On the mapping of partial differential equation computations onto distributed memory MIMD parallel machines

Nikos Panayiotis Chrisochoides, Purdue University

Abstract

This thesis deals with the automatic mapping strategies for the computations associated with the numerical solution of Partial Differential Equations (PDEs) onto distributed memory MIMD machines. In the case of PDE computations, such mappings can be formulated at three distinct levels: the discrete geometrical data structures associated with the PDE domain, the linear system of algebraic equations associated with some discretization of the PDE equations, and the data flow graph of the PDE solver. In the geometry mapping strategies we formulate the mapping problem at the discrete geometric data structures of the PDE domain. We describe these strategies in terms of three distinct phases: the partitioning, the allocation, and the message scheduling. In the partitioning phase, we decompose the geometric data structures in a specified number of subdomains or substructures so that: (i) the subdomains have the "same" number of elements or grid points, (ii) the number of interfaces among the subdomains is "small", (iii) the number of adjacent subdomains is "minimal", (iv) each sub domain is a "compact" domain. In the allocation phase the objective is to allocate these subdomains to processors, so that: (v) geometrically neighbor subdomains are allocated to neighbor processors in the interconnection network of a targeting parallel machine. In the message scheduling phase the objective is to decouple (color) the processors so that: (vi) the local synchronization among the processors introduces minimum edge contention in the network. First, we present an algorithmic and software infrastructure consisting of "fast" heuristics for determining optimal geometry based mappings of PDE data suitable for matrix and domain decomposition methods. Furthermore, we describe a software system which assists the user in visualizing and manipulating such mappings in the Parallel-ELLPACK environment. Second, we present a mapping methodology which consists of a set of well defined linear algebra primitives formulated on the PDE algebraic data structures and implemented on various targeting parallel architectures. In this methodology, known as BLAS (Basic Linear Algebra Subroutines), the solvers are implemented using these parallel BLAS primitives. In this thesis we have considered the parallelization of matrix vector and matrix matrix operations for banded matrices on distributed memory multiprocessor systems that support a mesh and ring interconnection topologies. (Abstract shortened by UMI.)

Degree

Ph.D.

Advisors

Elias, Purdue University.

Subject Area

Computer science

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