Stochastic Collocation via $l_1$-Minimisation on Low Discrepancy Point Sets with Application to Uncertainty Quantification

Abstract

Various numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method using $ℓ_1$- minimisation on low discrepancy point sets is investigated here. Halton and Sobol’ sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation via $ℓ_1$-minimisation.