Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics (Scientific Computation)

✍ Scribed by Olivier Le Maitre, Omar M Knio

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 542
• Category: Library

✦ Synopsis

This book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors’ interests in incompressible vortex-dominated ?ows and compressible ?ows 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 - ludes 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 ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid 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 suf?cient to address and reconstruct all but those highly elaborate examples.

✦ Table of Contents

Preface
Acknowledgements
Contents
Introduction: Uncertainty Quantification and Propagation
Introduction
Simulation Framework
Uncertainties
Uncertainty Propagation and Quantification
Objectives
Probabilistic Framework
Data Uncertainty
Approach to UQ
Monte Carlo Methods
Spectral Methods
Overview
Basic Formulations
Spectral Expansions
Karhunen-Loève Expansion
Problem Formulation
Properties of KL Expansions
Practical Determination
Rational Spectra
Non-rational Spectra
Numerical Resolution
Gaussian Processes
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
Stochastic Processes
Application to Uncertainty Quantification Problems
Non-intrusive Methods
Non-intrusive Spectral Projection
Orthogonal Basis
Orthogonal Projection
Simulation Approaches for NISP
Monte Carlo Method
Improved Sampling Strategies
Deterministic Integration Approach for NISP
Quadrature Formulas
Gauss Quadratures
Nested Quadratures
Tensor Product Formulas
Sparse Grid Cubatures for NISP
Sparse Grid Construction
Adaptive Sparse Grids
Dimension-Adaptive Sparse Grid
General Adaptive Sparse Grid Method
Least Squares Fit
Least Squares Minimization Problem
Selection of the Minimization Points
Weighted Least Squares Problem
Collocation Methods
Approximation Problem
Polynomial Interpolation
Sparse Collocation Method
Closing Remarks
Galerkin Methods
Stochastic Problem Formulation
Model Equations and Notations
Deterministic Problem
Stochastic Problem
Functional Spaces
Case of Discrete Deterministic Problems
Weak Form
Stochastic Discretization
Stochastic Basis
Data Parametrization and Solution Expansion
Spectral Problem
Stochastic Residual
Galerkin Method
Comments
Linear Problems
General Formulation
Structure of Linear Spectral Problems
Case of Deterministic Operator
General Case
Solution Methods for Linear Spectral Problems
Nonlinearities
Polynomial Nonlinearities
Galerkin Product
Higher-Order Polynomial Nonlinearity
Galerkin Inversion and Division
Square Root
Absolute Values
Min and Max Operators
Integration Approach
Other Types of Nonlinearities
Taylor Expansion
Non-intrusive Projection
Closing Remarks
Detailed Elementary Applications
Heat Equation
Deterministic Problem
Variational Formulation
Finite Element Approximation
Stochastic Problem
Stochastic Variational Formulation
Deterministic Discretization
Stochastic Discretization
Spectral Problem
Example 1: Uniform Conductivity
Trivial Cases
Validation
Example 2: Nonuniform Conductivity
Setup
Mean and Standard Deviation
Analysis of the Solution Modes
Probability Density Functions
Example 3: Uncertain Boundary Conditions
Treatment of Uncertain Boundary Conditions
Test Case
Simulations
Variance Analysis
Functional Decomposition
Application
Stochastic Viscous Burgers Equation
Deterministic Problem
Spatial Discretization
Discrete Deterministic Problem
Stochastic Problem
Stochastic Discretization
Stochastic Galerkin Projection
Numerical Example
Convergence of the Stochastic Approximation
Non-intrusive Spectral Projection
Quadrature Formula
Comparison with the Galerkin Projection
Monte-Carlo Method
Monte-Carlo Sampling
First- and Second-Order Estimates
Determination of Percentiles
Application to Navier-Stokes Equations
SPM for Incompressible Flow
Governing Equations
Intrusive Formulation and Solution Scheme
Numerical Examples
Example 1
Example 2
Example 3
Boussinesq Extension
Deterministic Problem
Stochastic Formulation
Stochastic Expansion and Solution Scheme
Boundary Conditions
Solution Method
Validation
Deterministic Prediction
Convergence Analysis
Analysis of Stochastic Modes
Velocity Modes
Temperature Modes
Comparison with NISP
Gauss-Hermite Quadrature
Latin Hypercube Sampling
Uncertainty Analysis
Low-Mach Number Solver
Zero-Mach-Number Model
Solution Method
Stochastic System
Boundary Conditions
Solution Method
Galerkin and Pseudo-spectral Evaluation of Nonlinear Terms
Pressure Solvability Constraints
Validation
Boussinesq Limit
Non-Boussinesq Regime
Uncertainty Analysis
Heat Transfer Characteristics
Mean Fields
Standard Deviations
Remarks
Stochastic Galerkin Projection for Particle Methods
Particle Method
Boussinesq Equations in Rotation Form
Particle Formulation
Approximation of Diffusion and Buoyancy Terms
Acceleration of Velocity Computation
Remeshing
Stochastic Formulation
Stochastic Basis and PC Expansion
Straightforward Particle Formulation
Particle Discretization of the Stochastic Flow
Validation
Diffusion of a Circular Vortex
Convection of a Passive Scalar
Application to Natural Convection Flow
Remarks
Mulitphysics Example
Physical Models
Momentum
Species Concentrations
Electrostatic Field Strength
Stochastic Formulation
Implementation
Data Structure
Spatial Discretization
Electroneutrality
Electrostatic Field Strength
Time Integration
Estimates of Nonlinear Transformations
Validation
Protein Labeling in a 2D Microchannel
Concluding Remarks
Advanced Topics
Solvers for Stochastic Galerkin Problems
Krylov Methods for Linear Models
Krylov Methods for Large Linear Systems
GMRes Method
Conjugate Gradient Method
Bi-Conjugate Gradient Method
Preconditioning
Jacobi Preconditioner
ILU Preconditioners
Preconditioners for Galerkin Systems
Block-Jacobi Preconditioners
Operator Expectation Preconditioning
Specialized Block Diagonal Preconditioners
Multigrid Solvers for Diffusion Problems
Spectral Representation
Continuous Formulation and Time Discretization
Stochastic Galerkin Projection
Boundary and Initial Conditions
Implicit Time Discretization
Finite Difference Discretization
Spatial Discretization
Treatment of Boundary Conditions
Iterative Method
Outer Iterations
Inner Iterations
Convergence of the Iterative Scheme
Multigrid Acceleration
Definition of Grid Levels
Projection and Prolongation Procedures
Multigrid Cycles
Implementation of the Multigrid Scheme
Results
Multigrid Acceleration
Influence of Stochastic Representation Parameters
Effects of Diffusivity Field Statistics
Selection of Multigrid Parameters
Stochastic Steady Flow Solver
Governing Equations and Integration Schemes
Stochastic Spectral Problem
Resolution of Steady Stochastic Equations
Newton Iterations
Stochastic Increment Problem
Matrix Free Solver
Test Problem
Problem Definition
Unsteady Simulations
Newton Iterations
Influence of the Stochastic Discretization
Computational Time
Unstable Steady Flow
Uncertainty Settings
Flow Equations and Stochastic Decoupling
Results
Closing Remarks
Wavelet and Multiresolution Analysis Schemes
The Wiener-Haar expansion
Preliminaries
Haar Scaling Functions
Haar Wavelets
Wavelet Approximation of a Random Variable
Multidimensional Case
Comparison with Spectral Expansions
Applications of WHa Expansion
Dynamical System
Solution Method
Results
Rayleigh-Bénard Instability
WLe Expansion
WHa Expansion
Continuous Problem
Multiresolution Analysis and Multiwavelet Basis
Change of Variable
Multiresolution Analysis
Vector Spaces
Multiwavelet Basis
Construction of the psij's
MW Expansion
Expansion of the Random Process
The Multidimensional Case
Mean and variance
Application to Lorenz System
h-p Convergence of the MW Expansion
Solution Method
Convergence Results
Comparison with Monte Carlo Sampling
Classical Sampling Strategy
Latin Hypercube Sampling
Closing Remarks
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
Two-Dimensional Problem
Higher Dimensional Problems
A posteriori Error Estimation
Variational Formulation
Deterministic Variational Problem
Stochastic Variational Problem
Probability Space
Stochastic Discretization
Spatial Discretization
Approximation Space Uh
Dual-based a posteriori Error Estimate
A posteriori Error
Posterior Error Estimation
Methodology
Refinement Procedure
Global and Local Error Estimates
Refinement Strategies
Application to Burgers Equation
Uncertainty Settings
Variational Problems
Isotropic hxi Refinement
Isotropic hxi,x Refinement
Anisotropic h/q Refinement
Generalized Spectral Decomposition
Variational Formulation
Stochastic Discretization
General Spectral Decomposition
Definition of an Optimal Pair (U,lambda)
A Progressive Definition of the Decomposition
Algorithms for Building the Decomposition
Extension to Affine Spaces
Application to Burgers Equation
Variational Formulation
Implementation of Algorithms 1 and 2
Spatial Discretization
Stochastic Discretization
Solvers
Results
Application to a Nonlinear Stationary Diffusion Equation
Application of GSD Algorithms
Results
Closing Remarks
Epilogue
Extensions and Generalizations
Open Problems
New Capabilities
Appendix A Essential Elements of Probability Theory and Random Processes
Probability Theory
Measurable Space
Probability Measure
Probability Space
Measurable Functions
Induced Probability
Random Variables
Measurable Transformations
Integration and Expectation Operators
Integrability
Expectation
L2 Space
Random Variables
Distribution Function of a Random Variable
Density Function of a Random Variable
Moments of a Random Variable
Convergence of Random Variables
Random Vectors
Joint Distribution and Density Functions
Independence of Random Variables
Moments of a Random Vector
Gaussian Vector
Stochastic Processes
Motivation and Basic Definitions
Properties of Stochastic Processes
Finite Dimensional Distributions and Densities
Second Moment Properties
Appendix B Orthogonal Polynomials
Classical Families of Continuous Orthogonal Polynomials
Legendre Polynomials
Hermite Polynomials
Laguerre Polynomials
Gauss Quadrature
Gauss-Legendre Quadrature
Gauss-Hermite Quadratures
Gauss-Laguerre Quadrature
Askey Scheme
Jacobi Polynomials
Discrete Polynomials
Appendix C Implementation of Product and Moment Formulas
One-Dimensional Polynomials
Moments of One-Dimensional Polynomials
Multidimensional PC Basis
Multi-Index Construction
Moments of Multidimensional Polynomials
Implementation Details
References
Index