Stochastic Collocation Methods for Nonlinear Parabolic Equations with Random Coe cients⇤

We evaluate the performance of global stochastic collocation methods for solving nonlinear parabolic and elliptic problems (e.g., transient and steady nonlinear di↵usion) with random coe cients. The robustness of these and other strategies based on a spectral decomposition of stochastic state variables depends on the regularity of the system’s response in outcome space. The latter is a↵ected by statistical properties of the input random fields. These include variances of the input parameters, whose e↵ect on the computational e ciency of this class of uncertainty quantification techniques has remained unexplored. Our analysis shows that if random coe cients have low variances and large correlation lengths, stochastic collocation strategies outperform Monte Carlo simulations (MCS). As variance increases, the regularity of the stochastic response decreases, which requires higherorder quadrature rules to accurately approximate the moments of interest and increases the overall computational cost above that of MCS.