Hierarchical parallelisation for the solution of stochastic finite element equations

As an example application the elliptic partial differential equation for steady groundwater flow is considered. Uncertainties in the conductivity may be quantified with a stochastic model. A discretisation by a Galerkin ansatz with tensor products of finite element functions in space and stochastic ansatz functions leads to a certain type of stochastic finite element system (SFEM). This yields a large system of equations with a particular structure. They can be efficiently solved by Krylov subspace methods, as here the main ingredient is the multiplication with the system matrix and the application of the preconditioner. We have implemented a “hierarchical parallel solver” on a distributed memory architecture for this. The multiplication and the preconditioning uses a—possibly parallel—deterministic solver for the spatial discretisation as a building block in a black-box fashion. This paper is concerned with a coarser grained level of parallelism resulting from the stochastic formulation. These coarser levels are implemented by running different instances of the deterministic solver in parallel. Different possibilities for the distribution of data are investigated, and the efficiencies determined. On up to 128 processors, systems with more than 5 × 10⁷ unknowns are solved.