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Deterministic and Stochastic Optimal Control and Inverse Problems



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ISBN 9780367506308
December 15, 2021 Forthcoming by CRC Press
394 Pages 25 Color & 11 B/W Illustrations

 
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Book Description

Inverse problems of identifying random parameters and random initial/boundary conditions in stochastic partial differential equations constitute a vibrant and emerging research area that has found numerous applications. A related problem of paramount importance is the optimal control problem for stochastic differential equations.

This edited volume comprises invited contributions from world-renowned researchers in the subject of stochastic control and inverse problems. There are several contributions on stochastic optimal control and stochastic inverse problems covering different aspects of the theory, numerical methods, and applications. Besides a unified presentation of the most recent and relevant developments, this volume also presents some survey articles to make the material self-contained. To maintain the highest level of scientific quality, all manuscripts have been thoroughly reviewed.

Table of Contents

Preface

All-At-Once Formulation Meets the Bayesian Approach: A Study of Two Prototypical Linear Inverse Problems

Anna Schlintl and Barbara Kaltenbacher

Introduction

Function Space Setting and Computation of Adjoints

Analysis of the Eigenvalues

Convergence Analysis

Choice of Joint Priors

Numerical Experiments

Conclusions and Remarks

References

On Iterated Tikhonov Kaczmarz Type Methods for Solving Systems of Linear Ill-posed Operator Equations

R. Filippozzi, J.C. Rabelo, and A. Leit˜ao

Introduction

A Range-relaxed Iterated Tikhonov Kaczmarz Method

A Convergence Result for Exact Data

Numerical Experiments

Conclusions

References

On Numerical Approximation of Optimal Control for Stokes Hemivariational Inequalities

Xiaoliang Cheng, Rongfang Gong, and Weimin Han

Introduction

Notation and Preliminaries

Stokes Hemivariational Inequality and Optimal Control

Numerical Approximation of the Optimal Control Problem

References

Tikhonov Regularization with Oversmoothing Penalty: The Balancing Principles

Bernd Hofmann, Christopher Hofmann, Peter Math´e, and Robert Plato

Introduction

General Error Estimate for Tikhonov Regularization in Hilbert Scales with Oversmoothing Penalty

Balancing Principles

Exponential Growth Model: Properties and Numerical Case Study

References

An Optimization Approach to Parameter Identification in Variational Inequalities of Second Kind - II

Joachim Gwinner

Introduction

Some Variational Inequalities and an Abstract Framework for Parameter Identification

The Regularization Procedure

The Optimization Approach

Concluding Remarks - An Outlook

References

 

 

 

 

 

Generalized Variational-hemivariational Inequalities in Fuzzy Environment

Shengda Zeng, Jinxia Cen, Stanisław Mig´orski, and Van Thien Nguyen

Introduction

Mathematical Prerequisites

Fuzzy Variational-hemivariational Inequalities

Optimal Control Problem

References

Boundary Stabilization of the Linear MGT Equation with Feedback Neumann Control

Marcelo Bongarti and Irena Lasiecka

Introduction

Wellposedness: proof of Theorem

References

Sweeping Process Arguments in the Analysis and Control of a Contact Problem

Mircea Sofonea and Yi-bin Xiao

Introduction

Notation and Preliminaries

The Contact Model

An Existence and Uniqueness Result

A Continuous Dependence Result

An Optimal Control Problem

Conclusion

References

Anderson Acceleration for Degenerate and Nondegenerate Problems Sara Pollock

Introduction

Nondegenerate Problems

Degenerate Problems

Conclusion

References

Approximate Coincidence Points for Single-valued Maps and Aubin Continuous Set-valued Maps

Mohamed Ait Mansour, Mohamed Amin Bahraoui, and Adham El Bekkali

Introduction

Notation

Coincidence and Approximate Coincidence Points of Singlevalued Maps

An Application: Parametric Abstract Systems of Equations

Approximate Local Contraction Mapping Principle and e-Fixed Points

Lyusternik-Graves Theorem and e-Fixed Points for Aubin Continuous

Set-valued Maps

References

Stochastic Variational Approach for Random Cournot-Nash Principle

Annamaria Barbagallo, Massimiliano Ferrara, and Paolo Mauro

Introduction

The Random Model

Existence Results

The Infinite-dimensional Duality Theory

The Lagrange Formulation of the Random Model

The Inverse Problem

A Numerical Example

Concluding Remarks

References

Augmented Lagrangian Methods for Optimal Control Problems Governed by Mixed Variational Hemivariational Inequalities Involving a Set-valued Mapping

O. Chadli and R.N. Mohapatra

Introduction

Problem Statement and Preliminaries

Existence Results for Solutions

Optimal Control

Application to Optimal Control of a Frictional Contact Problem

Remarks and Comments

References

Data Driven Reconstruction Using Frames and Riesz Bases

Andrea Aspri, Leon Frischauf, Yury Korolev, and Otmar Scherzer

Introduction

Gram-Schmidt Orthonormalization

Basics on Frames and Riesz-bases

Data Driven Regularization by Frames and Riesz Bases

Numerical Experiments

Conclusions

References

Antenna Problem Induced Regularization and Sampling Strategies

Willi Freeden

Uni-Variate Antenna Problem Induced Recovery Strategies

Multi-Variate Antenna Problem Induced Recovery Strategies

References

An Equation Error Approach for Identifying a Random Parameter in a Stochastic Partial Differential Equation

Baasansuren Jadamba, Akhtar A. Khan, Quinn Kolt, and Miguel Sama

Introduction

Solvability of the Direct Problem

Numerical Techniques for Stochastic PDEs

An Equation Error Approach

Discrete Formulae

Computational Experiments

Concluding Remarks

References

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Editor(s)

Biography

Baasansuren Jadamba is an Associate Professor at the Rochester Institute of Technology. She received her Ph.D. from Friedrich-Alexander University Erlangen-Nuremberg in 2004. Her research interests are numerical analysis of partial differential equations, finite element methods, parameter identification problems in partial differential equations, and stochastic equilibrium problems.

Akhtar A. Khan is a Professor at the Rochester Institute of Technology. His research deals with set-valued Optimization, inverse problems, and variational inequalities. He is a co-author of Set-valued Optimization, Springer (2015), and Co-editor of Nonlinear Analysis and Variational Problems, Springer (2009). He is Co-Editor in Chief of the Journal of Applied and Numerical Optimization, and Editorial Board member of Optimization, Journal of Optimization Theory and Applications, and Journal of Nonlinear and Variational Analysis.

Stanisław Migórski received his Ph.D. and Habilitation from Jagiellonian University in Krakow (JUK). Currently, he is a Full and Chair Professor of Mathematics at the Faculty of Mathematics and Computer Science at JUK. He published research work in the field of mathematical analysis and applications (partial differential equations, variational inequalities, optimal control, and Optimization). He is the co-author of two volumes on Nonlinear Analysis (Kluwer/Plenum), and monographs on Nonlinear Inclusions and Hemivariational Inequalities (Springer), and Variational-Hemivariational Inequalities with Applications (Chapman & Hall).

Miguel Sama is Associate Professor at Universidad Nacional Educación a Distancia (Madrid, Spain). His research is broadly on Optimization, focusing mainly on Applied Mathematics Models. His research interests are in infinite-dimensional optimization problems. They cover a wide range of theoretical and applied topics such as Ordered Vector Spaces, Set-Valued Analysis, Vector, and Set-Valued Optimization. PDE-constrained Optimization, Inverse Problems, Optimal Control Problems, and Uncertainty Quantification.