Jack Poulson

Elemental: A New Framework for Distributed Memory Dense Matrix Computations

Parallelizing dense matrix computations to distributed memory architectures is a well-studied subject and generally considered to be among the best understood domains of parallel computing. Two packages, developed in the mid 1990s, still enjoy regular use: ScaLAPACK and PLAPACK. With the advent of many-core architectures, which may very well take the shape of distributed memory architectures within a single processor, these packages must be revisited since the traditional MPI-based approaches will likely need to be extended. Thus, this is a good time to review lessons learned since the introduction of these two packages and to propose a simple yet effective alternative. Preliminary performance results show the new solution achieves competitive, if not superior, performance on large clusters.

A Parallel Sweeping Preconditioner for Heterogeneous 3D Helmholtz Equations

A parallelization of a sweeping preconditioner for three-dimensional Helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be

Designing Linear Algebra Algorithms by Transformation: Mechanizing the Expert Developer

O(\gamma^2 N^{4/3})

A Parallel Butterfly Algorithm

and

Programming many-core architectures - a case study: dense matrix computations on the Intel single-chip cloud computer processor

O(\gamma N \log N)

Parallel Matrix Multiplication: A Systematic Journey

, where

A Parallel Directional Fast Multipole Method

\gamma(\omega)

Mechanizing the expert dense linear algebra developer

denotes the modestly frequency-dependent number of grid points per perfectly matched layer. Several computational and memory improvements are introduced relative to using black-box sparse-direct solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for high-frequency problems distributed over thousands of cores. Two open-source packages are released along with this paper: Parallel Sweeping Preconditioner (PSP) and the underlying distributed multifrontal solver, Clique.

Sweeping preconditioner for the 3D Helmholtz equation

To implement dense linear algebra algorithms for distributed-memory computers, an expert applies knowledge of the domain, the target architecture, and how to parallelize common operations. This is often a rote process that becomes tedious for a large collection of algorithms. We have developed a way to encode this expert knowledge such that it can be applied by a system to generate mechanically the same (and sometimes better) highly-optimized code that an expert creates by hand. This paper illustrates how we have encoded a subset of this knowledge and how our system applies it and searches a space of generated implementations automatically.

Sweeping preconditioners for elastic wave propagation with spectral element methods

The butterfly algorithm is a fast algorithm which approximately evaluates a discrete analogue of the integral transform