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