Tridiagonal matrix algorithm

Abstract: We discuss the parallel implementation of the Cholesky factorization of a positive definite symmetric matrix when that matrix is block tridiagonal. While parallel implementations for this problem, and closely related problems like the factorization of banded matrices, have been previously reported in the literature, those implementations dealt with the special cases where the block size (bandwidth) was either very large (wide) or very small (narrow). We present a solution that can be used for the entire spectrum of cases, ranging from extremely large (wide) to very small (narrow). Preliminary performance results collected on a Cray T3E-600 distributed memory supercomputer show that our implementation attains respectable performance. Indeed, factorization of a matrix with block size b=1000 and a total dimension of more than 500,000 takes about 3.6 minutes on 128 processors.

Abstract: ‎In this paper the implementation of parallel algorithm of alternating direction implicit (ADI) method has been considered‎. ‎ADI parallel algorithm is used to solve a multiphase multicomponent fluid flow problem in porous media‎. ‎There are various technologies for implementing parallel algorithms on the CPU and GPU for solving hydrodynamic problems‎. ‎In this paper GPU-based (graphic processor unit) algorithm was used‎. ‎To implement the GPU-based parallel ADI method‎, ‎CUDA and OpenCL were used‎. ‎ADI is an iterative method used to solve matrix equations‎. ‎To solve the tridiagonal system of equations in ADI method‎, ‎the parallel version of cyclic reduction (CR) method was implemented‎. ‎The cyclic reduction is a method for solving linear equations by repeatedly splitting a problem as a Thomas method‎. ‎To implement of a sequential algorithm for solving the oil recovery problem‎, ‎the implicit Thomas method was used‎. ‎Thomas method or tridiagonal matrix algorithm is used to solve tridiagonal systems of equations‎. ‎To test parallel algorithms personal computer installed Nvidia RTX 2080 graphic card with 8 GB of video memory was used‎. ‎The computing results of parallel algorithms using CUDA and OpenCL were compared and analyzed‎. ‎The main purpose of this research work is a comparative analysis of the parallel algorithm computing results on different technologies‎, ‎in order to show the advantages and disadvantages each of CUDA and OpenCL for solving oil recovery problems‎.