A SPARSE GRID STOCHASTIC COLLOCATION METHOD FORPARTIAL DIFFERENTIAL EQUATIONS WITH RANDOMINPUT DATA

摘要

This work proposes and analyzes a Smolyak-type sparse grid stochastic collocationmethod for the approximation of statistical quantities related to the solution of partial differentialequations with random coefficients and forcing terms (input data of the model). To compute solutionstatistics, the sparse grid stochastic collocation method uses approximate solutions, produced hereby finite elements, corresponding to a deterministic set of points in the random input space. Thisnaturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If thenumber of random variables needed to describe the input data is moderately large, full tensor productspaces are computationally expensive to use due to the curse of dimensionality. In this case the sparsegrid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, itis of major practical relevance to understand in which situations the sparse grid stochastic collocationmethod is more efficient than Monte Carlo. This work provides error estimates for the fully discretesolution using L~qnorms and analyzes the computational efficiency of proposed method. Inparticular, it demonstrates algebraic convergence with respect to the total number of collocationpoints and quantifies the effect of the dimension of the problem (number of input random variables)in the final estimates. The derived estimates are then used to compare the method with Monte Carlo,indicating for which problems the former is more efficient than the latter. Computational evidencecomplements the present theory and shows the effectiveness of the sparse grid stochastic collocationmethod compared to full tensor and Monte Carlo approaches.