 ERM for Linear Complementarity Problems
213(9)
7.2 ERM for Nonlinear Complementarity Problems
222(2)
7.3 ERM for Variational Inequalities
224(6)
7.4 Comments and Bibliographical Notes
230(3)
8 Stochastic Approximation Approach
233(44)
8.1 Stochastic Approximation. An Overview
233(2)
8.2 Gradient and Subgradient Stochastic Approximation
235(10)
8.3 Stochastic Approximation for Variational Inequalities
245(6)
8.4 Stochastic Iterative Regularization
251(7)
8.5 Stochastic Extragradient Method
258(10)
8.6 Incremental Projection Method
268(5)
8.7 Comments and Bibliographical Notes
273(4)
III Applications
277(68)
9 Uncertainty Quantification in Electric Power Markets
279(14)
9.1 Introduction
279(1)
9.2 The Model
280(3)
9.2.1 Equilibrium Conditions for the Consumers
281(1)
9.2.2 Equilibrium Conditions for the Power Suppliers
281(1)
9.2.3 Equilibrium Conditions for the Power Generators
282(1)
9.3 Complete Supply Chain Equilibrium Conditions
283(6)
9.4 Numerical Experiments
289(1)
9.5 Comments and Bibliographical Notes
290(3)
10 Uncertainty Quantification in Migration Models
293(14)
10.1 Introduction
293(1)
10.2 A Simple Model of Population Distributions
294(3)
10.3 A More Refined Model
297(4)
10.4 Numerical Examples
301(4)
10.5 Comments and Bibliographical Notes
305(2)
11 Uncertainty Quantification in Nash Equilibrium Problems
307(14)
11.1 Introduction
307(1)
11.2 Stochastic Nash Games and Variational Inequalities
308(3)
11.3 The Stochastic Oligopoly Model
311(2)
11.4 Uncertainty Quantification in Utility Functions
313(6)
11.4.1 Linear-Quadratic Utility Functions
313(2)
11.4.2 Nonlinear Utility Functions
315(3)
11.4.3 Pointwise versus Mean-Value Constraints
318(1)
11.5 Comments and Bibliographical Notes
319(2)
12 Uncertainty Quantification in Traffic Equilibrium Problems
321(24)
12.1 Introduction
321(1)
12.2 Traffic Equilibrium Problems via Variational Inequalities
322(5)
12.3 Uncertain Traffic Equilibrium Problems
327(2)
12.4 Computational Results
329(9)
12.5 A Comparative Study of Various Approaches
338(4)
12.6 Comments and Bibliographical Notes
342(3)
Epilogue 345(2)
Bibliography 347(36)
Index 383
Joachim Gwinner is a retired Professor at the University of the Federal Army Munich. He earned his Ph.D. from University Mannheim in 1978. Then he was with Daimler-Benz company at Stuttgart for six years. After that, he became an Assistant Professor at Technical University Darmstadt and earned his Habilitation in 1989. His research interests lie in nonlinear and variational analysis, numerical analysis of partial differential equations, optimization theory and methods, and applications in continuum mechanics. He is the co-author of the monograph Advanced Boundary Element Methods: Treatment of Boundary Value, Transmission and Contact Problems.

Baasansuren Jadamba earned her Ph.D. in Applied Mathematics and Scientific Computing from Friedrich-Alexander University Erlangen-Nuremberg (Germany) in 2004, and she is an Associate Professor at the School of Mathematical Sciences at the Rochester Institute of Technology. Her research interests and publications are in the numerical analysis of partial differential equations, finite element methods, parameter identification in partial differential equations, and stochastic equilibrium problems.

Akhtar A. Khan is a Professor at the Rochester Institute of Technology. His research interests include inverse problems, optimal control, variational inequalities, and set-valued optimization. He is a co-author of the monograph Set-valued Optimization: An Introduction with Applications, Springer (2015) and co-editor of Nonlinear Analysis and Variational Problems: In Honor of George Isac, Springer (2009).

Fabio Raciti earned his Ph.D. in Theoretical Physics from the University of Catania (Italy), where he has been an Assistant Professor and then an Associate Professor of Mathematical Analysis. He is currently an Associate Professor of Operations Research at the University of Catania and has received the National (Italian) Habilitation as a Full Professor of Operations Research. He has published research work in the field of variational inequalities, optimization, inverse problems, and stochastic equilibrium problems.
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