Summer Proceedings 2018: The Center for Computing Research at Sandia National Laboratories

Abstract

Computational mathematics is concerned with the design, analysis, and implementation of algorithms to solve mathematical problems. Articles in this section describe the discretization of partial differential equations and methods to solve equations, couple multiphysics systems of equations, and quantify uncertainty.Chen, Rajamanickam, Boman, and Darve introduce a parallel, hierarchical solver for linear systems generated from the discretization of elliptic PDEs that is based on the LoRaSp hierarchical solver, which uses block Cholesky factorization and low-rank elimination. The parallel version makes use of the distance-2 connection property to partition the graph nodes based on their distances from processor boundaries and coloring to coordinate computation and communication for processor boundary nodes. They demonstrate that their algorithm achieves nearly linear scaling in solve time and memory usage for 3-D problems and constant iteration count when used as a preconditioner for Poisson problems, and good strong and weak scaling on multiple processors.Eller and Hoemmen derive scalable versions of the preconditioned conjugate gradient method that use a three-term recurrence. By overlapping two iterations of matrix-vector multiplies and preconditioner applications with a single non-blocking allreduce, they address scalability problems associated with blocking allreduce. They give full derivations of their scalable methods and compare them with two existing preconditioned conjugate gradient algorithms. Their approach provides the potential to develop further pipelined, non-blocking methods.Fisher and Cyr construct a discretization for PDE constrained optimization problems based on piecewise Hermite cubic orthogonal spline collocation in one and two dimensions. They also prove the optimality of a preconditioner for the MINRES iteration and confirm it computationally. They apply the method to the problems of distributed control of Poisson's equation and Burgers' equation.Lupo Pasini, Tuminaro, and Hu investigate an additive multigrid method adapted to a domain decomposition setting with the aim of achieving convergence properties comparable with multiplicative approaches while maintaining a computational cost comparable with additive approaches. The approach has the potential to enhance concurrency and improve performance on new and future architectures. They carefully construct the smoothed interpolation operator and use subdomain coloring to compress the prolongator to significantly decrease communication cost and memory storage requirements. They demonstrate that the approach can improve performance if the problem size and number of subdomains are carefully chosen to achieve good load balancing.Ohm and Tuminaro present a multigrid method for linear systems derived from PDEs discretized using meshes that have both structured and unstructured regions that ensures that the prolongation operator is consistent at the interfaces between structured and unstructured regions. They demonstrate significant reductions in th...