Comparison of reduced basis method and stochastic collocation method for parametric and stochastic elliptic problems

Reduced basis method and stochastic collocation method have been developed with many ideas in common to solve parametric and stochastic problems [1,2,3]. In this work, 1. we compare their convergence rate from the nonlinear approximation point of view and computational cost; 2. present similar computational reduction techniques such as sparse/adaptive/hierarchical collocation points and efficient greedy sampling strategies (e.g. “hp" type) in parameter space of reduced basis; 3. and analyze different strategies for dealing with high dimensional problems, e.g., dimension adaptation and analysis of sensitivity or variance (ANOVA).