scholarly journals Chaos expansion of uniformly distributed random variables and application to number theory

2021 ◽  
pp. 1-15
Author(s):  
Ciprian Tudor
Keyword(s):  
Number Theory ◽  
Random Variables ◽  
Chaos Expansion

UNCERTAINTY QUANTIFICATION IN STOCHASTIC SYSTEMS USING POLYNOMIAL CHAOS EXPANSION

2010 ◽  
Vol 02 (02) ◽  
pp. 305-353 ◽  
Author(s):  
K. SEPAHVAND ◽  
S. MARBURG ◽  
H.-J. HARDTKE
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Uncertainty Quantification ◽  
Stochastic Systems ◽  
Polynomial Chaos ◽  
Random Variables ◽  
Polynomial Chaos Expansion ◽  
Uncertain Parameters ◽  
Chaos Expansion ◽  
Generalized Polynomial

In recent years, extensive research has been reported about a method which is called the generalized polynomial chaos expansion. In contrast to the sampling methods, e.g., Monte Carlo simulations, polynomial chaos expansion is a nonsampling method which represents the uncertain quantities as an expansion including the decomposition of deterministic coefficients and random orthogonal bases. The generalized polynomial chaos expansion uses more orthogonal polynomials as the expansion bases in various random spaces which are not necessarily Gaussian. A general review of uncertainty quantification methods, the theory, the construction method, and various convergence criteria of the polynomial chaos expansion are presented. We apply it to identify the uncertain parameters with predefined probability density functions. The new concepts of optimal and nonoptimal expansions are defined and it demonstrated how we can develop these expansions for random variables belonging to the various random spaces. The calculation of the polynomial coefficients for uncertain parameters by using various procedures, e.g., Galerkin projection, collocation method, and moment method is presented. A comprehensive error and accuracy analysis of the polynomial chaos method is discussed for various random variables and random processes and results are compared with the exact solution or/and Monte Carlo simulations. The method is employed for the basic stochastic differential equation and, as practical application, to solve the stochastic modal analysis of the microsensor quartz fork. We emphasize the accuracy in results and time efficiency of this nonsampling procedure for uncertainty quantification of stochastic systems in comparison with sampling techniques, e.g., Monte Carlo simulation.

scholarly journals Wiener-Itô Chaos Expansion of Hilbert Space Valued Random Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
M. A. Alshanskiy
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Process ◽  
Calculus Of Variations ◽  
Hilbert Spaces ◽  
Malliavin Calculus ◽  
Random Variables ◽  
Random Variable ◽  
Chaos Expansion

The notion of n-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

Characteristic Functions

2021 ◽  
pp. 213-234
Author(s):  
James Davidson
Keyword(s):  
Number Theory ◽  
Characteristic Function ◽  
Complex Number ◽  
Random Variables ◽  
Infinite Divisibility ◽  
Characteristic Functions ◽  
Conditional Distributions ◽  
Inversion Theorem ◽  
Sums Of Random Variables

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.

scholarly journals A Flexible Polynomial Expansion Method for Response Analysis with Random Parameters

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Rugao Gao ◽  
Keping Zhou ◽  
Yun Lin
Keyword(s):  
Weight Function ◽  
Polynomial Chaos ◽  
Probability Distributions ◽  
Random Variables ◽  
Expansion Method ◽  
Response Analysis ◽  
Chaos Expansion ◽  
Generalized Polynomial Chaos ◽  
Generalized Polynomial ◽  
Polynomial Expansion Method

The generalized Polynomial Chaos Expansion Method (gPCEM), which is a random uncertainty analysis method by employing the orthogonal polynomial bases from the Askey scheme to represent the random space, has been widely used in engineering applications due to its good performance in both computational efficiency and accuracy. But in gPCEM, a nonlinear transformation of random variables should always be used to adapt the generalized Polynomial Chaos theory for the analysis of random problems with complicated probability distributions, which may introduce nonlinearity in the procedure of random uncertainty propagation as well as leading to approximation errors on the probability distribution function (PDF) of random variables. This paper aims to develop a flexible polynomial expansion method for response analysis of the finite element system with bounded random variables following arbitrary probability distributions. Based on the large family of Jacobi polynomials, an Improved Jacobi Chaos Expansion Method (IJCEM) is proposed. In IJCEM, the response of random system is approximated by the Jacobi expansion with the Jacobi polynomial basis whose weight function is the closest to the probability density distribution (PDF) of the random variable. Subsequently, the moments of the response can be efficiently calculated though the Jacobi expansion. As the IJCEM avoids the necessity that the PDF should be represented in terms of the weight function of polynomial basis by using the variant transformation, neither the nonlinearity nor the errors on random models will be introduced in IJCEM. Numerical examples on two random problems show that compared with gPCEM, the IJCEM can achieve better efficiency and accuracy for random problems with complex probability distributions.

THE GROSS DERIVATIVE AND GENERALIZED RANDOM VARIABLES

1999 ◽  
Vol 02 (03) ◽  
pp. 381-396 ◽  
Author(s):  
FRED ESPEN BENTH
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Random Variables ◽  
Random Variable ◽  
Representation Formula ◽  
Gaussian Random Variable ◽  
Wick Product ◽  
Chaos Expansion ◽  
Schwartz Distributions

We extend the Gross derivative to a space of generalized random variables which have a (formal) chaos expansion with kernels from the space of tempered Schwartz distributions. The extended derivative, which we call the Hida derivative, has to be interpreted in the sense of distributions. Many of the properties of the Gross derivative are proved to hold for the extension as well. In addition, we derive a representation formula for the Hida derivative involving the Wick product and a centered Gaussian random variable. We apply our results to calculate the Hida derivative of a class of stochastic differential equations of Wick type.

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