Multilevel Accelerated Quadrature for PDEs with Log-Normally Distributed Diffusion Coefficient

This article is dedicated to multilevel quadrature methods for the rapid solution of stochastic partial differential equations with a log-normally distributed diffusion coefficient. The key idea of such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's moments with focus on the mean and the variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments. This article is dedicated to multilevel quadrature methods forthe rapid solution of stochastic partial differential equationswith a log-normally distributed diffusion coefficient. The key ideaof such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's moments with focus on the mean and the variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments.