Comparison between reduced basis and stochastic collocation methods for elliptic problems

41/2012

Comparison between reduced basis and stochastic collocation methods for elliptic problems

Sunday 14th October 2012

Chen, P.; Quarteroni, A.; Rozza, G.

The stochastic collocation method has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method, primarily developed for solving parametric systems, has been recently used to deal with stochastic problems. In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: 1) convergence rate of each method referred to both a priori and a posteriori error estimate; 2) computational costs for oine construction and online evaluation of the two methods. Numerical experiments are performed in univariate problems as well as multivariate problems from low dimensions O(1) to moderate dimensions O(10) and to high dimensions O(100). The main result stemming from our comparison is that the reduced basis method converges no worse in theory and faster in practice than the stochastic collocation method, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs. keywords: stochastic elliptic problem, reduced basis method, stochastic collocation method, sparse grid, greedy algorithm, offline-online computational decomposition, convergence analysis