Evaluation of Designed Distributions for Stochastic Collocation Methods

Collocation methods refer to numerical solution through a finite-dimensional set of candidate solutions at specific locations within the domain. Quadrature is a collocation method for numerical solution of a defined integral, such as calculating the area of a surface, the volume of a solid, or the expectation of a function of a distributed variable. By evaluating a function at specified locations and summing the set of weighted solutions, a stochastic quantity of interest, such as the n^th moment of a function, can be determined to a predictable level of accuracy. Equally weighted locations may also be determined for a set of coefficients based on Walsh functions or Walsh-Hadamard matrix. Numerical solution of differential equations may also be categorized as collocation methods, for which explicit and implicit iterative methods, such as the Runge-Kutta family of methods, utilize weighted averages of specified increments of solution. For transient responses resulting from numerical solution of differential equations, uncertainty in the coefficients and boundary conditions that are represented as distributed variables may be propagated and stochastic characteristics of the resulting distribution of the states determined. This effort shall characterize the accuracy of truncated numerical solutions of known functions and compare to the accuracy provided by populations designed through Walsh functions, as well as Gauss quadrature. Several foundational examples shall demonstrate the efficacy of uniform, inverse uniform, normal, and lognormal distributions represented by designed distributions.