M. Croci, M. B. Giles, M. E. Rognes, and P. E. Farrell, Efficient white noise sampling and coupling for multilevel Monte Carlo, MCQMC 2018, Rennes, France. Download.
When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise the efficient sampling of white noise realizations can be challenging. In this talk we present a novel sampling technique that can be used to efficiently compute white noise samples in a finite element and multilevel Monte Carlo (MLMC) setting.
After discretization, the action of white noise on a test function yields a Gaussian vector with the FEM mass matrix as covariance. Sampling such a vector requires an expensive Cholesky factorization and for this reason P0 representations, for which the mass matrix is diagonal, are generally preferred in the literature. This however has other disadvantages. In this talk we introduce an alternative factorization that is naturally parallelizable and has linear cost and memory complexity (in the number of mesh elements).
Moreover, in a MLMC framework the white noise samples must be coupled between subsequent levels so as to respect the telescoping sum. We show how our technique can be used to enforce this coupling even in the case in which the hierarchy is non-nested via a supermesh construction. We conclude the talk with numerical experiments that demonstrate the efficacy of our method. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest. In a MLMC setting, a good coupling is enforced and the telescoping sum is respected.