Model Reduction of Complex Dynamical Systems

✍ Scribed by Peter Benner, Tobias Breiten, Heike Faßbender, Michael Hinze, Tatjana Stykel, Ralf Zimmermann

• Publisher: Birkhäuser
• Year: 2021
• Tongue: English
• Leaves: 416
• Series: International Series of Numerical Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This contributed volume presents some of the latest research related to model order reduction of complex dynamical systems with a focus on time-dependent problems.  Chapters are written by leading researchers and users of model order reduction techniques and are based on presentations given at the 2019 edition of the workshop series Model Reduction of Complex Dynamical Systems – MODRED, held at the University of Graz in Austria.  The topics considered can be divided into five categories:

• system-theoretic methods, such as balanced truncation, Hankel norm approximation, and reduced-basis methods; 
• data-driven methods, including Loewner matrix and pencil-based approaches, dynamic mode decomposition, and kernel-based methods;
• surrogate modeling for design and optimization, with special emphasis on control and data assimilation;
• model reduction methods in applications, such as control and network systems, computational electromagnetics, structural mechanics, and fluid dynamics; and
• model order reduction software packages and benchmarks.

This volume will be an ideal resource for graduate students and researchers in all areas of model reduction, as well as those working in applied mathematics and theoretical informatics.

✦ Table of Contents

Preface
Contents
-20pt Methods and Techniques of Model Order Reduction
On Bilinear Time-Domain Identification and Reduction in the Loewner Framework
1 Introduction
1.1 Outline of the Paper
2 System Theory Preliminaries
2.1 Linear Systems
2.2 Nonlinear Systems
3 The Loewner Framework
3.1 The Loewner Matrix
3.2 Construction of Interpolants
4 The Special Case of Bilinear Systems
4.1 The Growing Exponential Approach
4.2 The Kernel Separation Method
4.3 Identification of the Matrix N
4.4 A Separation Strategy for the second Kernel
4.5 The Loewner-Volterra Algorithm for Time-Domain Bilinear Identification and Reduction
4.6 Computational Effort of the Proposed Method
5 Numerical Examples
6 Conclusion
References
Balanced Truncation for Parametric Linear Systems Using Interpolation of Gramians: A Comparison of Algebraic and Geometric Approaches
1 Introduction
2 Balanced Truncation for Parametric Linear Systems and Standard Interpolation
2.1 Balanced Truncation
2.2 Interpolation of Gramians for Parametric Model Order Reduction
2.3 Offline-Online Decomposition
3 Interpolation on the Manifold mathcalS+(k,n)
3.1 A Quotient Geometry of mathcalS+(k,n)
3.2 Curve and Surface Interpolation on Manifolds
4 Numerical Examples
4.1 A model for heat conduction in solid material
4.2 An Anemometer Model
5 Conclusion
References
Toward Fitting Structured Nonlinear Systems by Means of Dynamic Mode Decomposition
1 Introduction
2 Dynamic Mode Decomposition
2.1 Dynamic Mode Decomposition with Control (DMDc)
2.2 Input-Output Dynamic Mode Decomposition
3 The Proposed Extensions
3.1 Bilinear Systems
3.2 Quadratic-Bilinear Systems
4 Numerical Experiments
4.1 The Viscous Burgers' Equation
4.2 Coupled van der Pol Oscillators
5 Conclusion
6 Appendix
6.1 Computation of the Reduced-Order Matrices for the Quadratic-Bilinear Case
References
Clustering-Based Model Order Reduction for Nonlinear Network Systems
1 Introduction
2 Preliminaries
2.1 Graph Theory
2.2 Graph Partitions
2.3 Linear Multi-agent Systems
2.4 Clustering-Based Model Order Reduction
2.5 Model Reduction for Non-asymptotically Stable Systems
3 Clustering for Linear Multi-agent Systems
4 Clustering for Nonlinear Multi-agent Systems
4.1 Nonlinear Multi-agent Systems
4.2 Clustering by Projection
5 Numerical Examples
5.1 Small Network Example
5.2 van der Pol Oscillators
6 Conclusions
References
Adaptive Interpolatory MOR by Learning the Error Estimator in the Parameter Domain
1 Introduction
2 Interpolatory MOR
3 Greedy Method for Choosing Interpolation Points
4 Adaptive Training by Learning the Error Estimator in the Parameter Domain
4.1 Radial Basis Functions
4.2 Learning the Error Estimator over the Parameter Domain
4.3 Adaptive Choice of Interpolation Points with Surrogate Error Estimator
5 Numerical Examples
5.1 RLC Interconnect Circuit
5.2 Thermal Model
5.3 Dual-Mode Circular Waveguide Filter
6 Conclusion
References
A Link Between Gramian-Based Model Order Reduction and Moment Matching
1 Introduction
1.1 Balancing of LTI Systems
1.2 Rational Interpolation
1.3 Organization of Paper
2 Gramian Quadrature Algorithm
2.1 Approximating the Gramian via Runge-Kutta Methods
2.2 Computation of mathcalHj in Algorithm 1
2.3 The Space Spanned by the Approximate Cholesky Factor Z
3 Approximate Balancing Transformation
4 Connection to Other Methods
4.1 Balanced POD
4.2 The ADI Iteration
5 Examples
6 Conclusion
References
Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms
1 Introduction
2 Empirical Gramians for Linear Systems
2.1 Empirical Controllability Gramian
2.2 Empirical Observability Gramian
2.3 Empirical Cross Gramian
2.4 Parametric Empirical Gramians
3 Empirical-Gramian-Based Model Reduction
3.1 Empirical Poor Man
3.2 Empirical Approximate Balancing
3.3 Empirical Dominant Subspaces
3.4 Empirical Balanced Truncation
3.5 Empirical Balanced Gains
4 Approximate Norms
4.1 Signal Norms
4.2 System Norms
4.3 Modified Induced Norms
4.4 Parametric Norms
5 MORscore
6 Benchmark Comparison
6.1 emgr – EMpirical GRamian Framework
6.2 Thermal Block Benchmark
6.3 Numerical Results
7 Conclusion
References
Optimization-Based Parametric Model Order Reduction for the Application to the Frequency-Domain Analysis of Complex Systems
1 Introduction
2 Basics of the Global Basis and Krylov Subspace Method
2.1 Krylov Subspaces
2.2 Affine Matrix Decomposition
3 OGPA: Optimization-based Greedy Parameter Sampling
3.1 Grid-Free Sampling
3.2 A-Posteriori Model Quality Evaluation
4 Numerical Examples
4.1 Cantilever Solid Beam
4.2 Rear Axle Carrier
5 Summary
References
On Extended Model Order Reduction for Linear Time Delay Systems
1 Introduction
2 Problem Statement
3 Observability and Controllability Inequalities
4 Model order reduction by truncation
5 Feasibility of the Matrix Inequalities
6 Example: Delay Neural Fields
7 Application to Parameterized Model Reduction
7.1 Example
8 Conclusions
References
-20pt Applications of Model Order Reduction
A Practical Method for the Reductionpg of Linear Thermo-Mechanical Dynamic Equations
1 Introduction
2 The Thermo-Mechanical Model
2.1 Structural Mechanics
2.2 Heat Transfer
2.3 Coupling of Equations
3 Derivation of the Reduction Algorithm
3.1 Model Order Reduction
3.2 Extraction of the Coupling Matrix
3.3 Algorithm
4 Implementation and Results
4.1 Modeling
4.2 Results
5 Conclusions
References
Reduced-Order Methods in Medical Imaging
1 Introduction
2 Methods
2.1 Medical Tomography
2.2 Proper Orthogonal Decomposition
2.3 Downsampled POD Method
2.4 Hybrid-POD Method
2.5 Implementation Details
3 Results
3.1 Test Tube with Fish Eggs
3.2 Down-Sampling Results
3.3 Hybrid-POD Method
4 Discussion
5 Conclusion
References
Efficient Krylov Subspace Techniques for Model Order Reduction of Automotive Structures in Vibroacoustic Applications
1 Introduction
2 Krylov-Based Model Order Reduction
2.1 Problem Definition
2.2 Reduction Framework
3 Numerical Implementation
4 Results
4.1 Generic System
4.2 Coupled System
5 Conclusions and Remarks
References
Model-Based Adaptive MOR Framework for Unsteady Flows Around Lifting Bodies
1 Introduction
2 Linear Reduced Basis Methods
3 Adaptive Approach
3.1 Physical Problem: Navier-Stokes Equations
3.2 Error Estimation
3.3 Sensitivity
4 Demonstration on Lifting Surfaces
4.1 Stalled NACA0012 Airfoil
4.2 High-Lift 30P30N Airfoil
5 Final Remarks and Outlook
References
Reduced Basis Methods for Quasilinear Elliptic PDEs with Applications to Permanent Magnet Synchronous Motors
1 Introduction
2 The Quasilinear Parametric Elliptic PDE
2.1 Abstract Formulation
3 Reduced Basis Approximation
3.1 An EIM-RB Method
3.2 Error Estimation
3.3 Computational Procedure
3.4 Numerical Results
4 Conclusion
References
Structure-Preserving Reduced- Order Modeling of Non-Traditional Shallow Water Equation
1 Introduction
2 Shallow Water Equation
3 Full- Order Model
4 Reduced- Order Model
5 Numerical Results
5.1 Single-Layer Geostrophic Adjustment
5.2 Single-Layer Shear Instability
6 Conclusions
References
*-20pt Benchmarks and Software of Model Order Reduction
A Non-stationary Thermal-Block Benchmark Model for Parametric Model Order Reduction
1 Introduction
2 Problem Description
3 Problem Variants
3.1 Four-Parameter LTI System
3.2 Single-Parameter LTI System
3.3 Non-parametric LTI System
4 Conclusion
References
Parametric Model Order Reduction Using pyMOR
1 Introduction
2 Software Design
3 Overview of Model Order Reduction Methods
3.1 Reduced Basis Method
3.2 System-Theoretic Methods
4 Numerical Results
4.1 Non-parametric Version
4.2 Single-Parameter Version
4.3 Four-Parameter Version
5 Conclusions
References
Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1—Philosophy, Features, and Application for (Parametric) Model Order Reduction
1 Introduction
1.1 A Brief History of M-M.E.S.S.
1.2 Structure of This Chapter
2 M-M.E.S.S.—Philosophy and Features
2.1 Available Solver Functions and Underlying Methods
3 Model Order Reduction in M-M.E.S.S.
3.1 IRKA and Classic Balanced Truncation
3.2 Further Variants of Balanced Truncation
4 Parametric Model Order Reduction Using M-M.E.S.S.
4.1 Piecewise MOR
4.2 Interpolation of Transfer Functions
5 Numerical Experiments
References
MORLAB—The Model Order Reduction LABoratory
1 Introduction
2 Code Design Principles
2.1 Toolbox Structure
2.2 Function Interfaces
2.3 Documentation
3 Additive System Decomposition Approach
3.1 Standard System Case
3.2 Descriptor System Case
4 Model Reduction with the MORLAB Toolbox
4.1 First-Order Methods
4.2 Second-Order Methods
5 Numerical Examples
5.1 Butterfly Gyroscope
5.2 Parametric Thermal Block Model
6 Conclusions
References