Optimal control of PDEs under uncertainty an introduction with application to optimal shape design of structures /
This book provides a direct and comprehensive introduction to theoretical and numerical concepts in the emerging field of optimal control of partial differential equations (PDEs) under uncertainty. The main objective of the book is to offer graduate students and researchers a smooth transition from...
| Main Author: | Martínez-Frutos, Jesús, |
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| Other Authors: | Esparza, Francisco Periago,, SpringerLink (Online service) |
| Format: | eBook |
| Language: | English |
| Published: |
Cham, Switzerland :
Springer,
[2018]
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| Physical Description: |
1 online resource. |
| Series: |
SpringerBriefs in mathematics.
BCAM SpringerBriefs. |
| Subjects: |
| Summary: |
This book provides a direct and comprehensive introduction to theoretical and numerical concepts in the emerging field of optimal control of partial differential equations (PDEs) under uncertainty. The main objective of the book is to offer graduate students and researchers a smooth transition from optimal control of deterministic PDEs to optimal control of random PDEs. Coverage includes uncertainty modelling in control problems, variational formulation of PDEs with random inputs, robust and risk-averse formulations of optimal control problems, existence theory and numerical resolution methods. The exposition focusses on the entire path, starting from uncertainty modelling and ending in the practical implementation of numerical schemes for the numerical approximation of the considered problems. To this end, a selected number of illustrative examples are analysed in detail throughout the book. Computer codes, written in MatLab, are provided for all these examples. This book is adressed to graduate students and researches in Engineering, Physics and Mathematics who are interested in optimal control and optimal design for random partial differential equations.-- |
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| Item Description: |
Includes bibliographical references and index. Intro; Preface; References; Contents; About the Authors; Acronyms and Initialisms; Abstract; 1 Introduction; 1.1 Motivation; 1.2 Modelling Uncertainty in the Input Data. Illustrative Examples; 1.2.1 The Laplace-Poisson Equation; 1.2.2 The Heat Equation; 1.2.3 The Bernoulli-Euler Beam Equation; References; 2 Mathematical Preliminaires; 2.1 Basic Definitions and Notations; 2.2 Tensor Product of Hilbert Spaces; 2.3 Numerical Approximation of Random Fields; 2.3.1 Karhunen-Loève Expansion of a Random Field; 2.4 Notes and Related Software; References. 3 Mathematical Analysis of Optimal Control Problems Under Uncertainty3.1 Variational Formulation of Random PDEs; 3.1.1 The Laplace-Poisson Equation Revisited I; 3.1.2 The Heat Equation Revisited I; 3.1.3 The Bernoulli-Euler Beam Equation Revisited I; 3.2 Existence of Optimal Controls Under Uncertainty; 3.2.1 Robust Optimal Control Problems; 3.2.2 Risk Averse Optimal Control Problems; 3.3 Differences Between Robust and Risk-Averse Optimal Control; 3.4 Notes; References; 4 Numerical Resolution of Robust Optimal Control Problems. 4.1 Finite-Dimensional Noise Assumption: From Random PDEs to Deterministic PDEs with a Finite-Dimensional Parameter4.2 Gradient-Based Methods; 4.2.1 Computing Gradients of Functionals Measuring Robustness; 4.2.2 Numerical Approximation of Quantities of Interest in Robust Optimal Control Problems; 4.2.3 Numerical Experiments; 4.3 Benefits and Drawbacks of the Cost Functionals; 4.4 One-Shot Methods; 4.5 Notes and Related Software; References; 5 Numerical Resolution of Risk Averse Optimal Control Problems; 5.1 An Adaptive, Gradient-Based, Minimization Algorithm. 5.2 Computing Gradients of Functionals Measuring Risk Aversion5.3 Numerical Approximation of Quantities of Interest in Risk Averse Optimal Control Problems; 5.3.1 An Anisotropic, Non-intrusive, Stochastic Galerkin Method; 5.3.2 Adaptive Algorithm to Select the Level of Approximation; 5.3.3 Choosing Monte Carlo Samples for Numerical Integration; 5.4 Numerical Experiments; 5.5 Notes and Related Software; References; 6 Structural Optimization Under Uncertainty; 6.1 Problem Formulation; 6.2 Existence of Optimal Shapes; 6.3 Numerical Approximation via the Level-Set Method. 6.3.1 Computing Gradients of Shape Functionals; 6.3.2 Mise en Scène of the Level Set Method; 6.4 Numerical Simulation Results; 6.5 Notes and Related Software; References; 7 Miscellaneous Topics and Open Problems; 7.1 The Heat Equation Revisited II; 7.2 The Bernoulli-Euler Beam Equation Revisited II; 7.3 Concluding Remarks and Some Open Problems; References; Index. This book provides a direct and comprehensive introduction to theoretical and numerical concepts in the emerging field of optimal control of partial differential equations (PDEs) under uncertainty. The main objective of the book is to offer graduate students and researchers a smooth transition from optimal control of deterministic PDEs to optimal control of random PDEs. Coverage includes uncertainty modelling in control problems, variational formulation of PDEs with random inputs, robust and risk-averse formulations of optimal control problems, existence theory and numerical resolution methods. The exposition focusses on the entire path, starting from uncertainty modelling and ending in the practical implementation of numerical schemes for the numerical approximation of the considered problems. To this end, a selected number of illustrative examples are analysed in detail throughout the book. Computer codes, written in MatLab, are provided for all these examples. This book is adressed to graduate students and researches in Engineering, Physics and Mathematics who are interested in optimal control and optimal design for random partial differential equations.-- Provided by publisher. |
| Physical Description: |
1 online resource. |
| Bibliography: |
Includes bibliographical references and index. |
| ISBN: |
9783319982106 3319982109 9783319982113 3319982117 |