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E-raamat: Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering

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An Accelerated Course with Applications in Computational Sciences and Engineering

This book presents the fundamental notions and advanced mathematical tools in the stochastic modeling of uncertainties and their quantification for large-scale computational models in sciences and engineering. In particular, it focuses in parametric uncertainties, and non-parametric uncertainties with applications from the structural dynamics and vibroacoustics of complex mechanical systems, from micromechanics and multiscale mechanics of heterogeneous materials. 

Resulting from a course developed by the author, the book begins with a description of the fundamental mathematical tools of probability and statistics that are directly useful for uncertainty quantification. It proceeds with a well carried out description of some basic and advanced methods for constructing stochastic models of uncertainties, paying particular attention to the problem of calibrating and identifying a stochastic model of uncertainty when experimental data is available. 

This book is intended to be a graduate-level textbook for students as well as professionals interested in the theory, computation, and applications of risk and prediction in science and engineering fields.

Arvustused

The book under review serves as an excellent reference for the uncertainty analysis community. the author has included an extensive bibliography in the end of the book that will be very useful to the interested reader. the book is an excellent reference for advanced users and practitioners of UQ and is strongly recommended. (Tujin Sahai, Mathematical Reviews, September, 2018)

1 Fundamental Notions in Stochastic Modeling of Uncertainties and Their Propagation in Computational Models
1(16)
1.1 Aleatory and Epistemic Uncertainties
1(1)
1.2 Sources of Uncertainties and Variabilities
2(1)
1.3 Experimental Illustration of Variabilities in a Real System
3(1)
1.4 Role Played by the Model-Parameter Uncertainties and the Modeling Errors in a Computational Model
4(1)
1.5 Major Challenges for the Computational Models
5(2)
1.5.1 Which Robustness Must Be Looked for the Computational Models
5(1)
1.5.2 Why the Probability Theory and the Mathematical Statistics Are Efficient
5(1)
1.5.3 What Types of Stochastic Analyses Can Be Done
6(1)
1.5.4 Uncertainty Quantification and Model Validation Must Be Carried Out
6(1)
1.5.5 What Are the Major Challenges
6(1)
1.6 Fundamental Methodologies
7(10)
1.6.1 A Partial Overview of the Methodology and What Should Never Be Done
8(5)
1.6.2 Summarizing the Main Steps of the Methodology for UQ
13(4)
2 Elements of Probability Theory
17(24)
2.1 Principles of Probability Theory and Vector-Valued Random Variables
17(6)
2.1.1 Principles of Probability Theory
17(1)
2.1.2 Conditional Probability and Independent Events
18(1)
2.1.3 Random Variable with Values in Rn and Probability Distribution
18(2)
2.1.4 Mathematical Expectation and Integration of Rn-Valued Random Variables
20(1)
2.1.5 Characteristic Function of an Rn-Valued Random Variable
21(1)
2.1.6 Moments of an Rn-Valued Random Variable
22(1)
2.1.7 Summary: How the Probability Distribution of a Random Vector X Can Be Described
22(1)
2.2 Second-Order Vector-Valued Random Variables
23(2)
2.2.1 Mean Vector and Centered Random Variable
23(1)
2.2.2 Correlation Matrix
23(1)
2.2.3 Covariance Matrix
24(1)
2.2.4 Cross-Correlation Matrix
24(1)
2.2.5 Cross-Covariance Matrix
25(1)
2.2.6 Summary: What Are the Second-Order Quantities That Describe Second-Order Random Vectors
25(1)
2.3 Markov and Tchebychev Inequalities
25(1)
2.3.1 Markov Inequality
25(1)
2.3.2 Tchebychev Inequality
26(1)
2.4 Examples of Probability Distributions
26(2)
2.4.1 Poisson Distribution on R with Parameter λ ε R+
26(1)
2.4.2 Gaussian (or Normal) Distribution on Rn
27(1)
2.5 Linear and Nonlinear Transformations of Random Variables
28(2)
2.5.1 A Method for a Nonlinear Bijective Mapping
28(1)
2.5.2 Method of the Characteristic Function
29(1)
2.5.3 Summary: What Is the Efficiency of These Tools for UQ
30(1)
2.6 Second-Order Calculations
30(1)
2.7 Convergence of Sequences of Random Variables
31(1)
2.7.1 Mean-Square Convergence or Convergence in L2 (θ, Rn)
31(1)
2.7.2 Convergence in Probability or Stochastic Convergence
31(1)
2.7.3 Almost-Sure Convergence
31(1)
2.7.4 Convergence in Probability Distribution
32(1)
2.7.5 Summary: What Are the Relationships Between the Four Types of Convergence
32(1)
2.8 Central Limit Theorem and Computation of Integrals in High Dimensions by the Monte Carlo Method
32(3)
2.8.1 Central Limit Theorem
32(2)
2.8.2 Computation of Integrals in High Dimension by Using the Monte Carlo Method
34(1)
2.9 Notions of Stochastic Processes
35(6)
2.9.1 Definition of a Continuous-Parameter Stochastic Process
35(1)
2.9.2 System of Marginal Distributions and System of Marginal Characteristic Functions
36(1)
2.9.3 Stationary Stochastic Process
36(1)
2.9.4 Fundamental Examples of Stochastic Processes
37(1)
2.9.5 Continuity of Stochastic Processes
37(1)
2.9.6 Second-Order Stochastic Processes with Values in En
38(2)
2.9.7 Summary and What Is the Error That Has to Be Avoided
40(1)
3 Markov Process and Stochastic Differential Equation
41(20)
3.1 Markov Process
41(3)
3.1.1 Notation
41(1)
3.1.2 Markov Property
42(1)
3.1.3 Chapman-Kolmogorov Equation
42(1)
3.1.4 Transition Probability
43(1)
3.1.5 Definition of a Markov Process
43(1)
3.1.6 Fundamental Consequences
44(1)
3.2 Stationary Markov Process, Invariant Measure, and Ergodic Average
44(2)
3.2.1 Stationary
44(1)
3.2.2 Invariant Measure
45(1)
3.2.3 Ergodic Average
45(1)
3.3 Fundamental Examples of Markov Processes
46(3)
3.3.1 Process with Independent Increments
46(1)
3.3.2 Poisson Process with Mean Function λ(t)
46(1)
3.3.3 Vector-Valued Normalized Wiener Process
47(2)
3.4 Ho Stochastic Integral
49(3)
3.4.1 Definition of an Ito Stochastic Integral and Why It Is Not a Classical Integral
49(1)
3.4.2 Definition of Nonanticipative Stochastic Processes with Respect to the Rm-Valued Normalized Wiener Process
50(1)
3.4.3 Definition of the Space M2w(n, m)
50(1)
3.4.4 Approximation of a Process in M2w(n, m) by a Step Process
50(1)
3.4.5 Definition of the Ito Stochastic Integral for a Stochastic Process in M2w(n, m)
51(1)
3.4.6 Properties of the Ito Stochastic Integral of a Stochastic Process in M2w(n, m)
51(1)
3.5 Ito Stochastic Differential Equation and Fokker-Planck Equation
52(2)
3.5.1 Definition of an Ito Stochastic Differential Equation (ISDE)
52(1)
3.5.2 Existence of a Solution of the ISDE as a Diffusion Process
52(1)
3.5.3 Fokker-Planck (FKP) Equation for the Diffusion Process
53(1)
3.5.4 Ito Formula for Stochastic Differentials
54(1)
3.6 Ito Stochastic Differential Equation Admitting an Invariant Measure
54(3)
3.6.1 ISDE with Time-Independent Coefficients
54(1)
3.6.2 Existence and Uniqueness of a Solution
55(1)
3.6.3 Invariant Measure and Steady-State FKP Equation
55(1)
3.6.4 Stationary Solution
56(1)
3.6.5 Asymptotic Stationary Solution
56(1)
3.6.6 Ergodic Average: Formulas for Computing Statistics by MCMC Methods
56(1)
3.7 Markov Chain
57(4)
3.7.1 Connection with the Markov Processes
57(1)
3.7.2 Definition of a Time-Homogeneous Markov Chain
58(1)
3.7.3 Properties of the Homogeneous Transition Kernel and Homogeneous Chapman-Kolmogorov Equation
58(1)
3.7.4 Asymptotic Stationarity for Time-Homogeneous Markov Chains
59(1)
3.7.5 Ergodic Average Using the Asymptotic Stationarity of a Time-Homogeneous Markov Chain: Formulas for Computing Statistics by MCMC Methods
59(2)
4 MCMC Methods for Generating Realizations and for Estimating the Mathematical Expectation of Nonlinear Mappings of Random Vectors
61(16)
4.1 Definition of the Problem to Be Solved
61(2)
4.1.1 Computation of Integrals in Any Dimension, in Particular in High Dimension
61(1)
4.1.2 Is the Usual Deterministic Approach Can Be Used
62(1)
4.1.3 The Computational Statistics Yield Efficient Approaches for the High Dimensions
62(1)
4.1.4 Algorithms in the Class of the MCMC Methods
63(1)
4.2 Metropolis-Hastings Method
63(5)
4.2.1 Metropolis-Hastings Algorithm
64(1)
4.2.2 Example and Analysis of the Metropolis-Hastings Algorithm
65(3)
4.3 Algorithm Based on an ISDE for the High Dimensions
68(9)
4.3.1 Summarizing the Methodology for Constructing the Algorithm
68(1)
4.3.2 First Case: The Support Is the Entire Space Rn
69(4)
4.3.3 Second Case: The Support Is a Known Bounded Subset of Rn (Rejection Method)
73(1)
4.3.4 Third Case: The Support Is a Known Bounded Subset of Rn (Regularization Method)
74(1)
4.3.5 Fourth Case: The Support Is Unknown and a Set of Realizations Is Known. Powerful Algorithm for the High Dimensions
75(2)
5 Fundamental Probabilistic Tools for Stochastic Modeling of Uncertainties
77(56)
5.1 Why a Probability Distribution Cannot Arbitrarily be Chosen for a Stochastic Modeling
77(2)
5.1.1 Illustration of the Problem with the Parametric Stochastic Approach of Uncertainties
77(1)
5.1.2 Impact of an Arbitrary Stochastic Modeling of Uncertain Parameter X
78(1)
5.1.3 What Is Important in UQ
79(1)
5.1.4 What Is the Objective of This
Chapter
79(1)
5.2 Types of Representation for Stochastic Modeling
79(2)
5.2.1 Direct Approach
80(1)
5.2.2 Indirect Approach
80(1)
5.3 Maximum Entropy Principle as a Direct Approach for Constructing a Prior Stochastic Model of a Random Vector
81(13)
5.3.1 Problem Definition
81(3)
5.3.2 Entropy as a Measure of Uncertainties for a Vector-Valued Random Variable
84(1)
5.3.3 Properties of the Shannon Entropy
84(1)
5.3.4 Maximum Entropy Principle
85(1)
5.3.5 Reformulation of the Optimization Problem by Using--Lagrange Multipliers and Construction of a Representation for the Solution
86(1)
5.3.6 Existence and Uniqueness of a Solution to the MaxEnt
87(1)
5.3.7 Analytical Examples of Classical Probability Distributions Deduced from the MaxEnt Principle
87(3)
5.3.8 MaxEnt as a Numerical Tool for Constructing Probability Distribution in Any Dimension
90(4)
5.4 Random Matrix Theory for Uncertainty Quantification in Computational Mechanics
94(23)
5.4.1 A Few Words on Fundamentals of the Random Matrix Theory
95(1)
5.4.2 Notations in Linear Algebra
95(1)
5.4.3 Volume Element and Probability Density Function for Random Matrices
96(1)
5.4.4 The Shannon Entropy as a Measure of Uncertainties for a Symmetric Real Random Matrix
97(1)
5.4.5 The MaxEnt Principle for Symmetric Real Random Matrices
97(1)
5.4.6 A Fundamental Ensemble for the Symmetric Real Random Matrices with an Identity Matrix as a Mean Value
98(2)
5.4.7 Fundamental Ensembles for Positive-Definite Symmetric Real Random Matrices
100(6)
5.4.8 Ensembles of Random Matrices for the Nonparametric Method in Uncertainty Quantification
106(8)
5.4.9 Illustration of the Use of Ensemble SEε+: Transient Wave Propagation in a Fluid-Solid Multilayer
114(1)
5.4.10 MaxEnt as a Numerical Tool for Constructing Ensembles of Random Matrices
115(2)
5.5 Polynomial Chaos Representation as an Indirect Approach for Constructing a Prior Probability Distribution of a Second-Order Random Vector
117(8)
5.5.1 What Is a Polynomial Chaos Expansion of a Second-Order Random Vector
117(1)
5.5.2 Polynomial Chaos (PC) Expansion with Deterministic Coefficients
118(4)
5.5.3 Computational Aspects for Constructing Realizations of Polynomial Chaos with High Degrees, for an Arbitrary Probability Distribution with a Separable or a Nonseparable pdf
122(2)
5.5.4 Polynomial Chaos Expansion with Random Coefficients
124(1)
5.6 Prior Algebraic Representation with a Minimum Number of Hyperparameters
125(1)
5.7 Statistical Reduction (PCA and KL Expansion)
126(7)
5.7.1 Principal Component Analysis (PCA) of a Random Vector X
126(3)
5.7.2 Karhunen-Loeve Expansion of a Random Field U
129(4)
6 Brief Overview of Stochastic Solvers for the Propagation of Uncertainties
133(8)
6.1 Why a Stochastic Solver Is Not a Stochastic Modeling
133(1)
6.2 Brief Overview on the Types of Stochastic Solvers
134(2)
6.3 Spectral Stochastic Approach by Using the PCE
136(2)
6.3.1 Description in a Simple Framework for Easy Understanding
136(1)
6.3.2 Illustration of the Spectral Stochastic Approach with PCE
136(2)
6.4 Monte Carlo Numerical Simulation Method
138(3)
7 Fundamental Tools for Statistical Inverse Problems
141(14)
7.1 Fundamental Methodology
141(3)
7.1.1 Description of the Problem
141(1)
7.1.2 Before Presenting the Statistical Tools, What Are the Fundamental Strategies Used in Uncertainty Quantification?
142(2)
7.2 Multivariate Kernel Density Estimation Method in Nonparametric Statistics
144(3)
7.2.1 Problem Definition
144(1)
7.2.2 Multivariate Kernel Density Estimation Method
145(2)
7.3 Statistical Tools for the Identification of Stochastic Models of Uncertainties
147(6)
7.3.1 Notation and Scheme for the Identification of Model-Parameter Uncertainties Using Experimental Data
147(1)
7.3.2 Least-Square Method for Estimating the Hyperparameter
148(1)
7.3.3 Maximum Likelihood Method for Estimating the Hyperparameters
149(2)
7.3.4 Bayes' Method for Estimating the Posterior Probability Distribution from a Prior Probability Model
151(2)
7.4 Application of the Bayes Method to the Output-Prediction-Error Method
153(2)
8 Uncertainty Quantification in Computational Structural Dynamics and Vibroacoustics
155(62)
8.1 Parametric Probabilistic Approach of Uncertainties in Computational Structural Dynamics
155(7)
8.1.1 Computational Model
156(1)
8.1.2 Reduced-Order Model and Convergence Analysis
157(2)
8.1.3 Parametric Probabilistic Approach of Model-Parameter Uncertainties
159(2)
8.1.4 Estimation of a Posterior Stochastic Model of Uncertainties Using the Output-Prediction-Error Method Based on the Use of the Bayesian Method
161(1)
8.2 Nonparametric Probabilistic Approach of Uncertainties in Computational Structural Dynamics
162(20)
8.2.1 What Is the Nonparametric Probabilistic Approach of Uncertainties
163(2)
8.2.2 Mean Computational Model
165(1)
8.2.3 Reduced-Order Model and Convergence Analysis
165(1)
8.2.4 Methodology for the Nonparametric Probabilistic Approach of Both the Modeling Errors and the Model-Parameter Uncertainties
166(2)
8.2.5 Estimation of the Hyperparameters of the Prior Stochastic Model of Uncertainties with the Nonparametric Probabilistic Approach
168(1)
8.2.6 Simple Example Showing the Capability of the Nonparametric Probabilistic Approach to Take into Account the Model Uncertainties Induced by the Modeling Errors in Structural Dynamics
169(3)
8.2.7 An Experimental Validation: Nonparametric Probabilistic Model of Uncertainties for the Vibration of a Composite Sandwich Panel
172(2)
8.2.8 Case of Nonhomogeneous Uncertainties Taken into Account by Using Substructuring Techniques
174(5)
8.2.9 Stochastic Reduced-Order Computational Model in Linear Structural Dynamics for Structures with Uncertain Boundary Conditions
179(3)
8.3 Nonparametric Probabilistic Approach of Uncertainties in Computational Vibroacoustics
182(7)
8.3.1 Mean Boundary Value Problem for the Vibroacoustic System
183(1)
8.3.2 Reduced-Order Model of the Vibroacoustic System
184(2)
8.3.3 Stochastic Reduced-Order Model of the Structural-Acoustic System Constructed with the Nonparametric Probabilistic Approach of Uncertainties
186(1)
8.3.4 Experimental Validation with a Complex Computational Vibroacoustic Model of an Automobile
187(2)
8.4 Generalized Probabilistic Approach of Uncertainties in a Computational Model
189(11)
8.4.1 Principle for the Construction of the Generalized Probabilistic Approach
190(1)
8.4.2 ROM Associated with the Computational Model and Convergence Analysis
191(1)
8.4.3 Methodology for the Generalized Probabilistic Approach
191(2)
8.4.4 Estimation of the Parameters of the Prior Stochastic Model of Uncertainties
193(1)
8.4.5 Posterior Stochastic Model of Model-Parameter Uncertainties in Presence of the Prior Stochastic Model of the Modeling Errors, Using the Bayesian Method
194(1)
8.4.6 Simple Example Showing the Capability of the Generalized Probabilistic Approach to Take into Account the Model Uncertainties in Structural Dynamics
195(1)
8.4.7 Simple Example Showing the Capability of the Generalized Probabilistic Approach Using Bayes' Method for Updating the Stochastic Model of an Uncertain Model Parameter in Presence of Model Uncertainties
196(4)
8.5 Nonparametric Probabilistic Approach of Uncertainties in Computational Nonlinear Elastodynamics
200(7)
8.5.1 Boundary Value Problem for Elastodynamics with Geometric Nonlinearities
200(1)
8.5.2 Weak Formulation of the Boundary Value Problem
201(1)
8.5.3 Nonlinear Reduced-Order Model
201(2)
8.5.4 Stochastic Reduced-Order Model of the Nonlinear Dynamical System Using the Nonparametric Probabilistic Approach of Uncertainties
203(1)
8.5.5 Simple Example Illustrating the Capability of the Approach Proposed
204(2)
8.5.6 Experimental Validation of the Theory Proposed by Using the Direct Computation
206(1)
8.6 A Nonparametric Probabilistic Approach for Quantifying Uncertainties in Low- and High-Dimensional Nonlinear Models
207(10)
8.6.1 Problem to Be Solved and Approach Proposed
208(1)
8.6.2 Nonparametric Probabilistic Approach for Taking into Account the Modeling Errors in a Nonlinear ROM
209(2)
8.6.3 Construction of the Stochastic Model of the SROB on a Subset of a Compact Stiefel Manifold
211(1)
8.6.4 Construction of a Stochastic Reduced-Order Basis (SROB)
212(1)
8.6.5 Numerical Validation in Nonlinear Structural Dynamics
213(4)
9 Robust Analysis with Respect to the Uncertainties for Analysis, Updating, Optimization, and Design
217(28)
9.1 Role of Statistical and Physical Reduction Techniques
217(2)
9.1.1 Why Taking into Account Uncertainties in the Computational Models Induces an Extra Numerical Cost
218(1)
9.1.2 What Type of Reduction Must Be Performed
218(1)
9.1.3 Reduction Techniques Must Be Taken Into Account
219(1)
9.2 Applications in Robust Analysis
219(13)
9.2.1 Robust Analysis of the Thermomechanics of a Complex Multilayer Composite Plate Submitted to a Thermal Loading
220(1)
9.2.2 Robust Analysis of the Vibrations of an Automobile in the Low-Frequency Range
221(3)
9.2.3 Robust Analysis of the Vibrations of a Spatial Structure
224(4)
9.2.4 Robust Analysis for the Computational Nonlinear Dynamics of a Reactor Coolant System
228(2)
9.2.5 Robust Analysis of the Dynamic Stability of a Pipe Conveying Fluid
230(2)
9.3 Application in Robust Updating
232(3)
9.4 Applications in Robust Optimization and in Robust Design
235(10)
9.4.1 Robust Design in Computational Dynamics of Turbomachines
236(4)
9.4.2 Robust Design in Computational Vibroacoustics
240(5)
10 Random Fields and Uncertainty Quantification in Solid Mechanics of Continuum Media
245(56)
10.1 Random Fields and Their Polynomial Chaos Representations
246(5)
10.1.1 Definition of a Random Field
247(1)
10.1.2 System of Marginal Probability Distributions
247(1)
10.1.3 Summary of the Karhunen-Loeve Expansion of the Random Field
248(1)
10.1.4 Polynomial Chaos Expansion of a Random Field
249(2)
10.2 Setting the Statistical Inverse Problem to Be Solved for the High Stochastic Dimensions
251(3)
10.2.1 Stochastic Elliptic Operator and Boundary Value Problem
252(2)
10.2.2 Stochastic Computational Model and Available Data Set
254(1)
10.2.3 Statistical Inverse Problem to Be Solved
254(1)
10.3 Parametric Model-Based Representation for the Model Parameters and for the Model Observations
254(6)
10.4 Methodology for Solving the Statistical Inverse Problem for the High Stochastic Dimensions
260(6)
10.5 Prior Stochastic Model for an Elastic Homogeneous Medium
266(5)
10.6 Algebraic Prior Stochastic Model for a Heterogeneous Anisotropic Elastic Medium
271(7)
10.7 Statistical Inverse Problem Related to the Elasticity Random Field for Heterogeneous Microstructures
278(18)
10.7.1 Multiscale Statistical Inverse Problem to Be Solved
279(1)
10.7.2 Adaptation to the Multiscale Case of the First Two Steps of the General Methodology
279(1)
10.7.3 Introduction of a Family of Prior Stochastic Models for the Non-Gaussian Tensor-Valued Random Field at the Mesoscale and Its Generator
280(3)
10.7.4 Multiscale Identification of the Prior Stochastic Model Using a Multiscale Experimental Digital Image Correlation, at the Macroscale and at the Mesoscale
283(4)
10.7.5 Example of Application of the Method for Multiscale Experimental Measurements of Cortical Bone in 2D Plane Stresses
287(3)
10.7.6 Example of the Construction of a Bayesian Posterior of the Elasticity Random Field for a Heterogeneous Anisotropic Microstructure
290(6)
10.8 Stochastic Continuum Modeling of Random Interphases from Atomistic Simulations for a Polymer Nanocomposite
296(5)
References 301(18)
Glossary 319(4)
Index 323
Christian Soize is professor at Universite Paris-Est Marne-la-Valee.  His research interests include stochastic modeling of uncertainties in computational mechanics, their propagation and their quantification.
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