Advances In Dynamics, Optimization And Computation: A Volume Dedicated To Michael Dellnitz On The Occasion Of His 60th Birthday (studies In Systems, Decision And Control (304))

This book presents a collection of papers on recent advances in problems concerning dynamics, optimal control and optimization. In many chapters, computational techniques play a central role. Set-oriented techniques feature prominently throughout the book, yielding state-of-the-art algorithms for computing general invariant sets, constructing globally optimal controllers and solving multi-objective optimization problems. Preface Acknowledgements Contents List of Contributors I Dynamics A Continuation Approach to Computing Phase Resetting Curves 1 Introduction 2 Basic Setting and Definitions 3 Algorithm for Computing a Phase Resetting Curve 3.1 Continuation Set-Up for Rotated Representation of 3.2 Continuation Set-Up for the Phase Reset 4 Illustration of the Method with a Model Example 4.1 Computing the PTC 4.2 Loss of Invertibility 5 Phase Resetting in the FitzHugh-Nagumo Model 6 Phase Resetting in a Seven-Dimensional Sinoatrial Node Model 7 Conclusions References Input-Output Networks, Singularity Theory, and Homeostasis 1 Introduction 2 Thermoregulation: A Motivation for Homeostasis 3 Biochemical Input-Output Networks 3.1 Feedforward Excitation 3.2 Product Inhibition 3.3 Substrate Inhibition 3.4 Negative Feedback Loop 4 Infinitesimal Homeostasis 5 Input-Output Networks 6 Core Networks 7 Types of Infinitesimal Homeostasis 8 Low Degree Homeostasis Types 9 Singularity Theory of Input-Output Functions 9.1 Chair Points for Blocks of Degree 1 and 2 9.2 Elementary Catastrophe Theory and Homeostasis 10 Evolving Towards Homeostasis 11 Input-Output Maps with Two Inputs 11.1 Catastrophe Theory Classification 11.2 Normal Forms and Plateaus 11.3 The Hyperbolic Umbilic 12 Gene Regulatory Networks and Housekeeping Genes 12.1 Gene Regulatory Networks and Homeostasis 12.2 Basic Structural Elements of GRNs 12.3 The Gene Regulatory Network for PGA2 References The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems 1 Introduction 2 Infinite Dimensional Embedding Techniques 3 The Core Dynamical System 4 Computation of Embedded Invariant Sets 4.1 A Subdivision Scheme for the Approximation of Embedded Attractors 4.2 A Continuation Technique for the Approximation of Embedded Unstable Manifolds 5 Numerical Realization of the CDS 5.1 Delay Differential Equations 5.2 Partial Differential Equations 6 Numerical Results 6.1 The Mackey-Glass Equation 6.2 The Kuramoto-Sivashinsky Equation 7 Conclusion References Set-Oriented and Finite-Element Study of Coherent Behavior in Rayleigh-Bénard Convection 1 Introduction 2 Nonautonomous Dynamics, Transfer Operators and Transport 3 Set-Oriented Numerical Framework 3.1 Approximation of Transfer Operator 3.2 Extracting Finite-Time Coherent Sets 3.3 Set-Oriented Computation of FTE 4 Finite-Element Framework 4.1 Disentangling Multiple Features with SEBA 5 Application to Rayleigh-Bénard Convection 5.1 2D System 5.2 3D System 6 Conclusion and Outlook References Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces 1 Introduction 2 Preliminaries 3 Decompositions of RKHS Operators 3.1 RKHS 3.2 RKHS Operators 3.3 Basis Orthonormalization and Kernel QR Decomposition 3.4 Eigendecomposition via Auxiliary Problem 3.5 Singular Value Decomposition via Auxiliary Problem 4 Applications 4.1 Low-Rank Approximation, Pseudoinverse and Optimization 4.2 Kernel Covariance and Cross-Covariance Operator 4.3 Conditional Mean Embedding 4.4 Kernel Transfer Operators 5 Conclusion A Appendix A.1 Proof of Block SVD A.2 Derivation of the Empirical CCA Operator References A Weak Characterization of Slow Variables in Stochastic Dynamical Systems 1 Introduction 2 Good Reaction Coordinates 2.1 Timescales of Molecular Dynamics 2.2 Reaction Coordinates 2.3 Preservation of Time Scales 3 Weak Reducibility of Stochastic Systems 3.1 Condition for Good Reaction Coordinates Based on Transfer Operator Eigenfunctions 3.2 Weak Reducibility and Weak Transition Manifolds 4 Numerical Example: A Weakly Reducible System 5 Conclusion and Outlook References Analysis and Simulation of Extremes and Rare Events in Complex Systems 1 Extremes and Rare Event Computation. 2 The Four Methods 2.1 Generalized Extreme Value Distribution (GEV) 2.2 Brute Force Monte Carlo 2.3 Importance Sampling Techniques 3 Numerical Results 3.1 The Generalized Extreme Value (GEV) Model for Numerical Comparison 3.2 Ornstein-Uhlenbeck Process 3.3 Lorenz Model 3.4 Planet Simulator (PlaSim) 4 Discussion References Dynamical Systems Theory and Algorithms for NP-hard Problems 1 Introduction 2 Novel Algorithm Construction: Decentralized Graph Clustering 3 Novel Algorithm Construction: Invariant Manifolds and the Traveling Salesman Problem 4 Novel Algorithm Construction: Network of Duffing Oscillators for the MAX-CUT Problem 5 Analysis of Algorithms: Koopman Operators Based Analysis of Algorithms 6 Analysis of Algorithms: Chaos and Dynamical Systems for Analyzing the Satisfiability (SAT) Problem 7 Conclusion References I Optimal Control Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach 1 Introduction 2 Preliminaries 2.1 Multiobjective Optimal Control 2.2 Model Predictive Control 2.3 Symmetry 3 Symmetries in Multiobjective Optimal Control 4 Symmetry Exploiting Model Predictive Control 5 Results for the Mobile Robot 6 Concluding Remarks References POD-Based Mixed-Integer Optimal Control of Evolution Systems 1 Introduction 2 Problem Formulation 3 Relaxation Method 3.1 Convexification 4 Numerical Solution Method 4.1 Solution of the Relaxed Problem 4.2 Sum-Up Rounding 4.3 Redefine the Time Discretization 4.4 The POD Method 5 Numerical Experiments 5.1 Full Finite Element Method Model 5.2 The Reduced POD Model 6 Conclusions and Outlook References From Bellman to Dijkstra: Set-Oriented Construction of Globally Optimal Controllers 1 Introduction 2 Problem Formulation 3 The Optimality Principle 4 A Discrete Optimality Principle 4.1 The Discrete Value Function 4.2 The Discrete Feedback 5 The Optimality Principle for Perturbed Systems 6 A Discrete Optimality Principle for Perturbed Systems 6.1 Convergence 7 The Discretization as a Perturbation 8 Hybrid, Event and Quantized Systems 9 Lazy Feedbacks References An Optimal Control Derivation of Nonlinear Smoothing Equations 1 Introduction 2 Preliminaries and Background 2.1 The Smoothing Problem 2.2 Solution of the Smoothing Problem 2.3 Path-Wise Representation of the Zakai Equations 2.4 The Finite State-Space Case 3 Optimal Control Problem 3.1 Variational Formulation 3.2 Optimal Control: Euclidean State-Space 3.3 Optimal Control: Finite State-Space 3.4 Derivation of the Smoothing Equations 3.5 Relationship to the Log Transformation 3.6 Linear Gaussian Case 4 Conclusions A Appendix A.1 Derivation of Lagrangian: Euclidean Case A.2 Derivation of Lagrangian: Finite State-Space Case A.3 Proof of Proposition 1 A.4 Proof of Proposition 2 A.5 Proof of Proposition 3 A.6 Proof of Proposition 4 References I Optimization Structural Properties of Pareto Fronts: The Occurrence of Dents in Classical and Parametric Multiobjective Optimization Problems 1 Introduction 2 Theoretical Background 2.1 Multiobjective Optimization 2.2 Parametric Multiobjective Optimization Problems 2.3 Bifurcation Theory 3 Dents in Non-parametric Pareto Fronts 4 Evolution of Dents in Parameter-Dependent Pareto Fronts 4.1 Properties of Dent Border Points 4.2 Numerical Examples 5 Conclusion and Outlook References An Image Set-Oriented Method for the Numerical Treatment of Bi-Level Multi-objective Optimization Problems 1 Introduction 2 Background and Related Work 3 Algorithm and Realization 4 Convergence 5 Numerical Results 5.1 Example 1 5.2 Example 2 6 Conclusions References The Gradient Subspace Approximation and Its Application to Bi-objective Optimization Problems 1 Introduction 2 Gradient Subspace Approximation 2.1 Background and Related Work 2.2 The Basic Idea 2.3 Gradient Subspace Approximation 3 Bi-objective Optimization 3.1 Background and Related Work 3.2 A Descent Direction for Constrained BOPs 3.3 Equality Constrained BOPs 3.4 Inequality Constrained BOPs 3.5 A Gradient Free Approximation of a MODD for CBOPs 4 Application: Use of GFDD Within NSGA-II 5 Conclusions References Author Index