The impact of global communication latency at extreme scales on Krylov methods

Krylov Subspace Methods (KSMs) are popular numerical tools for solving large linear systems of equations. We consider their role in solving sparse systems on future massively parallel distributed memory machines, by estimating future performance of their constituent operations. To this end we construct a model that is simple, but which takes topology and network acceleration into account as they are important considerations. We show that, as the number of nodes of a parallel machine increases to very large numbers, the increasing latency cost of reductions may well become a problematic bottleneck for traditional formulations of these methods. Finally, we discuss how pipelined KSMs can be used to tackle the potential problem, and appropriate pipeline depths

Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained