Model Order Reduction and Applications: Cetraro, Italy 2021

✍ Scribed by Michael Hinze, J. Nathan Kutz, Olga Mula, Karsten Urban, Maurizio Falcone, Gianluigi Rozza

• Publisher: Springer-CIME
• Year: 2023
• Tongue: English
• Leaves: 241
• Series: Lecture Notes in Mathematics, 2328
• Category: Library

✦ Synopsis

This book addresses the state of the art of reduced order methods for modelling and computational reduction of complex parametrised systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in various fields.

Consisting of  four contributions presented at the CIME summer school, the book presents several points of view and techniques to solve demanding problems of increasing complexity. The focus is on theoretical investigation and applicative algorithm development for reduction in the complexity – the dimension, the degrees of freedom, the data – arising in these models.

The book is addressed to graduate students, young researchers and people interested in the field. It is a good companion for graduate/doctoral classes.

✦ Table of Contents

Preface
Acknowledgement
Contents
List of Symbols
1 The Reduced Basis Method in Space and Time: Challenges, Limits and Perspectives
1.1 Introduction
1.2 Some Industrial Challenges: Why Do We Need Model Reduction?
1.2.1 Ship Design, Steering and Propulsion
1.2.1.1 Optimal Steering
1.2.1.2 Optimal Design and Control
1.2.2 Financial and Energy Markets, Trading
1.2.3 Medicine: Biomechanics, Fracture Healing, Flow Simulation, Optics
1.2.4 Lessons Learnt
1.3 The Reduced Basis Method (RBM)
1.3.1 Parameterized Linear PDEs
1.3.2 A Detailed Approximation: The "Truth"
1.3.3 Offline Training: The Reduced Problem
1.3.3.1 Offline-Online Decomposition
1.3.3.2 A Posteriori Error Estimation
1.3.3.3 Greedy Selection of the Reduced Basis
1.3.4 What Is the Benchmark?
1.3.5 Weak-Greedy Convergence
1.4 Guiding Examples
1.4.1 The Thermal Block
1.4.2 The Heat Equation
1.5 Beyond Elliptic and Parabolic Problems
1.5.1 Some More Guiding Examples
1.5.2 Ultraweak Formulations
1.5.3 Stable Ultraweak Petrov-Galerkin Discretization
1.5.4 The Ultraweak Reduced Model
1.5.5 Guiding Examples Revisited
1.5.5.1 The Linear Transport Problem
1.5.5.2 The Wave Equation
1.5.5.3 The Schrödinger Equation
1.5.6 Ultraweak "Truth" Discretization
1.5.6.1 Linear Transport
1.5.6.2 The Wave Equation
1.5.6.3 The Schrödinger Equation
1.5.6.4 Common Challenges
1.5.7 The Kolmogorov N-Width Again
1.5.7.1 Linear Transport
1.5.7.2 The Parameterized Wave Equation
1.5.7.3 Schrödinger Equation
1.5.7.4 Common Challenges
1.6 Numerical Aspects
1.6.1 "Truth" Approximation in Space and Time
1.6.1.1 The Parameterized Heat Equation
1.6.1.2 The Parameterized Wave Equation
1.6.2 Reduced Basis Method
1.6.2.1 Thermal Block/The Parameterized Heat Equation
1.6.2.2 The Parameterized Transport Equation
1.6.2.3 The Parameterized Wave Equation
1.7 Conclusions and Outlook
References
2 Inverse Problems: A Deterministic Approach Using Physics-Based Reduced Models
2.1 Introduction
2.2 Forward and Inverse Problems
2.3 Optimality Benchmarks for State Estimation
2.4 Optimal Affine Algorithms
2.4.1 Definition and Preliminary Remarks
2.4.2 Characterization of Affine Algorithms
2.4.3 A Practical Algorithm for Optimal Affine Recovery
2.4.3.1 Discretization and Truncation
2.4.3.2 Optimization Algorithms
2.4.3.3 Final Remark About the Primal-Dual Algorithm
2.5 Sensor Placement
2.5.1 A Collective OMP Algorithm
2.5.1.1 Description of the Algorithm
2.5.1.2 Convergence Analysis
2.5.2 A Worst Case OMP Algorithm
2.5.2.1 Description of the Algorithm
2.5.2.2 Convergence Analysis
2.5.3 Application to Point Evaluation
2.6 Joint Selection of Vn and Wm
2.6.1 Optimality Benchmark
2.6.2 A General Nested Greedy Algorithm
2.6.3 The Generalized Empirical Interpolation Method
2.7 A Piece-Wise Affine Algorithm to Reach the Benchmark Optimality
2.7.1 Optimality Benchmark Under Perturbations
2.7.2 Piecewise Affine Reduced Models
2.7.3 Constructing Admissible Reduced Model Families
2.7.4 Reduced Model Selection and Recovery Bounds
2.8 Bibliographical Remarks/Connections with Other Works
2.8.1 A Bit of History on the Use of Reduced Models to Solve Inverse Problems
2.8.2 For Further Reading
Appendix 1: Practical Computation of An, the Linear PBDW Algorithm
Appendix 2: Practical Computation of β(Vn, Wm)
References
3 Model Order Reduction for Optimal Control Problems
3.1 Outline
3.2 Introduction and Preliminaries
3.2.1 Motivation
3.2.1.1 Examples of PDE Systems
3.3 Lecture 1: The Model Order Reduction (MOR) Techniques
3.3.1 The Beginning of Snapshot POD with Sirovich in 1987
3.3.2 Marry POD with Adaptivity
3.4 Lecture 2: POD in Optimal Control
3.4.1 Cahn-Hilliard Optimization with Spatial Adaptivity
3.4.2 Other Algorithmic Developments
3.4.3 Certification: Error Estimates for Surrogate-Based Optimal Control
3.4.4 Data Quality in Surrogate Based Optimal Control
3.4.5 Test 1: Solution with Steep Gradient Towards Final Time
3.4.6 Test 2: Solution with Steep Gradient in the Middle of the Time Interval
3.4.7 Test 3: Control Constrained Problem
3.4.7.1 Conclusion
3.4.7.2 How Many Snapshots?
3.4.7.3 Where to Take Snapshots?
3.5 Lecture 3: A Fully Certified Reduced Basis Method for PDE Constrained Optimization
3.5.1 General Setting and Model Problem
3.6 The Reduced Problem and the Greedy Procedure
3.6.1 A Relative Error Bound
3.6.2 Convergence of the Method
3.6.3 Numerical Experiments
References
4 Machine Learning Methods for Reduced Order Modeling
4.1 Introduction
4.2 Mathematical Formulation
4.3 Reduced Order Modeling
4.3.1 Dynamic Mode Decomposition
4.3.2 Sparse Identification of Nonlinear Dynamics
4.3.3 Neural Networks
4.4 Discovery of Coordinates and Models for ROMs
4.5 Conclusions
References