Ruth E. Shaw

Parallel Gaussian elimination using OpenMP and MPI

In this paper, we present a parallel algorithm for Gaussian elimination: in both a shared memory environment using OpenMP, and in a distributed memory environment using MPI. Parallel LU and Gaussian algorithms for linear systems are studied extensively, and the the results of examining various load balancing schemes on both platforms are presented. The results show an improvement in many cases over the default implementation.

A parallel algorithm for solving Toeplitz linear systems

Numerical methods of solution are considered for systems which are Toeplitz and symmetric. In our case, the coefficient matrix is essentially tridiagonal and sparse. There are two distinct approaches to be considered each of which is efficient in its own way. Here we will combine the two approaches which will allow application of the cyclic reduction method to coefficient matrices of more general forms. The convergence of the approximations to the exact solution will also be examined. Solving linear systems by the adapted cyclic reduction method can be parallel processed.

A fast method for solving second Order boundary value volterra Integro-differential equations

Using finite difference methods to solve nonlinear Volterra integro-differential equations with two point boundary conditions give rise to a symmetric banded coefficient matrix. A typical method for solving systems of this form involves the LU method. In this paper the original system is modified to allow the implementation of a special fast algorithm for solving tridiagonal systems. Numerical examples are given to compare an efficient form of the LU method with the new approach.

A split-correct parallel algorithm for solving tridiagonal symmetric toeplitz systems

In 1994, Yan and Chung produced a fast algorithm for solving a diagonally dominant symmetric Toeplitz tridiagonal system of linear equations Ax = b. In this work a method will be presented which will allow for problems of the above nature to be split into two separate systems which can be solved in parallel, and then combined and corrected to obtain a solution to the original system. An error analysis will be provided along with example cases and time comparison results.

Solving banded and near symmetric systems

In an earlier work by the authors [8], finite difference methods applied to obtain numerical approximations for the solution of nonlinear Volterra integro-differential equations with two point boundary conditions were considered. In [8], the auxiliary conditions required to complete the system, were chosen to preserve the symmetry of the banded system. In general, applying natural auxiliary conditions (NAC) would give rise to a nonsymmetric banded system. In this article, we consider the efficient solving of a linear banded system with a near-symmetric coefficient matrix.

A parallel numerical algorithm for fredholm integro-differential two-point boundary value problems

Numerical methods for second order differential equations with two-point boundary conditions are incorporated into a three part method for the solution of a second order nonlinear Fredholm integro-differential equation. The interest in this paper is the development of an algorithm for parallel processing the discrete nonlinear system.Numerical examples are given.

A parallel algorithm for nonlinear Volterra integro-differential equations

Direct methods of solution for solving nonlinear Volterra integral and integro-differential equations are inherently serial and therefore have not received much attention for use on a parallel computer. It is possible, however, to make significant gains in speedup by employing some novel approaches to existing methods. It can be observed that when approximating the integral term, all approximations up to and including the kth step in the interval can be evaluated at the same time. The resulting algorithm is described here and several numerical examples illustrate the results. With the four-processor multicomputer utilized, speedups of 3 to 4 were realized.

A parallel shooting method for second order volterra integro-differential equations with two point boundary conditions

The method of parallel shooting for ordinary differential equations is adapted for solving second order Volterra integro-differential equations (VIDEs) with two point boundary conditions. The modifications incorporate the integral term from the VIDE into the original method as described in Na [5]. Numerical examples illustrate the results.

A parallel QR factorization algorithm for solving Toeplitz tridiagonal systems

Summary form only given. QR methods for solving Toeplitz tridiagonal systems are well developed with applications in numerous interdisciplinary fields. There is a strong motivation to develop faster, more efficient and, more importantly, scalable algorithms to factor such systems due to their significance in many scientific applications. We present two parallel QR factorization algorithms used to solve Toeplitz tridiagonal systems. QR factorization is accomplished using Householder reflections and Givens rotations. These parallel algorithms exhibit high scalability and near linear to superlinear speedup on large system sizes when implemented on a distributed system.

The IBiCGStab method on bulk synchronous parallel architectures

In this paper, an improved version of the BiCGStab method for the solutions of large and sparse linear systems of equations with unsymmetric coefficient matrices is proposed. The method combines elements of numerical stability and parallel algorithm design without increasing the computational costs. The algorithm is derived such that all inner products of a single iteration step are independent and communication time required for inner product can be overlapped efficiently with computation time of vector updates. Therefore, the cost of global communication can be significantly reduced. In this paper, the bulk synchronous parallel (BSP) model is used to design a fully efficient, scalable and portable parallel proposed algorithm and to provide accurate performance prediction of the algorithm for a wide range of architectures including the Cray T3D, the Parsytec, and a cluster of workstations connected by an Ethernet. This performance model provides us useful insight in the time complexity of the method using only a few system dependent parameters based on a simple and accurate cost modelling. The theoretical performance prediction are compared with some preliminary measured timing results of a numerical application from ocean flow simulation.