Model Order Reduction and Applications: Cetraro, Italy 2021

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