UNITEXT La Matematica per il 3+2

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1 UNITEXT La Matematica per il 3+2 Volume 92 Editor-in-chief A. Quarteroni Series editors L. Ambrosio P. Biscari C. Ciliberto M. Ledoux W.J. Runggaldier

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3 Alfio Quarteroni Andrea Manzoni Federico Negri Reduced Basis Methods for Partial Differential Equations An Introduction

4 Alfio Quarteroni Ecole Polytechnique Fédérale de Lausanne Lausanne, Switzerland Andrea Manzoni Ecole Polytechnique Fédérale de Lausanne Lausanne, Switzerland Federico Negri Ecole Polytechnique Fédérale de Lausanne Lausanne, Switzerland ISSN ISSN (electronic) UNITEXT La Matematica per il 3+2 ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer Cham Heidelberg New York Dordrecht London Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Cover illustration: The cover figure displays a set of reduced basis functions for an advectiondiffusion-reaction boundary value problem in a rectangular computational domain. Cover Design: Simona Colombo, Giochi di Grafica, Milano, Italy Springer International Publishing AG Switzerland is part of Springer Science+Business Media (

5 Preface Reduced basis (RB) methods represent a very efficient approach for the numerical approximation of problems involving the repeated solution of differential equations arising from engineering and applied sciences. Noteworthy examples include partial differential equations (PDEs) depending on several parameters, PDE-constrained optimization, and optimal control and inverse problems. In all these cases, reducing the severe computational complexity is crucial. With this in mind, over the past four decades, reduced-order models (ROMs) have been developed aiming at replacing the original large-dimension numerical problem (typically called high-fidelity approximation) by a reduced problem of substantially smaller dimension. Strategies to generate the reduced problem from the high-fidelity one can be manifold, depending on the context. The strategy adopted in RB methods consists in the projection of the high-fidelity problem upon a subspace made of specially selected basis functions, representing a set of high-fidelity solutions corresponding to suitably chosen parameters. Pioneering works in this area date back to the late 1970s (e.g., B.O. Almroth et al. [5, 6], D. Nagy [193], A.K. Noor and J.M. Peters [201, 202, 203, 204] and address linear and nonlinear structural analysis problems. The first theoretical analysis of RB methods in connection with the use of the continuation method for parametrized equations was presented by J.P. Fink and W.C. Rheinboldt [109, 110] in the mid 1980s. Extensions to problems in fluid dynamics are primarily due to the contributions of Peterson [210] and Gunzburger [124] in the late 1980s. The method was set on a more general and sound mathematical ground in the early 2000s thanks to the seminal work of A.T. Patera, Y. Maday and coauthors [214, 255]. Their work has led to a decisive improvement in the computational aspects of RB methods owing to an efficient criterion for the selection of the basis functions, a systematic splitting of the computational procedure into an offline (parameterindependent) and an online (parameter-dependent) phase, and the use of a posteriori error estimates that guarantee certified numerical solutions for the reduced problem. These have become the essential constituents of the RB methods now most widely used. Often, they are also embedded into more general reduced-order models. v

6 vi Preface RB methods have witnessed a spectacular effervescence in the past decade. Additional achievements during that time relate to the treatment of nonlinear and/or parametrically nonaffine problems by the so-called empirical interpolation method and its several extensions. This has substantially improved RB methods, making possible their application to a broad variety of complex problems such as time-dependent problems, optimal control and design problems, and real-time computing. This is the first textbook to provide a basic mathematical introduction to RB methods. We present a general formulation of RB methods, analyze their fundamental theoretical properties, and discuss their algorithmic and implementation aspects, highlighting their built-in algebraic and geometric structures. More specifically, we carry out both a priori and a posteriori error analysis, formulate strategies for the construction of accurate reduced basis spaces, and analyze offline-online decomposition strategies to ensure the reduction of computational complexity. The entire mathematical discussion is made more stimulating by the use of several representative examples of applicative interest, in the context of both linear and nonlinear PDEs. The authors are grateful to Charbel Farhat, Yvon Maday, and Anthony Patera for being source of inspiration and for many fruitful discussions on reduced-order models. We also acknowledge David Amsallem, Luca Dedè, Simone Deparis and Toni Lassila for the great amount of time that they have spent with the authors talking about different subjects covered in this book and, last but not least, Gianluigi Rozza for having introduced the last two authors to the subject. In addition, special thanks are due to Francesca Bonadei and Francesca Ferrari of Springer Italia for their invaluable help in the preparation of the manuscript. Lausanne, Switzerland May 2015 Alfio Quarteroni Andrea Manzoni Federico Negri

7 Contents 1 Introduction Numerical Simulation and Beyond The Need for Reduction Reduced Basis Methods for PDEs at a Glance Accuracy and Computational Efficiency of RB Methods Content of the Book Representative Problems: Analysis and (High-Fidelity) Approximation Four Problems Advection-Diffusion-Reaction Equation Linear Elasticity Equations Stokes Equations Navier-Stokes Equations Formulation and Analysis of Variational Problems Strongly Coercive Problems Weakly Coercive (or Inf-Sup Stable) Problems Saddle-Point Problems Analysis of Three (out of Four) Problems Advection-Diffusion-Reaction Equation Linear Elasticity Equations Stokes Equations On the Numerical Approximation of Variational Problems Strongly Coercive Problems Algebraic Form of (P1 h ) Computation of the Discrete Coercivity Constant Weakly Coercive Problems Algebraic Form of (P2 h ) Computation of the Discrete Inf-Sup Constant Saddle-Point Problems Algebraic Form of (P3 h ) vii

8 viii Contents 2.5 Finite Element Spaces Exercises RB Methods: Basic Principles, Basic Properties Parametrized PDEs: Formulation and Assumptions High-Fidelity Discretization Techniques Reduced Basis Methods Galerkin RB Method Least-Squares RB Method Algebraic Form of Galerkin and Least-Squares RB Problems Galerkin RB Case Least-Squares RB Case Reduction of Computational Complexity: Offline/Online Decomposition A Posteriori Error Estimation A Relationship between Error and Residual Error Bound Practical (and Efficient) Computation of Error Bounds Computing the Norm of the Residual Computing the Stability Factor by the Successive Constraint Method Computing the Stability Factor by Interpolatory Radial Basis Functions An Illustrative Numerical Example Exercises On the Algebraic and Geometric Structure of RB Methods Algebraic Construction and Interpretation Algebraic Interpretation of the G-RB Problem Algebraic properties of the G-RB Problem Least-Squares and Petrov-Galerkin RB Problems Geometric Interpretation Projection and Bases Matrix Characterization of Projection Operators Orthogonal and Oblique Projection Operators The Galerkin Case The Petrov-Galerkin Case Exercises The Theoretical Rationale Behind The Solution Manifold When is a Problem Reducible? Smoothness of the Solution Set Continuity and Compactness... 90

9 Contents ix Differentiability of the Solution Map and Sensitivity Equations Dimensionality of the Solution Set Dimensionality and Analiticity Analiticity of the Solution Map: an Instance Kolmogorov n-width and Analiticity Kolmogorov n-width and Parametric Complexity Lagrange, Taylor and Hermite RB Spaces Exercises Construction of RB Spaces by SVD-POD Basic Notions on Singular Value Decomposition SVD and Low-Rank Approximations Interlude Image Compression Principal Component Analysis Proper Orthogonal Decomposition POD for Parametrized Problems POD with Energy Inner Product P-continuous Analogue of POD Back to the Discrete Setting Our Illustrative Numerical Example Revisited More on Reducibility Exercises Construction of RB Spaces by the Greedy Algorithm Greedy Algorithm: an Algebraic Perspective The Idea Behind Greedy Algorithms The Weak Greedy Algorithm Our Illustrative Numerical Example Revisited An Abstract Formulation of the Greedy Algorithm A Priori Error Analysis Numerical Assessment of a Priori Convergence Results Exercises RB Methods in Action: Setting up the Problem Going from the Original to the Reference Domain Change of Variables Formulas Extension to the Vector Case Advection-Diffusion-Reaction, Case I: Heat Transfer Reference Configuration and Affine Transformations Weak Formulation on the Reference Domain Dealing with Nonhomogeneous Boundary Conditions Advection-Diffusion-Reaction, Case II: Mass Transfer with Parametrized Source

10 x Contents 8.5 Advection-Diffusion-Reaction, Case III: Mass Transfer in a Parametrized Domain More on the Transformation of Vector Fields Linear Elasticity: An Elastic Beam Fluid Flows, Case I: Backward-Facing Step Channel Reference Domain and Affine Transformation Weak Formulation on the Reference Domain Fluid Flows, Case II: Sudden Expansion Channel Problems Features at a Glance Exercises RB Methods in Action: Computing the Solution Heat Transfer: Results An Elastic Beam: Results Backward-Facing Step Channel, Stokes Flow: Results RB Approximation of Parametrized Stokes Equations A Posteriori Error Estimation Numerical Results: Backward-Facing Step Channel Extension to Nonaffine Problems Empirical Interpolation Method Polynomial Interpolation vs. Empirical Interpolation Empirical Interpolation EIM Algorithm Error Analysis for the Empirical Interpolation Practical Implementation Discrete Empirical Interpolation EIM-G-RB Approximation of Nonaffine Problems Mass Transfer with Parametrized Source: Results Comparison of EIM and DEIM (D)EIM-G-RB Approximation Mass Transfer in a Parametrized Domain: Results Exercises Extension to Nonlinear Problems Parametrized Nonlinear PDEs Navier-Stokes Equations A Semilinear Elliptic PDE High-Fidelity Approximation Newton s Method Algebraic Formulation Reduced Basis Approximation Algebraic Formulation Galerkin Projection LS-RB: Newton then Least-Squares

11 Contents xi LS-RB Revisited: Least-Squares then Gauss-Newton Reduction of Computational Complexity A Posteriori Error Estimation for Nonlinear Problems Application to the Steady Navier-Stokes Equations RB Approximation of the Navier-Stokes Equations A posteriori Error Estimation Numerical Results: Backward-Facing Step Channel Numerical Results: Sudden Expansion Channel Numerical results: a Simplified Bypass Graft Exercises Reduction and Control Parameter-Dependent PDE-Constrained Optimization Parametric Optimization Problems Reduction Strategies Application to an Optimal Flow Control Problem Parametrized Optimal Control Problems Reduction Strategies A Posteriori Error Estimation Application to an Optimal Heat Transfer Problem Appendix A Basic Theoretical Tools A.1 Linear Maps, Functionals and Bilinear Forms A.2 Hilbert Spaces A.3 Adjoint Operators A.4 Compact Operators A.5 Differentiation in Linear Spaces A.6 Sobolev Spaces A.6.1 Square-Integrable Functions A.6.2 The Spaces H 1 (Ω) and H0 1 (Ω) A.7 Bochner Spaces A.8 Polynomial Interpolation and Orthogonal Polynomials References Index...293