دانلود کتاب Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization (International Series of Numerical Mathematics, 172)

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Preface
Contents
Error Bounds for Discretized Optimal Transport and Its Reliable Efficient Numerical Solution
1 Introduction
2 Discretized Optimal Transport
2.1 General Formulation
2.2 Discretization
2.3 Optimality Conditions
2.4 Sparsity
3 Error Analysis
4 Active-Set Strategy
5 Numerical Experiments
5.1 Problem Specifications
5.2 Complexity Considerations
5.3 Experimental Convergence Rates
References
Numerical Methods for Diagnosis and Therapy Design of Cerebral Palsy by Bilevel Optimal Control of Constrained Biomechanical Multi-Body Systems
1 Introduction
1.1 Cerebral Palsy
1.2 Modeling Approach
2 Modeling the Human Body
2.1 Rigid Multi-Body Systems
2.2 Detailed Submodules
2.2.1 Foot Modeling and Ground Contact
2.2.2 Muscle Modeling
2.3 Biomechanical Model for p03-cp Patients
3 Modeling the Human Gait
3.1 A Multi-Phase Optimal Control Approach
3.2 A Mixed-Integer Optimal Control Approach
4 Two Bilevel Problems for Diagnosis and Therapy Design of Cerebral Palsy
4.1 An Inverse Optimal Control Problem for Diagnosis of Cerebral Palsy
4.2 A Robustified Optimal Control Problem for Therapy Design of Cerebral Palsy
5 Conclusions and Outlook
References
ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation
1 Introduction
2 A Continuation Method for MOPs with Inexact Objective Gradients
2.1 Multiobjective Optimization
2.2 Continuation Method with Exact Gradients
2.3 Continuation Method with Inexact Gradients
2.3.1 Strategy 1
2.3.2 Strategy 2
2.4 Globalization Approach
3 Multiobjective Optimization of an Elliptic PDE Using the RB Method
3.1 Multiobjective Optimization of an Elliptic PDE
3.2 The Reduced Basis Method
3.3 Error Estimation for the Gradients
4 Numerical Results
4.1 Generation of the Reduced Basis
4.2 Application of the Continuation Methods to an MPOP
5 Conclusion and Outlook
Appendix A: Proof of Theorem 2.10
References
Analysis and Solution Methods for Bilevel Optimal Control Problems
1 Introduction
2 Two Example Problems
3 Optimality Conditions
3.1 Definition of Optimality Systems for (IOCf)
3.2 Regularization Approach
3.2.1 Assumptions and Properties of the Lower Level Problem
3.2.2 C-Stationarity for Local Minimizers
3.3 Relaxation Approach
3.3.1 The Optimal Value Reformulation and Its Relaxation
3.3.2 C-Stationarity for Local Minimizers
3.4 Variational Analysis Approach and Mordukhovich-Stationarity
3.5 Comments on Biactivity and S-Stationarity
4 Numerical Solution
4.1 Global Solution Algorithm for ([eq:upperlevel]IOCf2)
4.2 Numerical Example
5 Future Perspectives
References
A Calculus for Non-smooth Shape Optimization with Applications to Geometric Inverse Problems
1 Introduction
2 Image Reconstruction on Surfaces
2.1 Functions of Bounded Variation on Surfaces
2.2 Dual Representation
2.3 Implementation Details and Numerical Results
2.4 Discrete Total Variation
3 Shape Optimization Using Total Variation of the Normal Vector Field
3.1 Total Variation of Normal
3.2 Mesh Denoising
3.3 Inverse Problem
4 Conclusion and Outlook
References
Rate-Independent Systems and Their Viscous Regularizations: Analysis, Simulation, and Optimal Control
1 Introduction
2 Rate-Independent Systems and Solution Concepts
3 Discretization Schemes for Rate-Independent Systems and Their Convergence
3.1 The Semilinear Setting
3.2 Discretization Schemes, Abstract Semilinear Setting
3.3 A Priori Estimates, Abstract Semilinear Setting
3.4 Finite-Element Discretization and Numerical Realization
4 Optimal Control of Rate-Independent Systems
5 Optimal Control of Thermo-Viscoplasticity
References
Generalized Nash Equilibrium Problems with Partial Differential Operators: Theory, Algorithms, and Risk Aversion
1 Introduction
2 Nash Games Involving Nonlinear Operator Equations
2.1 On the Convexity of Optimal Control Problems Involving Nonlinear Operator Equations
3 Nash Games Using Penalization Techniques
3.1 -Convergence
4 PDE-Constrained GNEPs Under Uncertainty
4.1 Motivation
4.2 Additional Notation and Preliminary Results
4.3 Risk-Averse PDE-Constrained Optimization: Theory
4.4 A Risk-Averse PDE-Constrained Nash Equilibrium Problem
4.5 Risk-Averse PDE-Constrained Decision Problems: Smooth Approximation
4.6 Risk-Averse PDE-Constrained Optimization: Solution Methods
5 Outlook
References
Stability and Sensitivity Analysis for Quasi-Variational Inequalities
1 Introduction
2 QVIs: Mathematical Setting and Existence
2.1 Existence of Solutions: Order Approach
2.2 Existence of Solutions: Iteration Approach
2.3 Miscellaneous: A Pitfall
3 Sensitivities
3.1 Stability for Minimal and Maximal Solution Maps
3.2 Directional Differentiability
3.3 Parabolic QVIs
3.4 Application to Thermoforming
3.4.1 The Model
3.4.2 Properties and Existence for the System
3.4.3 Numerical Implementation Details
3.4.4 Numerical Results
4 Control of QVIs
5 Outlook
References
Simulation and Control of a Nonsmooth Cahn–Hilliard Navier–Stokes System with Variable Fluid Densities
1 Introduction
2 Problem Setting
3 Optimal Control of the Semi-Discrete CHNS System
3.1 The Semi-Discrete CHNS System and the Optimal Control Problem
3.2 Existence of Feasible and Globally Optimal Points
3.3 E-Almost C-Stationary Points
3.4 Strong Stationarity
3.5 Adaptive Mesh Refinement
3.6 Penalization Algorithm
3.7 Bundle-Free Implicit Programming Approach
4 Model Order Reduction with Proper Orthogonal Decomposition
4.1 POD in Hilbert Spaces with Space-Adapted Snapshots
4.2 POD Reduced-Order Modeling for the Cahn–Hilliard System
4.3 Numerical Example of POD-MOR for the Cahn–Hilliard System
4.4 Stable POD Reduced-Order Modeling for Navier–Stokes with Space-Adapted Snapshots
5 Outlook
References
Safeguarded Augmented Lagrangian Methods in Banach Spaces
1 Introduction
2 Background Material
2.1 Cones
2.2 Convex Functions and Concave Operators
2.3 Pseudomonotone Operators
2.4 KKT-Type Conditions
3 Motivation and Statement of the Algorithm
3.1 The Original Method of Multipliers
3.2 Inequality Constraints and Slack Variables
3.3 The Algorithm
4 Global Convergence Theory
4.1 Existence of Penalized Solutions
4.2 Convergence to Global Minimizers
4.3 Stationarity of Limit Points
5 Local Convergence
5.1 Existence of Local Minima und Strong Convergence
5.2 Rate of Convergence
6 Numerical Results
6.1 State-Constrained Optimal Control Problems
6.2 Bratu\'s Obstacle Problem
6.3 C-Minimization
7 Final Remarks
References
Decomposition and Approximation for PDE-Constrained Mixed-Integer Optimal Control
1 Introduction
1.1 Outline of the Remaining Sections
1.2 Notation
2 Approximation Arguments for the CIA Decomposition
2.1 Properties of Rounding Meshes and Algorithms
2.2 Weak Control Approximation
2.3 State Vector Approximation
2.4 Optimality and Feasibility in the Absence of Mixed Constraints
2.5 Optimality and Feasibility in the Presence of Mixed Constraints
3 Approximation Quality of Roundings
3.1 Sum-up Rounding Algorithms
3.2 Combinatorial Integral Approximation Problems
4 Solving the CIA Problem
5 Illustration of the Multidimensional Control Approximation
5.1 Test Problem
5.2 Mesh Structure and Sierpinski Curve
5.3 Numerical Results Obtained with the CIA Decomposition
6 Conclusion
References
Strong Stationarity for Optimal Control of Variational Inequalities of the Second Kind
1 Introduction
2 Strong Stationarity in an Abstract Framework
3 Application to Concrete Settings
3.1 The Obstacle Problem
3.2 Static Elastoplasticity
3.3 The Lasso Problem in Sobolev Spaces
3.4 Non-Newtonian Fluids: The Mosolov Problem
4 Conclusion
Appendix A: Auxiliary Results
References
Optimizing Fracture Propagation Using a Phase-Field Approach
1 Introduction
2 Problem Setting
2.1 Model Problem, Notation, and Assumptions
2.2 The Phase-Field Equation
3 The Limiting First-Order Necessary Conditions
4 An SQP Method for (NLPγ)
4.1 The SQP Algorithm
4.2 First-Order Optimality Conditions for (QPγ) and Its Limit
4.3 Approximation of (QPγ) by Finite Elements
5 An SQP Method for (NLPVI)
5.1 SQP Algorithm for (NLPVI)
5.2 Convergence of FE Approximation to (QPVI)
References
Algorithms for Optimal Control of Elastic Contact Problems with Finite Strain
1 Introduction
2 Contact Problems in Hyperelasticity
3 Optimal Control of Nonlinear Elastic Contact Problems
4 Numerical Optimization Algorithms
4.1 An Affine Covariant Composite Step Method
4.2 Computation of Steps by Iterative Solvers
4.3 Inexact Constraint Preconditioning
4.4 Accuracy Matching
4.5 Choice of Functional Analytic Framework
4.6 Non-convexity of Objective and Energy
4.7 Non-convexity of the Energy
4.8 Path Following
5 Conclusion and Outlook
References
Algorithms Based on Abs-Linearization for Non-smooth Optimization with PDE Constraints
1 Motivation and Introduction
2 The Abs-Linearization
3 The SALMIN Algorithm
4 The SCALi Algorithm
5 Conclusion and Outlook
References
Shape Optimization for Variational Inequalities of Obstacle Type: Regularized and Unregularized Computational Approaches
1 Introduction
2 Model Problem and Its Challenges
3 Optimization Based on the Steklov–Poincaré Metric
4 Solution Techniques Based on the Regularized Problem
4.1 Linearized Adapted Primal-dual Active Set Algorithm
4.2 Deformation Equation
4.3 Summary
5 Toward the Unregularized Problem
5.1 Analytical Investigations: Existence of Adjoints and Shape Derivatives
5.2 Optimization Algorithm
6 Numerical Results
References
Extensions of Nash Games in Finite and Infinite Dimensions with Applications
1 Introduction and State-of-the-Art
2 Games in Finite Dimensions
2.1 A GNEP with Vanishing Constraints for Computation Offloading
2.2 Quadratic Multi-Leader–Follower Game
3 Games in Infinite Dimensions
3.1 Stationarity Concepts for MLFGs in Banach Spaces
3.2 An MLFG with Quadratic Lower Level Problem
3.3 Stackelberg Game with Infinitely Many Followers
3.4 Dynamic Boundary Control Games with Networks of Strings
4 Outlook
References
Stress-Based Methods for Quasi-Variational Inequalities Associated with Frictional Contact
1 Introduction
2 The Dual Stress-Based Formulation of Contact with Coulomb Friction
3 A Posteriori Error Estimation by Displacement Reconstruction
4 A Norm Equivalence in H1/2 ()
5 Numerical Experiments
References
An Inexact Bundle Method and Subgradient Computations for Optimal Control of Deterministic and Stochastic Obstacle Problems
1 Introduction
2 Inexact Bundle Method
2.1 Trial Iterates, Function Values, and Subgradients
2.2 The Cutting Plane Model
2.3 Proximity Control
2.4 The Subproblem of the Bundle Method
2.5 Global Convergence Result
3 Generalized Derivatives
4 Properties of the Obstacle Problem
4.1 Differentiability of the Solution Operator
5 An Element of the Bouligand Generalized Differential
5.1 The Set-valued Map u →H01(I(u))
5.2 Existence of Points of Gâteaux Differentiability in the Positive Cone
5.3 Characterization of a Generalized Derivative
6 Characterization of the Entire Generalized Differentials
6.1 Capacitary Measures and the Differentials Involving the Weak Operator Topologies
7 The Stochastic Obstacle Problem
7.1 Problem Setting
7.2 Approximate Subgradients of the Stochastic Reduced Objective Function
7.3 Computation of Exact Subgradients
7.4 Construction of a Subgradient for
References
Maxwell Variational Inequalities in Type-II Superconductivity
1 Introduction
2 Preliminaries
3 Analysis
3.1 Direct Approach
3.2 General Hyperbolic Maxwell VIs of the Second Kind
4 Numerical Analysis
4.1 Fully Discrete Scheme
5 Computations
6 Numerical Experiments
References