As the cost of data movement increasingly dominates performance, developers of finite-volume and finite-difference solutions for partial differential equations (PDEs) are exploring novel higher-order stencils that increase numerical accuracy and computational intensity. This paper describes a new compiler reordering transformation applied to stencil operators that performs partial sums in buffers, and reuses the partial sums in computing multiple results. This optimization has multiple effect son improving stencil performance that are particularly important to higher-order stencils: exploits data reuse, reduces floating-point operations, and exposes efficient SIMD parallelism to backend compilers. We study the benefit of this optimization in the context of Geometric Multigrid (GMG), a widely used method to solve PDEs, using four different Jacobi smoothers built from 7-, 13-, 27- and 125-point stencils. We quantify performance, speedup, and numerical accuracy, and use the Roofline model to qualify our results. Ultimately, we obtain over 4× speedup on the smoothers themselves and up to a 3× speedup on the multigrid solver. Finally, we demonstrate that high-order multigrid solvers have the potential of reducing total data movement and energy by several orders of magnitude.

We present an Adaptive Mesh Refinement benchmark for evaluating programmability and performance of modern parallel programming languages. Benchmarks employed today by language developing teams, originally designed for performance evaluation of computer architectures, do not fully capture the complexity of state-of-the-art computational software systems running on today’s parallel machines or to be run on the emerging ones from the multi-cores to the peta-scale High Productivity Computer Systems. This benchmark, extracted from a real application framework, presents challenges for a programming language in both expressiveness and performance. It consists of an infrastructure for finite difference calculations on block-structured adaptive meshes and a solver for elliptic Partial Differential Equations built on this infrastructure. Adaptive Mesh Refinement algorithms are challenging to implement due to the irregularity introduced by local mesh refinement. We describe those challenges posed by this benchmark through two reference implementations (C++/Fortran/MPI and Titanium) and in the context of three programming models. Categories and Subject Descriptors

SCALLOP is a highly scalable solver and library for elliptic partial differential equations on regular block-structured domains. SCALLOP avoids high communication overheads algorithmically by taking advantage of the locality properties inherent to solutions to elliptic PDEs. Communication costs are small, on the order of a few percent of the total running time on up to 1024 processors of NPACI's and NERSC's IBM Power-3 SP sytems. SCALLOP trades off numerical overheads against communication. These numerical overheads are independent of the number of processors for a wide range of problem sizes. SCALLOP is implicitly designed for infinite domain (free space) boundary conditions, but the algorithm can be reformulated to accommodate other boundary conditions. The SCALLOP library is built on top of the KeLP programming system and runs on a variety of platforms.

Titanium is a language and system for high-performance parallel scientific computing. Titanium uses Java as its base, thereby leveraging the advantages of that language and allowing us to focus attention on parallel computing issues. The main additions to Java are immutable classes, multidimensional arrays, an explicitly parallel SPMD model of computation with a global address space, and zone-based memory management. We discuss these features and our design approach, and report progress on the development of Titanium, including our current driving application: a three-dimensional adaptive mesh refinement parallel Poisson solver. © 1998 John Wiley & Sons, Ltd.