Spectral Methods for Uncertainty Quantification

Transcription

1 Spectral Methods for Uncertainty Quantification

2 Scientific Computation Editorial Board J.-J.Chattot,Davis,CA,USA P. Colella, Berkeley, CA, USA W. E, Princeton, NJ, USA R. Glowinski, Houston, TX, USA Y. Hussaini, Tallahassee, FL, USA P. Joly, Le Chesnay, France J.E. Marsden, Pasadena, CA, USA D.I. Meiron, Pasadena, CA, USA O. Pironneau, Paris, France A. Quarteroni, Lausanne, Switzerland and Politecnico of Milan, Milan, Italy J.Rappaz,Lausanne,Switzerland R.Rosner,Chicago,IL,USA P. Sagaut, Paris, France J.H. Seinfeld, Pasadena, CA, USA A. Szepessy, Stockholm, Sweden M.F. Wheeler, Austin, TX, USA For other titles published in this series, go to

3 O.P. Le Maître O.M. Knio Spectral Methods for Uncertainty Quantification With Applications to Computational Fluid Dynamics

4 Prof. Dr. O.P. Le Maître LIMSI-CNRS Université Paris-Sud XI Orsay cedex France Prof. Dr. O.M. Knio Department of Mechanical Engineering The Johns Hopkins University 3400 North Charles Street 223 Latrobe Hall Baltimore MD USA ISBN e-isbn DOI / Springer Dordrecht Heidelberg London New York Library of Congress Control Number: Springer Science+Business Media B.V No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: estudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To the Ladies, certainly, Marie-Christine & May

6 Preface This book deals with the application of spectral methods to problems of uncertainty propagation and quantification in model-based computations. It specifically focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special attention to models arising in simulations of fluid flows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors interests in incompressible vortex-dominated flows and compressible flows at low Mach numbers. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the rich mathematical foundation associated with probability and measure spaces. Despite the authors fascination with this foundation, the discussion only alludes to those theoretical aspects needed to set the stage for subsequent applications. The book is authored by practitioners, and is primarily intended for researchers or graduate students in computational mathematics, physics, or fluid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral methods is naturally helpful though not essential. Full appreciation of elaborate examples in computational fluid dynamics (CFD) would require familiarity with key, and in some cases delicate, features of the associated numerical methods. Besides these shortcomings, our aim is to treat algorithmic and computational aspects of spectral stochastic methods with details sufficient to address and reconstruct all but those highly elaborate examples. This book is composed of 10 chapters. Chapter 1 discusses the relevance and (ever increasing) role of uncertainty propagation and quantification in model-based predictions. This is followed with brief comments on various approaches used to deal with model data uncertainties, focusing in particular on a probabilistic framework that forms the foundation for subsequent discussion. The remaining nine chapters are divided into two parts. Part I (Chaps. 2 6) focuses on basic formulations and mechanics, providing diverse illustrations based on elementary examples. Chapter 2 discusses fundamentals of spectral expansions of random parameters and processes. Treated in detail are the classical concepts underlying Karhunen-Loève (KL) expansions, homogeneous vii

7 viii Preface chaos, and polynomial chaos (PC). An outline is also provided of the application of these concepts to the representation of uncertain model data, and to the representation of the corresponding uncertain model outputs. Chapter 3 discusses so-called non-intrusive spectral methods of uncertainty propagation. These resemble collocation methods used in the numerical solution of PDEs, and are termed non-intrusive since they generally do not require modification of existing or legacy simulation codes. The discussion covers several approaches falling within this class of spectral methods, including stochastic quadratures, as well as cubature and regression methods. In Chap. 4, we discuss Galerkin (intrusive) approaches to uncertainty propagation, focusing in particular on weak formulations of stochastic problems involving data uncertainty. Stochastic basis function expansions are introduced, and the setup of the resulting stochastic problem is discussed in detail. Special attention is paid to the estimation of nonlinearities, and a brief outline of solution methods is provided. Chapter 5 provides detailed illustration of the implementation of PC methods to simple problems, namely through application to transient diffusion equations in two space dimensions, and to the steady Burgers equation in one space dimension. Chapter 6 then provides several examples illustrating the application of various approaches introduced in Chaps. 3 and 4 to flows governed by the time-dependent Navier-Stokes equations. Examples include incompressible flows, variable-density flows at low-mach-number, and electrokinetically driven flows. Part II (Chaps. 7 10) focuses exclusively on Galerkin methods, and deals with more advanced topics, more recent developments, or more elaborate applications. Chapter 7 discusses the application of specialized solution methods that are of general interest in stochastic flow computations. These include methods for finding stochastic stationary flow solutions, stochastic multigrid solvers, and a brief discussion of pre-conditioning and Krylov methods for the resolution of large systems of linear equations arising in Galerkin projections. Chapter 8 deals with generalized spectral representation concepts, particularly wavelet and multiwavelet representations, as well as multi-resolution analysis of stochastic problems. The applicability of these schemes to problems exhibiting discontinuous dependence on model data is emphasized, and is illustrated using applications to simple dynamical problems and to flow computations. Chapter 9 deals with adaptive representations, stochastic domain decomposition techniques, stochastic error estimation and refinement, and reduced basis approximations. New challenges, open questions, and closing remarks are mentioned in Chap. 10. Orsay, France Baltimore, Maryland O.P. Le Maître O.M. Knio

8 Acknowledgements We wish to thank Prof. Roger Ghanem for his persistence in conveying his passion in the current subject matter. OMK, in particular, discovered that he had already learned quite a bit from Prof. Ghanem even before deliberately charging into the uncertain, by osmosis and random collisions that have spanned multiple years. Much of our initial work took place within the framework of two focused projects that have brought us together a number of colleagues and collaborators, including Prof. Ghanem of the University of Southern California, and Drs. Habib Najm, Bert Debusschere, and Matthew Reagan of Sandia National Laboratories. Interactions and exchanges with these colleagues had made tremendous contributions to our appreciation of the subject matter, as well as developments outlined in this monograph. These exchanges have been made possible through the support of the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under Agreement F , and by the Laboratory Directed Research and Development Program at Sandia National Laboratory, funded by the US Department of Energy. OLM wishes to acknowledge the support of the two institutions that hosted him along the past years when working on stochastic spectral methods: the Laboratoire de Mécanique et d Energétique at the Université d Evry Val d Essonne (LMEE) and the Laboratoire d Informatique pour la Mécanique et les Sciences de l Ingénieur (LIMSI) of the Centre National de la Recherche Scientifique (CNRS). The directors of these two institutions, Olivier Daube (LMEE) and Patrick Le Quéré (LIMSI), deserve special thanks for having provided OLM with the best possible working conditions and the necessary freedom to start new adventurous researches in the uncertainty world. OLM is also grateful to the Johns Hopkins University who supported him on many occasions over the last decade while visiting OMK: a large part of the materials presented in this monograph was initiated, and sometime performed, during stays at the Johns Hopkins University. Different financial supports from the French office for nuclear energy (CEA), funding agencies ANR (JCJC ) and Digiteo, and research network MoMaS were also benefical to OLM. Working on these projects and others, OLM was involved in collaborations with French colleagues; he wishes to particularly acknowledge numerous and fruitful ix

9 x Acknowledgements discussions with Drs. Lionel Mathelin (LIMSI), Jean-Marc Martinez (CEA) and Profs. Anthony Nouy (Université de Nantes), Christian Soize (Université de Paris Est), Alexandre Ern (Ecole des Ponts) and Serge Huberson (Université de Poitiers). OMK wishes to express his gratitude to Prof. Rupert Klein of the Free University of Berlin for helpful contributions to his recent work on uncertainty. Exchanges with Prof. Klein have been supported by the Humboldt Foundation under a Friechrich Wilhelm Bessel research award. He also wishes to acknowledge support from the US Department of Energy under Awards DE-SC and DE-SC These collaborative efforts involving Prof. Roger Ghanem, Prof. Youssef Marzouk of Massachusetts Institute of Technology, Prof. Kevin Long of Texas Tech University, and Dr. Habib Najm, Dr. Bert Debusschere and Dr. Helgi Adalsteinsson of Sandia National Laboratories have inspired some of the material presented in Part II and many of ideas outlined in the Epilogue. He finally wishes to articulate his indebtedness to Prof. Serge Huberson of the Université de Poitiers for connecting him with OLM, and for his unwavering support. We are grateful to Prof. Pierre Sagaut for suggesting the preparation of this monograph. We are also grateful to Dr. Ramon Khanna, Mr. Tobias Schwaibold and the Springer staff for their encouragement and assistance during this project. During the initial conception stages, we had anticipated delivering about a 300-page manuscript in April Consequently, we also wish to express our gratitude to the Springer editors and staff for their patience and persistence, along with our commitment to incorporate experiences and knowledge gained during this project into in future endeavors.

10 Contents 1 Introduction: Uncertainty Quantification and Propagation Introduction SimulationFramework Uncertainties Uncertainty Propagation and Quantification Objectives Probabilistic Framework Data Uncertainty Approach to UQ Monte Carlo Methods Spectral Methods Overview Spectral Expansions Karhunen-Loève Expansion ProblemFormulation Properties of KL Expansions PracticalDetermination GaussianProcesses Polynomial Chaos Expansion Polynomial Chaos System One Dimensional PC Basis Multidimensional PC Basis Truncated PC Expansion Generalized Polynomial Chaos Independent Random Variables Chaos Expansions Dependent Random Variables Spectral Expansions of Stochastic Quantities Random Variable Random Vectors xi

11 xii Contents Stochastic Processes Application to Uncertainty Quantification Problems Non-intrusive Methods Non-intrusive Spectral Projection Orthogonal Basis Orthogonal Projection Simulation Approaches for NISP MonteCarloMethod ImprovedSamplingStrategies Deterministic Integration Approach for NISP Quadrature Formulas Tensor Product Formulas Sparse Grid Cubatures for NISP Sparse Grid Construction Adaptive Sparse Grids Least Squares Fit Least Squares Minimization Problem SelectionoftheMinimizationPoints Weighted Least Squares Problem Collocation Methods ApproximationProblem Polynomial Interpolation Sparse Collocation Method ClosingRemarks Galerkin Methods Stochastic Problem Formulation Model Equations and Notations Functional Spaces CaseofDiscreteDeterministicProblems WeakForm Stochastic Discretization Stochastic Basis Data Parametrization and Solution Expansion Spectral Problem Stochastic Residual GalerkinMethod Comments Linear Problems General Formulation Structure of Linear Spectral Problems Solution Methods for Linear Spectral Problems Nonlinearities Polynomial Nonlinearities GalerkinInversionandDivision... 92

12 Contents xiii Square Root AbsoluteValues MinandMaxOperators Integration Approach Other Types of Nonlinearities ClosingRemarks Detailed Elementary Applications HeatEquation DeterministicProblem Stochastic Problem Example 1: Uniform Conductivity Example 2: Nonuniform Conductivity Example 3: Uncertain Boundary Conditions Variance Analysis Stochastic Viscous Burgers Equation DeterministicProblem Stochastic Problem NumericalExample Non-intrusive Spectral Projection Monte-CarloMethod Application to Navier-Stokes Equations SPM for Incompressible Flow GoverningEquations Intrusive Formulation and Solution Scheme NumericalExamples BoussinesqExtension DeterministicProblem Stochastic Formulation Stochastic Expansion and Solution Scheme Validation Analysis of Stochastic Modes ComparisonwithNISP Uncertainty Analysis Low-MachNumberSolver Zero-Mach-Number Model SolutionMethod Validation Uncertainty Analysis Remarks Stochastic Galerkin Projection for Particle Methods ParticleMethod Stochastic Formulation Validation...245

13 xiv Contents ApplicationtoNaturalConvectionFlow Remarks Mulitphysics Example Physical Models Stochastic Formulation Implementation Validation Protein Labeling in a 2D Microchannel Concluding Remarks Solvers for Stochastic Galerkin Problems Krylov Methods for Linear Models Krylov Methods for Large Linear Systems Preconditioning Preconditioners for Galerkin Systems Multigrid Solvers for Diffusion Problems Spectral Representation Continuous Formulation and Time Discretization Finite Difference Discretization IterativeMethod Convergence of the Iterative Scheme Multigrid Acceleration Results Stochastic Steady Flow Solver Governing Equations and Integration Schemes Stochastic Spectral Problem Resolution of Steady Stochastic Equations TestProblem Unstable Steady Flow ClosingRemarks Wavelet and Multiresolution Analysis Schemes The Wiener-Haar expansion Preliminaries Wavelet Approximation of a Random Variable Multidimensional Case Comparison with Spectral Expansions Applications of WHa Expansion Dynamical System Rayleigh-Bénard Instability Multiresolution Analysis and Multiwavelet Basis Change of Variable Multiresolution Analysis Expansion of the Random Process The Multidimensional Case ApplicationtoLorenzSystem...382

14 Contents xv h p Convergence of the MW Expansion ComparisonwithMonteCarloSampling ClosingRemarks Adaptive Methods Adaptive MW Expansion Algorithm for Iterative Adaptation Application to Rayleigh-Bénard Flow Adaptive Partitioning of Random Parameter Space Partition of the Random Parameter Space Local Expansion Basis Error Indicator and Refinement Strategy Example A posteriori ErrorEstimation Variational Formulation Dual-based a posteriori ErrorEstimate Refinement Procedure ApplicationtoBurgersEquation Generalized Spectral Decomposition Variational Formulation General Spectral Decomposition Extension to Affine Spaces ApplicationtoBurgersEquation Application to a Nonlinear Stationary Diffusion Equation ClosingRemarks Epilogue Extensions and Generalizations OpenProblems New Capabilities Appendix A Essential Elements of Probability Theory and Random Processes A.1 Probability Theory A.1.1 Measurable Space A.1.2 Probability Measure A.1.3 Probability Space A.2 Measurable Functions A.2.1 Induced Probability A.2.2 Random Variables A.2.3 MeasurableTransformations A.3 Integration and Expectation Operators A.3.1 Integrability A.3.2 Expectation A.3.3 L 2 Space A.4 Random Variables...489

15 xvi Contents A.4.1 Distribution Function of a Random Variable A.4.2 Density Function of a Random Variable A.4.3 Moments of a Random Variable A.4.4 Convergence of Random Variables A.5 Random Vectors A.5.1 Joint Distribution and Density Functions A.5.2 Independence of Random Variables A.5.3 Moments of a Random Vector A.5.4 GaussianVector A.6 Stochastic Processes A.6.1 MotivationandBasicDefinitions A.6.2 Properties of Stochastic Processes A.6.3 Second Moment Properties Appendix B Orthogonal Polynomials B.1 Classical Families of Continuous Orthogonal Polynomials B.1.1 Legendre Polynomials B.1.2 Hermite Polynomials B.1.3 Laguerre Polynomials B.2 Gauss Quadrature B.2.1 Gauss-Legendre Quadrature B.2.2 Gauss-Hermite Quadratures B.2.3 Gauss-Laguerre Quadrature B.3 Askey Scheme B.3.1 Jacobi Polynomials B.3.2 Discrete Polynomials Appendix C Implementation of Product and Moment Formulas C.1 One-Dimensional Polynomials C.1.1 Moments of One-Dimensional Polynomials C.2 Multidimensional PC Basis C.2.1 Multi-IndexConstruction C.2.2 Moments of Multidimensional Polynomials C.2.3 ImplementationDetails References Index...531