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Parallel asynchronous algorithms: A survey

“…Bai and also Chinese scientists (see [55]- [56]) who, to accelerate the speed of convergence, have introduced several relaxation parameters. In addition, J. Bahi J.C. Miellou and K. Rhofir [54] have extended by contraction techniques the previous results for the resolution of pseudo-linear problems without or with constraints corresponding in the latter case to the solution of multivalued problems.…”

“…Note also that for a particular choice of the weighting matrices D lk , we can obtain various iterative methods and particularly on one hand a subdomain method without overlapping and on the other hand the classical Schwarz alternating method (see [54]); then this last point establish a link between these subdomain methods and the multisplitting methods. According with [54] the block -Jacobi method corresponds to the following choice of M l M l = diag(I 1 , . .…”

“…A formal multisplitting associated with problem (2) or problem (3) is defined by the collection of fixed point problems (see [54])…”

“…A particular case of multisplitting method is constituted by the two-level methods where an asynchronous parallel external iteration corresponding to a coordination of the subsystems to be solved is considered, these latter subsystems being themselves solved by an iterative method. A particular case of a two-level method is constituted by the coupling of a block relaxation method, each block being solved by a point relaxation method leading then to the two-stage method (see for example [65]); another example of a two-level method is constituted by an external iteration of Newton and an internal iteration of relaxation (see for example [52], [54], [57], [65]).…”

“…Proposition 9. Assume that assumption (54) holds and moreover that assumptions (6) to (8) are satisfied. If the fixed point U verify U > 0, which imply that U 0, then for any given positive real number η their exists an index r such that for all r ≥ r, the stopping criterion (67) is satisfied.…”