We introduce non-local dynamics on directed networks through the construction of a fractional version of a non-symmetric Laplacian for weighted directed graphs. Furthermore, we provide an analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.
Linear solvers for large and sparse systems are a key element of scientific applications, and their efficient implementation is necessary to harness the computational power of current computers. Algebraic MultiGrid (AMG) preconditioners are a popular ingredient of such linear solvers; this is the motivation for the present work where we examine some recent developments in a package of AMG preconditioners to improve efficiency, scalability and robustness on extreme scale problems. The main novelty is the design and implementation of a parallel coarsening algorithm based on aggregation of unknowns employing weighted graph matching techniques; this is a completely automated procedure, requiring no information from the user, and applicable to general symmetric positive definite (s.p.d.) matrices. The new coarsening algorithm improves in terms of numerical scalability at low operator complexity over decoupled aggregation algorithms available in previous releases of the package. The preconditioners package is built on the parallel software framework PSBLAS, which has also been updated to progress towards exascale. We present weak scalability results on one of the most powerful supercomputer in Europe for linear systems with sizes up to O(10 10 ) unknowns.
In this paper we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The numerical experiments on some fractional partial differential equation models confirm that the proposed approach is promising.
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