Asynchronous algorithms for poisson's equation with nonlinear boundary conditions

Abstract

--ZusammenfassungAsynchronous Algorithms for Poisson's Equation with Nonlinear Boundary Conditions. Poisson's equation with nonlinear boundary conditions is discretized with the method of lines to obtain a system of second order differential equations with multi-point boundary conditions. This differential system is converted, using invariant imbedding for each one-dimensional problem, into a fixed point problem and then the asynchronous algorithms are applied. AMS Subject Classifications: 65H 10, 65M20; CR: 5.15, 5.17Asynehrone Algoritlunen far die Poisson-Gleiehung mit nichtlinearen Randbedingungen. Die PoissonGleichung mit nichtlinearen Randbedingungen wird mittels der Linienmethode diskretisiert und ergibt ein System von gew6hnlichen Differentialgleichungen zweiter Ordnung mit Randbedingungen. Durch invariante Einbettung for jedes eindimensionale Problem wird dieses System in ein Fixpunktproblem verwandelt, auf das dann die asynchronen Algorithmen angewandt werden.

An asynchronous incomplete block LU preconditioner for computational fluid dynamics on unstructured grids

“…In the event of asynchronous updates, some diagonal entries may sometimes be set to zero. To take this into account, Chow and Patel define a 'modified Jacobi-type iteration' corresponding to the iteration in equation (15), in which whenever a zero diagonal entry is encountered, it is replaced by an arbitrary non-zero value. They showed [14, theorem 3.7] that a synchronized modified Jacobi-type iteration corresponding to equation ( 17) converges in at most m iterations irrespective of the initial guess.…”

“…There have been some efforts to apply asynchronous iterations to solution of scalar partial differential equations (PDEs). In the 1980's, Anwar and El Tarazi used a non-linear asynchronous iteration to solve a Poisson problem with non-linear boundary conditions [15]. More recently, Chow and Patel [14] demonstrated their asynchronous ILU factorization for solving the Poisson equation and the linear convection-diffusion equation with promising results.…”

The performance of asynchronous algorithms on hypercubes

“…We now turn our attention to an implementation of Algorithm 3.1. We assume that there are p processors available for the solution of the linear system (1). We also assume, for ease of exposition, that n, the number of unknowns, is an integer multiple of p. Thus, each processor is responsible for evaluating m = nip ofthe GA;(X), the coordinatefunctionals of G(x).…”

“…Many authors have used the results of [3,7,4] or have used similar methods to establish the convergence of specific algorithms. These include [1,15,16,11,12].…”

“…A notable exception is the work of Baudet [3], who studied the implementations of several algorithms on the Carnegie-Mellon multiprocessor computer. Experimental results including comparisons with synchronous algorithms can also be found in [2,1]. All of these experiments were carried out on shared memory machines with a small number of processors.…”