Parallel Orthomin Equation Solver on the Cell Broadband Engine

Abstract

This paper presents our parallelization and implementation of the ORTHOMIN solver on the Cell Broadband Engine. The solution of linear systems of equations is one of the most central processing unit-intensive steps in many engineering and simulation applications and can greatly benefit from the multitude of SIMD-capable synergistic processor element (SPE) cores in the Cell processor. We report the serial ORTHOMIN implementation on the Cell's PowerPC processor element (PPE), and the parallelization and performance analysis of ORTHOMIN across 8 SPEs for Tridiagonal (1-D reservoir grid) and Heptadiagonal (3-D reservoir grid) matrices. Our implementation is shown to scale well with data size, and grid dimensionality.Parallel linear equation solvers are used in the solution of many engineering and simulation applications. These solvers find the solution of a system of linear equations taking advantage of parallel processing methods. Parallelism is essential to reduce the execution time for large systems of linear equations with thousands of equations and thousands of unknown variables. In prior work [1-2], we parallelized and implemented the Conjugate Gradient solver on the STI (Sony, Toshiba, IBM) Cell Broadband Engine (Cell) platform [8][9][10]. The Cell processor is a multi-core processor with nine cores on a single chip. One core is a PowerPC RISC processor known as the PPE (power processing element). Eight computational cores with vector capabilities are known as the SPEs (synergistic processing elements). With its multi-core parallelism and vector and SIMD processing capabilities, the STI Cell processor was shown to deliver top computation performance levels on graphics and image processing applications [3] and video surveillance applications [4]. Because of the Cell's unique architecture, shorter simulator run times are expected on the Cell platform compared to existing systems. Shorter simulation times translate into faster solutions (in terms of days, depending on the size of the problem) and allow for more simulation runs to be performed on the same hardware resources during fixed time durations.A key component of a simulator in engineering disciplines is a linear equation solver which computes the solution to a system of difference equations. ORTHOMIN is an example of minimal-residual method used in engineering simulations such as oil reservoir simulation. In ORTHOMIN, a series of vectors is generated for x(1), x(2), …, x(k+1) for k+1 <= n starting from an initial estimate of x(0), where n is the number linear equations or the number of unknown variables (unknowns). The linear system equation is of the form A X = B where X is the vector of unknowns, and A and B are scalar vectors. A is known as the coefficient matrix and X is known as the unknown variable matrix. The exact solution is obtained in n iterations assuming no round off errors occur, while an approximate solution is obtained in less than n iterations. The drawback of ORTHOMIN is that it requires a large memory which provides a huge bottleneck on ...