Certified Reduced Basis Methods for Parametrized Partial Differential Equations (eBook)

Preface 7
References 8
Contents 10
1 Introduction 13
1.1 Historical Background and Perspectives 14
1.2 About this Book 17
1.3 Software Libraries with Support for Reduced Basis Algorithms and Applications 18
References 20
2 Parametrized Differential Equations 26
2.1 Parametrized Variational Problems 26
2.1.1 Parametric Weak Formulation 27
2.1.2 Inner Products, Norms and Well-Posedness of the Parametric Weak Formulation 27
2.2 Discretization Techniques 28
2.3 Toy Problems 30
2.3.1 Illustrative Example 1: Heat Conduction Part 1 31
2.3.2 Illustrative Example 2: Linear Elasticity Part 1 33
References 36
3 Reduced Basis Methods 37
3.1 The Solution Manifold and the Reduced Basis Approximation 38
3.2 Reduced Basis Space Generation 41
3.2.1 Proper Orthogonal Decomposition (POD) 42
3.2.2 Greedy Basis Generation 44
3.3 Ensuring Efficiency Through the Affine Decomposition 47
3.4 Illustrative Examples 49
3.4.1 Illustrative Example 1: Heat Conduction Part 2 49
3.4.2 Illustrative Example 2: Linear Elasticity Part 2 51
3.5 Summary of the Method 52
References 53
4 Certified Error Control 54
4.1 Introduction 54
4.2 Error Control for the Reduced Order Model 55
4.2.1 Discrete Coercivity and Continuity Constants of the Bilinear Form 55
4.2.2 Error Representation 56
4.2.3 Energy and Output Error Bounds 57
4.2.4 mathbbV-Norm Error Bounds 59
4.2.5 Efficient Computation of the a Posteriori Estimators 61
4.2.6 Illustrative Examples 1 and 2: Heat Conduction and Linear Elasticity Part 3 63
4.3 The Stability Constant 64
4.3.1 Min-?-approach 65
4.3.2 Multi-parameter Min-?-approach 66
4.3.3 Illustrative Example 1: Heat Conduction Part 4 66
4.3.4 The Successive Constraint Method (SCM) 68
4.3.5 A Comparitive Discussion 72
References 75
5 The Empirical Interpolation Method 76
5.1 Motivation and Historical Overview 76
5.2 The Empirical Interpolation Method 77
5.3 EIM in the Context of the RBM 81
5.3.1 Non-affine Parametric Coefficients 81
5.3.2 Illustrative Example 1: Heat Conduction Part 5 83
5.3.3 Illustrative Example 1: Heat Conduction Part 6 87
References 93
6 Beyond the Basics 95
6.1 Time-Dependent Problems 95
6.1.1 Discretization 96
6.1.2 POD-greedy Sampling Algorithm 98
6.1.3 A Posteriori Error Bounds for the Parabolic Case 100
6.1.4 Illustrative Example 3: Time-Dependent Heat Conduction 102
6.2 Geometric Parametrization 104
6.2.1 Illustrative Example 4: A 2D Geometric Parametrization for an Electronic Cooling Component 107
6.2.2 Illustrative Example 5: A 3D Geometric Parametrization for a Thermal Fin 111
6.3 Non-compliant Output 113
6.3.1 Illustrative Example 6: A 2D Graetz Problem with Non-compliant Output 115
6.4 Non-coercive Problems 118
6.5 Illustrative Example 7: A 3D Parametrized Graetz Channel 121
References 125
Appendix A Mathematical Preliminaries 127
Index 137