Uncertainty Quantification in Stochastic Systems Using Polynomial Chaos Expansion

Abstract

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.

Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system

“…First, one can always use the same startpoint x 0 that is provided by the user to initialize the algorithm. Another possibility is to convert the final decomposition x (i) f over S i into equivalent decompositions over each of the sub-cell using formulas (26) and (27). Finally, one can extrapolate a decomposition from neighboring cells: considering a one dimensional problem, once the decomposition over one cell is validated, the startpoint for each of the two neighbors is the decomposition that matches constant values for eigenmodes over each sub-cell.…”

“…Otherwise, a cell S i is defined in the remaining space as well as a startpoint x (i) i , depending on the chosen method. The collection of matrices A (i) n and B (i) n defining the problem over the current cell are evaluated using (26) and (27) and the problem (12) and (13) is then solved, returning a new set of coefficients x (i) f . Accuracy of the decomposition is tested as detailed in Section 2.4.3.…”

“…The second family of methods relies on the so-called Polynomial Chaos Expansions and was first introduced in the field of structural dynamics by Ghanem and Spanos [25]. It basically consists in the decomposition of quantities of interest on an orthogonal basis of polynomials (see for example [26] for a detailed presentation of such methods). The procedure proposed in this paper derives from PCE and its further developments.…”

Survey and Evaluate Uncertainty Quantification Methodologies

“…Such an approach converges for any arbitrary stochastic process with finite second moment. Extensions of this approach (Xiu 2010;Lin and Tartakovsky 2009;Sepahvand et al 2010) are also known as the generalized polynomial chaos framework.…”

Homogenization with uncertain input data

“…In this work it is a Gaussian variable which is best approximated by Hermite polynomials. For more Information see [3,4].…”