## Certified Reduced Basis Methods for Parametrized Partial Differential EquationsThis book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples. |

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### Contents

1 | |

2 Parametrized Differential Equations | 14 |

3 Reduced Basis Methods | 27 |

4 Certified Error Control | 44 |

5 The Empirical Interpolation Method | 67 |

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### Common terms and phrases

A.T. Patera affine assumption affine decomposition applications approach approximation space arg max basis functions bilinear form C.R. Math Cauchy–Schwarz inequality Certified Reduced Basis coefficient coercivity constant Comput convergence defined denote discrete discussed efficient eigenvalue elliptic Empirical Interpolation Method ESAIM evaluate field variable finite element method Galerkin geometric parametrization given greedy algorithm Heat Conduction Hilbert space Illustrative Example inf-sup inner product interpolation points J.S. Hesthaven Linear algebra box lower bound Maday Min-6-approach multi-parameter N-dimensional N.C. Nguyen norm number of basis offline online procedure online stage optimal output functional output of interest parameter space parameter value Parametrized Partial Differential partial differential equations POD-greedy posteriori error estimation proper orthogonal decomposition Quarteroni reduced basis approximation reduced basis method reduced basis space reduced-basis Rozza SIAM solution manifold Springer truth approximation truth model truth problem truth solution vector