Model, identification & analysis of complex stochastic systems: Applications in stochastic partial differential equations and multiscale mechanics

xi 1 Chapter 1: Introduction 1 1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Chapter 2: Asymptotic Distribution for Polynomial Chaos Representation from Data 5 2.1 Motivation and Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Representation and Characterization of the Random Process from Measurements . . . . 10 2.2.1 Karhunen-Loeve Decomposition: Reduced Order Representation of the Random Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Polynomial Chaos Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 Polynomial Chaos Representation from Data . . . . . . . . . . . . . . . . . . . 15 2.2.4 Asymptotic Probability Distribution Function of hxq(λn) . . . . . . . . . . . . 19 2.3 Estimations of the mjpdf of the nKL Vector, the Fisher Information Matrix and the Gradient Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Multivariate Joint Probability Density Function of the nKL Vector . . . . . . . . 21 2.3.2 Relationship between MaxEnt and Maximum Likelihood Probability Models . . 24 2.3.3 MEDE Technique and Some Remarks on the Form of pZ(Z) . . . . . . . . . . 25 2.3.4 Computation of the Fisher Information Matrix, Fn(λ) . . . . . . . . . . . . . . 27 2.3.5 Computation of the Gradient Matrix, h ′ xq (λ) . . . . . . . . . . . . . . . . . . . 28 2.4 Numerical Illustration and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.1 Measurement of the Stochastic Process . . . . . . . . . . . . . . . . . . . . . . 30 2.4.2 Construction and MaxEnt Density Estimation of nKL Vector . . . . . . . . . . . 33 2.4.3 Simulation of the nKL vector and Estimation of the Fisher Information Matrix . 35 2.4.4 Estimation of PC coefficients of Z and Y . . . . . . . . . . . . . . . . . . . . . 37 2.4.5 Determination of Asymptotic Probability Distribution Function of hxq(λn) . . . 39 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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