Extending adaptive sparse grids for stochastic collocation to hybrid architectures

We are developing an adaptive sparse grid library tailored for emerg-ing architectures that will allow the solution of very large stochastic problems. In here, we give a brief overview of the problem at hand and present first results for a GPU-based small cluster. Stochastic collocation In a stochastic simulation, one is interested in the relationship between the input random variables Z and the outputs g of the simulation state u = u(x, t;Z) that depends on Z and the deterministic variables x and (possibly) time t. The mapping from Z to g(u) can be given by Z 7 → G(Z;x, t) , g(u(x, t;Z)). (1) The key idea behind stochastic collocation (SC) is to select a set of nodes in the random space and then conduct repetitive deterministic simulation at each node. Denoting with {G(Zi,j)mij=1} the deterministic approximation at discrete points, we can approximate the one-dimensional component of the solution u over the range of Zi by Gi[G] = mi∑ j=1 G(Zi,j) · ψi,j(Zi) (2) where ψi,j is the interpolating basis. Using tensor products, the entire space can be constructed as Gl = Gi1 ⊗ · · · ⊗ Gid. (3) Yet, Eq. (3) suffers from the curse of dimensionality. It can be delayed by employing sparse grids that are based on th