Parallel language constructs for tensor product computations on loosely coupled architectures

Distributed memory architectures offer high levels of performance and flexibility, but have proven awkard to program. Current languages for nonshared memory architectures provide a relatively low level programming environment, and are poorly suited to modular programming, and to the construction of libraries. A set of language primitives designed to allow the specification of parallel numerical algorithms at a higher level is described. Tensor product array computations are focused on along with a simple but important class of numerical algorithms. The problem of programming 1-D kernal routines is focused on first, such as parallel tridiagonal solvers, and then how such parallel kernels can be combined to form parallel tensor product algorithms is examined

The improved robustness of multigrid elliptic solvers based on multiple semicoarsened grids

Multigrid convergence rates degenerate on problems with stretched grids or anisotropic operators, unless one uses line or plane relaxation. For 3-D problems, only plane relaxation suffices, in general. While line and plane relaxation algorithms are efficient on sequential machines, they are quite awkward and inefficient on parallel machines. A new multigrid algorithm is presented based on the use of multiple coarse grids, that eliminates the need for line or plane relaxation in anisotropic problems. This algorithm was developed and the standard multigrid theory was extended to establish rapid convergence for this class of algorithms. The new algorithm uses only point relaxation, allowing easy and efficient parallel implementation, yet achieves robustness and convergence rates comparable to line and plane relaxation multigrid algorithms. The algorithm described is a variant of Mulder's multigrid algorithm for hyperbolic problems. The latter uses multiple coarse grids to achieve robustness, but is unsuitable for elliptic problems, since its V-cycle convergence rate goes to one as the number of levels increases. The new algorithm combines the contributions from the multiple coarse grid via a local switch, based on the strength of the discrete operator in each coordinate direction

Mapping implicit spectral methods to distributed memory architectures

Spectral methods were proven invaluable in numerical simulation of PDEs (Partial Differential Equations), but the frequent global communication required raises a fundamental barrier to their use on highly parallel architectures. To explore this issue, a 3-D implicit spectral method was implemented on an Intel hypercube. Utilization of about 50 percent was achieved on a 32 node iPSC/860 hypercube, for a 64 x 64 x 64 Fourier-spectral grid; finer grids yield higher utilizations. Chebyshev-spectral grids are more problematic, since plane-relaxation based multigrid is required. However, by using a semicoarsening multigrid algorithm, and by relaxing all multigrid levels concurrently, relatively high utilizations were also achieved in this harder case

Programming distributed memory architectures using Kali

Programming nonshared memory systems is more difficult than programming shared memory systems, in part because of the relatively low level of current programming environments for such machines. A new programming environment is presented, Kali, which provides a global name space and allows direct access to remote data values. In order to retain efficiency, Kali provides a system on annotations, allowing the user to control those aspects of the program critical to performance, such as data distribution and load balancing. The primitives and constructs provided by the language is described, and some of the issues raised in translating a Kali program for execution on distributed memory systems are also discussed

Floating shock fitting via Lagrangian adaptive meshes

In recent works we have formulated a new approach to compressible flow simulation, combining the advantages of shock-fitting and shock-capturing. Using a cell-centered Roe scheme discretization on unstructured meshes, we warp the mesh while marching to steady state, so that mesh edges align with shocks and other discontinuities. This new algorithm, the Shock-fitting Lagrangian Adaptive Method (SLAM) is, in effect, a reliable shock-capturing algorithm which yields shock-fitted accuracy at convergence. Shock-capturing algorithms like this, which warp the mesh to yield shock-fitted accuracy, are new and relatively untried. However, their potential is clear. In the context of sonic booms, accurate calculation of near-field sonic boom signatures is critical to the design of the High Speed Civil Transport (HSCT). SLAM should allow computation of accurate N-wave pressure signatures on comparatively coarse meshes, significantly enhancing our ability to design low-boom configurations for high-speed aircraft

Object-oriented numerical computing C++

An object oriented language is one allowing users to create a set of related types and then intermix and manipulate values of these related types. This paper discusses object oriented numerical computing using C++

Semi-automatic process partitioning for parallel computation

On current multiprocessor architectures one must carefully distribute data in memory in order to achieve high performance. Process partitioning is the operation of rewriting an algorithm as a collection of tasks, each operating primarily on its own portion of the data, to carry out the computation in parallel. A semi-automatic approach to process partitioning is considered in which the compiler, guided by advice from the user, automatically transforms programs into such an interacting task system. This approach is illustrated with a picture processing example written in BLAZE, which is transformed into a task system maximizing locality of memory reference

Supporting shared data structures on distributed memory architectures

Programming nonshared memory systems is more difficult than programming shared memory systems, since there is no support for shared data structures. Current programming languages for distributed memory architectures force the user to decompose all data structures into separate pieces, with each piece owned by one of the processors in the machine, and with all communication explicitly specified by low-level message-passing primitives. A new programming environment is presented for distributed memory architectures, providing a global name space and allowing direct access to remote parts of data values. The analysis and program transformations required to implement this environment are described, and the efficiency of the resulting code on the NCUBE/7 and IPSC/2 hypercubes are described

Triangular spectral elements for incompressible fluid flow

We discuss the use of triangular elements in the spectral element method for direct simulation of incompressible flow. Triangles provide much greater geometric flexibility than quadrilateral elements and are better conditioned and more accurate when small angles arise. We employ a family of tensor product algorithms for triangles, allowing triangular elements to be handled with comparable arithmetic complexity to quadrilateral elements. The triangular discretizations are applied and validated on the Poisson equation. These discretizations are then applied to the incompressible Navier-Stokes equations and a laminar channel flow solution is given. These new triangular spectral elements can be combined with standard quadrilateral elements, yielding a general and flexible high order method for complex geometries in two dimensions

ICASE/LaRC Workshop on Adaptive Grid Methods

Solution-adaptive grid techniques are essential to the attainment of practical, user friendly, computational fluid dynamics (CFD) applications. In this three-day workshop, experts gathered together to describe state-of-the-art methods in solution-adaptive grid refinement, analysis, and implementation; to assess the current practice; and to discuss future needs and directions for research. This was accomplished through a series of invited and contributed papers. The workshop focused on a set of two-dimensional test cases designed by the organizers to aid in assessing the current state of development of adaptive grid technology. In addition, a panel of experts from universities, industry, and government research laboratories discussed their views of needs and future directions in this field