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

Various numerical methods have been developed in order to solve complexsystems 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 arecompared. The tests discussed involve a given target function in polynomial form,high-dimensional functions and a random ODE model. Our numerical resultsshow that the low discrepancy point sets perform as well or better than randomsampling for stochastic collocation via $ℓ_1$-minimisation.