High Performance Computing Symposium (HPC 2007)

SCS Spring Simulation Multiconference (SpringSim'07)

HPC 2007 Keynote Address

Mathematical Software for High-end Computational Science and Engineering

David Keyes
Fu Foundation Professor of Applied Mathematics,
Columbia University

Multiscale, multirate scientific and engineering applications based on systems of partial differential equations possess resolution requirements that are typically inexhaustible and demand execution on the highest-capability computers available, which will soon reach the petascale. While the variety of applications is enormous, their needs for mathematical software infrastructure are surprisingly coincident. Domains with complex geometry require versatile meshing and discretization tools. Resolution requirements that evolve with the solution require dynamic adaptivity. Implicit methods for stable and accurate integration of transient problems and efficient treatments for equilibrium problems lead to large, ill-conditioned algebraic systems that must be solved with an algorithmic complexity that is close to linear in problem size or storage complexity. Distributed memory architectures demand efficient means of creating and managing load-balanced partitions of unstructured objects. These and other algorithmic challenges that are generic to nearly all mesh- and particle-based applications are addressed in the SciDAC Institute and Centers for Enabling Technologies in mathematics, which we briefly overview in this talk.

The chief to bottleneck to scalability is often the solver. At their current scalability limits, many applications spend a vast majority of their operations in solvers, due to solver algorithmic complexity that is superlinear in the problem size, whereas other phases scale linearly. Furthermore, the solver may be the phase of the simulation with the poorest parallel scalability, due to intrinsic global dependencies. The Towards Optimal PDE Simulations (TOPS) center focuses on relieving this bottleneck while providing a multilevel programming interface that allows users to advance from initial concerns of correctness and robustness to ultimate concerns of efficiency and performance portability.

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