Deterministic and Stochastic Optimal Control and Inverse Problems

Cover
Title Page
Copyright Page
Dedication
Preface
Table of Contents
Contributors
1. All-At-Once Formulation Meets the Bayesian Approach: A Study of Two Prototypical Linear Inverse Problems
	1.1 Introduction
		1.1.1 Examples
	1.2 Function Space Setting and Computation of Adjoints
		1.2.1 Inverse Source Problem
		1.2.2 Backwards Heat Problem
	1.3 Analysis of the Eigenvalues
		1.3.1 Inverse Source Problem
			1.3.1.1 Analytic Computation of the Eigenvalues
			1.3.1.2 Numerical Computation of the Eigenvalues
		1.3.2 Backwards Heat Equation
			1.3.2.1 Analytic Computation of the Eigenvalues
			1.3.2.2 Numerical Computation of the Eigenvalues
	1.4 Convergence Analysis
		1.4.1 Fulfillment of the Link Condition for the All-At-Once-Formulation
	1.5 Choice of Joint Priors
		1.5.1 Block Diagonal Priors Satisfying Unilateral Link Estimates
		1.5.2 Heuristic Choice of C0 for the Backwards Heat Problem
		1.5.3 Priors for the Inverse Source Problem
		1.5.4 Prior for the State Variable of the Backwards Heat Problem
	1.6 Numerical Experiments
		1.6.1 Lagrangian Method for Computing the Adjoint Based Hessian and Gradient
			1.6.1.1 Inverse Source Problem
			1.6.1.2 Backwards Heat Problem
		1.6.2 Implementation
			1.6.2.1 Inverse Source Problem
			1.6.2.2 Backwards Heat Equation, Sampled Initial Condition
			1.6.2.3 Backwards Heat Equation, Chosen Initial Condition
			1.6.2.4 Backwards Heat Equation, Chosen Initial Condition, Prior Motivated by the Link Condition
	1.7 Conclusions and Remarks
	References
2. On Iterated Tikhonov Kaczmarz Type Methods for Solving Systems of Linear Ill-posed Operator Equations
	2.1 Introduction
	2.2 A Range-relaxed Iterated Tikhonov Kaczmarz Method
		2.2.1 Main Assumptions
		2.2.2 Description of the Method
		2.2.3 Preliminary Results
	2.3 A Convergence Result for Exact Data
	2.4 Numerical Experiments
	2.5 Conclusions
	References
3. On Numerical Approximation of Optimal Control for Stokes Hemivariational Inequalities
	3.1 Introduction
	3.2 Notation and Preliminaries
	3.3 Stokes Hemivariational Inequality and Optimal Control
	3.4 Numerical Approximation of the Optimal Control Problem
	References
4. Nonlinear Tikhonov Regularization in Hilbert Scales with Oversmoothing Penalty: Inspecting Balancing Principles
	4.1 Introduction
		4.1.1 Hilbert Scales with Respect to an Unbounded Operator
		4.1.2 Tikhonov Regularization with Smoothness Promoting Penalty
		4.1.3 State of the Art
		4.1.4 Goal of the Present Study
	4.2 General Error Estimate for Tikhonov Regularization in Hilbert Scales with Oversmoothing Penalty
		4.2.1 Smoothness in Terms of Source Conditions
		4.2.2 Error Decomposition
	4.3 Balancing Principles
		4.3.1 Quasi-optimality
		4.3.2 The Balancing Principles: Setup and Formulation
		4.3.3 Discussion
		4.3.4 Specific Impact on Oversmoothing Penalties
	4.4 Exponential Growth Model: Properties and Numerical Case Study
		4.4.1 Properties
		4.4.2 Numerical Case Study
	References
5. An Optimization Approach to Parameter Identification in Variational Inequalities of Second Kind-II
	5.1 Introduction
	5.2 Some Variational Inequalities and an Abstract Framework for Parameter Identification
	5.3 The Regularization Procedure
		5.3.1 Smoothing the Modulus Function
		5.3.2 Regularizing the VI of Second Kind
		5.3.3 An Estimate of the Regularization Error
	5.4 The Optimization Approach
	5.5 Concluding Remarks—An Outlook
	References
6. Generalized Variational-hemivariational Inequalities in Fuzzy Environment
	6.1 Introduction
	6.2 Mathematical Prerequisites
	6.3 Fuzzy Variational-hemivariational Inequalities
	6.4 Optimal Control Problem
	References
7. Boundary Stabilization of the Linear MGT Equation with Feedback Neumann Control
	7.1 Introduction
		7.1.1 The Linearized PDE Model with Space-dependent Viscoelasticity
		7.1.2 Main Results and Discussion
	7.2 Wellposedness: Proof of Theorem 7.1.2
		7.2.1 Stabilization in H: Proof of Theorem 7.1.5
	References
8. Sweeping Process Arguments in the Analysis and Control of a Contact Problem
	8.1 Introduction
	8.2 Notation and Preliminaries
	8.3 The Contact Model
	8.4 An Existence and Uniqueness Result
	8.5 A Continuous Dependence Result
	8.6 An Optimal Control Problem
	8.7 Conclusion
	References
9. Anderson Acceleration for Degenerate and Nondegenerate Problems
	9.1 Introduction
		9.1.1 Mathematical Setting and Algorithm
		9.1.2 The Nondegeneracy Condition
	9.2 Nondegenerate Problems
		9.2.1 Relating Residuals to Differences Between Consecutive Iterates
		9.2.2 Full Residual Bound
		9.2.3 Numerical Examples (Nondegenerate)
	9.3 Degenerate Problems
		9.3.1 Scalar AA-Newton
		9.3.2 Methods for Higher-order Roots
		9.3.3 Analysis of the AA-Newton Rootfinding Method
		9.3.4 Numerical Examples (Degenerate)
			9.3.4.1 Example 1
			9.3.4.2 Example 2
	9.4 Conclusion
	References
10. Approximate Coincidence Points for Single-valued Maps and Aubin Continuous Set-valued Maps
	10.1 Introduction
	10.2 Notation
	10.3 Coincidence and Approximate Coincidence Points of Single-valued Maps
	10.4 An Application: Parametric Abstract Systems of Equations
	10.5 Approximate Local Contraction Mapping Principle and ε-Fixed Points
	10.6 Lyusternik-Graves Theorem and ε-Fixed Points for Aubin Continuous Set-valued Maps
	References
11. Stochastic Variational Approach for Random Cournot-Nash Principle
	11.1 Introduction
	11.2 The Random Model
	11.3 Existence Results
	11.4 The Infinite-dimensional Duality Theory
	11.5 The Lagrange Formulation of the Random Model
	11.6 The Inverse Problem
	11.7 A Numerical Example
	11.8 Concluding Remarks
	References
12. Augmented Lagrangian Methods For Optimal Control Problems Governed by Mixed Variational-Hemivariational Inequalities Involving a Set-valued Mapping
	12.1 Introduction
	12.2 Problem Statement and Preliminaries
	12.3 Existence Results for Solutions
	12.4 Optimal Control
	12.5 Application to Optimal Control of a Frictional Contact Problem
	12.6 Remarks and Comments
	References
13. Data Driven Reconstruction Using Frames and Riesz Bases
	13.1 Introduction
	13.2 Gram-Schmidt Orthonormalization
		13.2.1 Weak Convergence
	13.3 Basics on Frames and Riesz-Bases
	13.4 Data Driven Regularization by Frames and Riesz Bases
		13.4.1 Weak Convergence
	13.5 Numerical Experiments
		13.5.1 Orthonormalization Procedures
		13.5.2 Comparison Between Frames and Decomposition Algorithms
	13.6 Conclusions
	References
14. Antenna Problem Induced Regularization and Sampling Strategies
	14.1 Uni-Variate Antenna Problem Induced Recovery Strategies
	14.2 Multi-Variate Antenna Problem Induced Recovery Strategies
	References
15. An Equation Error Approach for Identifying a Random Parameter in a Stochastic Partial Differential Equation
	15.1 Introduction
	15.2 Solvability of the Direct Problem
	15.3 Numerical Techniques for Stochastic PDEs
		15.3.1 Monte Carlo Finite Element Type Methods
		15.3.2 The Stochastic Collocation Method
		15.3.3 The Stochastic Galerkin Method
	15.4 An Equation Error Approach
	15.5 Discrete Formulae
	15.6 Computational Experiments
	15.7 Concluding Remarks
	References
Index