On multilevel quadrature for elliptic stochastic partial differential equations

In this article, we show that the multilevel Monte Carlo method for elliptic stochastic partial differential equations can be interpreted as a sparse grid approximation. By using this interpretation, the method can straightforwardly be generalized to any given quadrature rule for high dimensional integrals like the quasi Monte Carlo method or the polynomial chaos approach. Besides the multilevel quadrature for approximating the solution's expectation, a simple and efficient modification of the approach is proposed to compute the stochastic solution's variance. Numerical results are provided to demonstrate and quantify the approach. In this article, we show that the multilevel Monte Carlo method for elliptic stochastic partial differential equations can be interpreted as a sparse grid approximation.By using this interpretation, the method can straightforwardly be generalized to any given quadrature rule for high dimensional integrals like the quasi Monte Carlo method or the polynomial chaos approach. Besides the multilevel quadrature for approximatingthe solution's expectation, a simple and efficient modification of the approach is proposed to compute the stochastic solution's variance. Numerical results are provided to demonstrate and quantify the approach.