Cílem článku je naznačit úlohu matematiky a efektivnost nových algoritmů pro řešení rozsáhlých soustav lineárních rovnic na současných masívně paralelních superpočítačích. Na příkladu řešení Poissonovy rovnice je popsána základní varianta metody rozložení oblasti typu FETI (finite element tearing and interconnecting) s projektorem na přirozenou hrubou síť, jsou odvozeny základní kvalitativní výsledky demonstrující asymptoticky lineární (optimální) složitost řešení a jsou popsána prakticky důležitá...
Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements...
The numerical solution of granular dynamics problems with Coulomb friction leads to the problem of minimizing a convex quadratic function with semidefinite Hessian subject to a separable conical constraints. In this paper, we are interested in the numerical solution of this problem. We suggest a modification of an active-set optimal quadratic programming algorithm. The number of projection steps is decreased by using a projected Barzilai-Borwein method. In the numerical experiment, we compare our...
An algorithm for quadratic minimization with simple bounds is introduced, combining, as many well-known methods do, active set strategies and projection steps. The novelty is that here the criterion for acceptance of a projected trial point is weaker than the usual ones, which are based on monotone decrease of the objective function. It is proved that convergence follows as in the monotone case. Numerical experiments with bound-constrained quadratic problems from CUTE collection show that the modified...
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