Stopping criteria, forward and backward errors for perturbed asynchronous linear fixed point methods in finite precision

^** Email: pierre.spiteri{at}enseeiht.fr This paper deals with perturbed linear fixed point methods inthe presence of round-off errors. Successive approximationsas well as the more general asynchronous iterations are treated.Forward and backward error estimates are presented, and theseare used to propose theoretical stopping criteria for thesemethods. In the case of asynchronous iterations, macro-iterationsare used as a tool in order to obtain estimates.     

A new stopping criterion for linear perturbed asynchronous iterations

A new stopping criterion is proposed for asynchronous linear fixed point methods in finite precision. The case of absolute error is considered. The originality of this stopping criterion relies on the fact that all tests are made within the same macroiteration.     

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Parallel Asynchronous Schwarz and Multisplitting Methods for a Nonlinear Diffusion Problem

Parallel asynchronous subdomain algorithms with flexible communication for the numerical solution of nonlinear diffusion problems are presented. The discrete maximum principle is considered and the Schwarz alternating method and multisplitting methods are studied. A connection is made with M-functions for a classical nonlinear diffusion problem. Finally, computational experiments carried out on a shared memory multiprocessor are presented and analyzed.     

Optimization of the Relaxation Parameter for S.S.O.R. and A.D.I. Preconditioning

We present and compare several approaches for the optimization of the relaxation parameter both for A.D.I. and S.S.O.R. basic iteration and preconditioning conjugate gradient method. For each kind of preconditioning a detailed link between estimates of the spectral radius of the iteration matrix and of the condition number resulting from preconditioning is proposed. It allows to choose the best approach in order to obtain the optimal relaxation parameter and the corresponding optimal estimates either of the spectral radius of the iteration matrix and of the resulting condition mumber of the S.S.O.R. and A.D.I. preconditioning.     

Asynchronous multisplitting methods for nonlinear fixed point problems

Our aim is to present for nonlinear problems asynchronous multisplitting algorithms including both the basic situation of O'Leary and White and the discrete analogue of Schwarz's alternating method and its multisubdomain extensions and moreover their two-stage counterparts. The analysis of these methods is based on El Tarazi’s convergence theorem for asynchronous iterations and leads to a good level of asynchronism in each of the considered situations. This revised version was published online in August 2006 with corrections to the Cover Date.     

A new class of asynchronous iterative algorithms with order intervals

This paper deals with a new class of parallel asynchronous iterative algorithms for the solution of nonlinear systems of equations. The main feature of the new class of methods presented here is the possibility of flexible communication between processors. In particular partial updates can be exchanged. Approximation of the associated fixed point mapping is also considered. A detailed convergence study is presented. A connection with the Schwarz alternating method is made for the solution of nonlinear boundary value problems. Computational results on a shared memory multiprocessor IBM 3090 are briefly presented.

     

Evaluations of Parameters for the Optimization of S.S.O.R. and A.D.I. Preconditioning

The main goal of this paper is to give two ways to estimate the needed parameters in order to obtain the condition number of S.S.O.R. preconditioned matrices, namely, the algebraic matricial formulation of convexity Riesz theorem and the tridiagonal Fourier analysis. The improvement with respect to Axelsson's approach is explicitly given. Estimations of the condition number in the case of A.D.I. preconditioning is also considered.