Vincent A. Barker

LAPACK95 users' guide

LAPACK95 is a Fortran 95 interface to the Fortran 77 LAPACK library. It is relevant for anyone who writes in the Fortran 95 language and needs reliable software for basic numerical linear algebra. It improves upon the original user-interface to the LAPACK package, taking advantage of the considerable simplifications that Fortran 95 allows.

Solving the Stokes Problem on a Massively Parallel Computer

Abstract We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distri...

Comparison of iterative methods for computing the pressure field in a dynamic network model

In dynamic network models, the pressure map (the pressure in the pores) must be evaluated at each time step. This calculation involves the solution of a large number of nonlinear algebraic systems of equations and accounts for more than 80 of the total CPU–time. Each nonlinear system requires at least the partial solution of a sequence of linear systems. We present a comparative study of iterative methods for solving these systems, where we apply both standard routines from the public domain package ITPACK 2C and our own routines tailored to the network problem. The conjugate gradient method, preconditioned by symmetric successive overrelaxation, was found to be consistently faster and more robust than the other solvers tested. In particular, it was found to be much superior to the successive overrelaxation technique currently used by many researchers.

Spectral element simulation of ultrafiltration

Abstract A spectral element method for simulating stationary 2-D ultrafiltration is presented. The mathematical model is comprised of the Navier–Stokes equations for the velocity field of the fluid and a transport equation for the concentration of the solute. In addition to the presence of the velocity vector in the transport equation, the system is coupled by the dependency of the fluid viscosity on the solute concentration and by a concentration-dependent boundary condition for the Navier–Stokes equations at the membrane surface. The spectral element discretization yields a nonlinear algebraic system for the unknowns at the mesh nodes. This system is solved via a technique combining the penalty method, Newton–Raphson iterations, static condensation, and a solver for banded linear systems. In addition, a smoothing technique is used to handle a singularity in the boundary condition at the membrane. The performance of the spectral element code when applied to several ultrafiltration problems is reported.

Parallel computation of rotating flows

Abstract This paper deals with the simulation of 3‐D rotating flows based on the velocity‐vorticity formulation of the Navier‐Stokes equations in cylindrical coordinates. The governing equations are discretized by a finite difference method. The solution is advanced to a new time level by a two‐step process. In the first step, the vorticity at the new time level is computed using the velocity at the previous time level. In the second step, the velocity at the new time level is computed using the new vorticity. We discuss here the second part which is by far the most time‐consuming. The numerical problem is that of solving a singular, large, sparse, over‐determined linear system of equations, and the iterative method CGLS is applied for this purpose. We discuss some of the mathematical and numerical aspects of this procedure and report on the performance of our software on a wide range of parallel computers.

Computing Eigenvalues of Sparse Matrices on the Connection Machine

This paper discusses two Fortran subroutines, LANSYM and LANUSM, for computing eigenvalues of real sparse matrices on the CM-200. These subroutines are designed for symmetric and unsymmetric matrices, respectively. Both are adaptations of single-vector Lanczos algorithms developed by Cullum and Willoughby. The eigenvalues are computed in a region prescribed by the user. In the case of LANSYM, this is a real interval [a,b]. In the case of LANUSM, it is a quadrilateral, Q, in the complex plane. The main attractions of the Cullum and Willoughby approach are the absence of both the reorthogonalization of the Lanczos vectors and factorizations of the (shifted) input matrix.

Wavelets and Differential Equations

Wavelet applications to date have been dominated by signal and image processing. While perhaps not immediately appealing as a means of solving differential equations, the growing body of literature in this area indicates that wavelets have a role to play here, too. We give here some of the basic background and an example illustrating how wavelets can be used to solve differential equations.

Similarity transformations of MAPs

We introduce the notion of similar Markovian Arrival Processes (MAPs) and show that the event stationary point processes related to two similar MAPs are stochastically equivalent. This holds true for the time stationary point processes too. We show that several well known stochastical equivalences as e.g. that between the H2 renewal process and the Interrupted Poisson Process (IPP) can be expressed by the similarity transformations of MAPs. In the appendix the valid region of similarity transformations for two-state MAPs is characterized.

Using program package NSPCG to analyze the trunk reservation service protection method

Abstract Unlike certain service protection methods for mixed traffic streams, such as the class-limitation method, the trunk reservation scheme cannot be based on a product form property of a stationary probability distribution vector. Rather, the analysis of the trunk reservation scheme requires solving, by purely numerical methods, a set of balance equations, Ax = 0, often of very high order. Since the coefficient matrix is typically sparse, it is natural to apply iterative methods to this task. Many such methods have been incorporated in program package NSPCG, developed at the Center for Numerical Analysis at the University of Texas at Austin. In this paper we report our experience in applying the NSPCG package to a typical system arising from the trunk reservation scheme.