CEE 598UQ Spring 2020: HW Assignment 6 Due on 5/3/2020 11pm Instructor: Hadi Meidani ([email protected]) Note: For all the problems, email your code(s) and solutions to [email protected] and [email protected] by the due date. Problem 1. (75 points) Consider the following target function f(x) = f(x1, x2, · · · , x5) = Π5i=1 sin(ixi), where xi ∈ [0, 1],∀i. We would like to approximate ∫ [0,1]5 f(x)dx using quadrature. Compare the following multidimensional quadrature rules: (a) Tensor product using Clenshaw-Curtis (CC) points at levels 1, 2 and 3. (b) Sparse grid using CC points at levels 1, 2 and 3. (c) Sparse grid using Gauss-Legendre quadrature points with the number of points in each dimension being equal to 1, 2 and 3. compare it with one that does so using sparse grid quadratures at levels 1, 3 and 5. Note: Clenshaw-Curtis points and their weights are given on Slide 15 of Lecture 20. Problem 2. (75 points) Using the Stochastic Galerkin approach, find a PCE approximation for y given by ay = bc + d where a = 3−0.4Ψ1(z1)+0.02Ψ2(z1), b = −5+0.4Ψ1(z2)+0.4Ψ2(z2), c = 3+0.02Ψ1(z3) and d = 11−3Ψ1(z3) are four random parameters, where {zi} are independent uniform random variables in [0, 1], and {Ψi} are Legendre polynomials. 1
