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CivilComp Proceedings
ISSN 17593433 CCP: 112
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON PARALLEL, DISTRIBUTED, GPU AND CLOUD COMPUTING FOR ENGINEERING Edited by: P. Iványi and B.H.V. Topping
Paper 15
Asynchronous parallel multisplitting mixed methods P. Spiteri^{1}, L. ZianeKhodja
^{2} and R. Couturier
^{2}
^{1}IRIT INPT, University of Toulouse, Toulouse, France P. Spiteri, L. ZianeKhodja, R. Couturier, "Asynchronous parallel multisplitting mixed methods", in P. Iványi, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Parallel, Distributed, GPU and Cloud Computing for Engineering", CivilComp Press, Stirlingshire, UK, Paper 15, 2019. doi:10.4203/ccp.112.15
Keywords: asynchronous parallel algoritm, high performance computing, multisplitting methods,
Krylov method, Newton method, discretized pseudolinear problem, large scale systems,
nonlinear boundary value problems.
Summary
The present study is related to the analysis and application of mixed multisplitting methods
to solve pseudo  linear stationary problems. These problems are stationary either intrinsically
or as the result of the discretization of time evolution problems by implicit or semiimplicit
time marching schemes. The considered problems are defined as an affine application AUF
perturbed by an increasing diagonal operator . In the following A has the property of being
a large dimensional Mmatrix, F is a vector, U is the unknown vector. Note that this type of
problem occurs when solving elliptic, parabolic or hyperbolic second order boundaryvalue
problems. Note also that the Mmatrix property is well verified after discretization by classical
finite differences, finite volumes or finite elements provided that the angle condition is
verified. In this case the problem will be solved by a specific method corresponding to a local
linearization and to the implementation of the iterative Newton method. Thus, a large sparse
linear system must be solved. This linear system is then associated with a fixed point problem
and we intend to solve it by asynchronous parallel iterations. Taking into account the properties
of the matrix A and the operator’s monotony properties , it is shown that fixed point
applications are contractive with respect to an uniform weighted norm, which ensures on the
one hand the existence and uniqueness of the solution of the algebraic system to be solved
and on the other hand the convergence of asynchronous parallel iterations towards the solution
of the problem. In addition, in order to unify the presentation and analysis of the algorithm
behavior, we consider multisplitting methods that unify the presentation of subdomain methods,
either to model subdomain methods without overlap, or to model subdomain methods
with overlap such as Schwarz’s alternating method. These multisplitting methods are then
applied to solve the target problem, the convergence analysis being still carried out by contraction
techniques with respect to a weighted uniform norm. From an implementation point of
view, we consider a mixed algorithm constituted by a two stage iterative algorithm, where the
outer iteration is the multisplitting method and the inner iteration is the Krylov based iterative
method, typically the GMRES method. As applications, we consider a diffusion convection
problem perturbed by an increasing diagonal operator, the problem being solved by a mixed
Newton  multisplitting method.
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