A partitioning strategy for nonuniform problems on multiprocessors

The partitioning of a problem on a domain with unequal work estimates in different subddomains is considered in a way that balances the work load across multiple processors. Such a problem arises for example in solving partial differential equations using an adaptive method that places extra grid points in certain subregions of the domain. A binary decomposition of the domain is used to partition it into rectangles requiring equal computational effort. The communication costs of mapping this partitioning onto different microprocessors: a mesh-connected array, a tree machine and a hypercube is then studied. The communication cost expressions can be used to determine the optimal depth of the above partitioning

Distributed memory compiler methods for irregular problems: Data copy reuse and runtime partitioning

Outlined here are two methods which we believe will play an important role in any distributed memory compiler able to handle sparse and unstructured problems. We describe how to link runtime partitioners to distributed memory compilers. In our scheme, programmers can implicitly specify how data and loop iterations are to be distributed between processors. This insulates users from having to deal explicitly with potentially complex algorithms that carry out work and data partitioning. We also describe a viable mechanism for tracking and reusing copies of off-processor data. In many programs, several loops access the same off-processor memory locations. As long as it can be verified that the values assigned to off-processor memory locations remain unmodified, we show that we can effectively reuse stored off-processor data. We present experimental data from a 3-D unstructured Euler solver run on iPSC/860 to demonstrate the usefulness of our methods

Rectilinear partitioning of irregular data parallel computations

New mapping algorithms for domain oriented data-parallel computations, where the workload is distributed irregularly throughout the domain, but exhibits localized communication patterns are described. Researchers consider the problem of partitioning the domain for parallel processing in such a way that the workload on the most heavily loaded processor is minimized, subject to the constraint that the partition be perfectly rectilinear. Rectilinear partitions are useful on architectures that have a fast local mesh network. Discussed here is an improved algorithm for finding the optimal partitioning in one dimension, new algorithms for partitioning in two dimensions, and optimal partitioning in three dimensions. The application of these algorithms to real problems are discussed

Parallelization of irregularly coupled regular meshes

Regular meshes are frequently used for modeling physical phenomena on both serial and parallel computers. One advantage of regular meshes is that efficient discretization schemes can be implemented in a straight forward manner. However, geometrically-complex objects, such as aircraft, cannot be easily described using a single regular mesh. Multiple interacting regular meshes are frequently used to describe complex geometries. Each mesh models a subregion of the physical domain. The meshes, or subdomains, can be processed in parallel, with periodic updates carried out to move information between the coupled meshes. In many cases, there are a relatively small number (one to a few dozen) subdomains, so that each subdomain may also be partitioned among several processors. We outline a composite run-time/compile-time approach for supporting these problems efficiently on distributed-memory machines. These methods are described in the context of a multiblock fluid dynamics problem developed at LaRC

Run-time scheduling and execution of loops on message passing machines

Sparse system solvers and general purpose codes for solving partial differential equations are examples of the many types of problems whose irregularity can result in poor performance on distributed memory machines. Often, the data structures used in these problems are very flexible. Crucial details concerning loop dependences are encoded in these structures rather than being explicitly represented in the program. Good methods for parallelizing and partitioning these types of problems require assignment of computations in rather arbitrary ways. Naive implementations of programs on distributed memory machines requiring general loop partitions can be extremely inefficient. Instead, the scheduling mechanism needs to capture the data reference patterns of the loops in order to partition the problem. First, the indices assigned to each processor must be locally numbered. Next, it is necessary to precompute what information is needed by each processor at various points in the computation. The precomputed information is then used to generate an execution template designed to carry out the computation, communication, and partitioning of data, in an optimized manner. The design is presented for a general preprocessor and schedule executer, the structures of which do not vary, even though the details of the computation and of the type of information are problem dependent

An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver

We propose an efficient algorithm for the immersed boundary method on distributed-memory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudo-compressibility method recently proposed by Guermond and Minev [Comptes Rendus Mathematique, 348:581-585, 2010] that uses a directional splitting strategy to discretize the incompressible Navier-Stokes equations, thereby reducing the linear systems to a series of one-dimensional tridiagonal systems. We perform numerical simulations of several fluid-structure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. For these problems, we compare the proposed algorithm against other second-order projection-based fluid solvers. Lastly, the strong and weak scaling properties of the proposed algorithm are investigated

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions