Analysis of a new class of Forward Semi-Lagrangian schemes for the 1D Vlasov-Poisson Equations. (arXiv:0912.4952v1 [math.NA])

The Vlasov equation is a kinetic model describing the evolution of charged
particles, and is coupled with Poisson's equation, which rules the evolution of
the self-consistent electric field. In this paper, we introduce a new class of
forward Semi-Lagrangian schemes for the Vlasov-Poisson system based on a Cauchy
Kovalevsky (CK) procedure for the numerical solution of the characteristic
curves. Exact conservation properties of the first moments of the distribution
function for the schemes are derived and a convergence study is performed that
applies as well for the CK scheme as for a more classical Verlet scheme. The
convergence in L1 norm of the schemes is proved and error estimates are
obtained.

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Original source : http://arxiv.org/abs/0912.4952...

Periodic reordering

For many networks in nature, science and technology, it is possible to order the nodes so that most links are short-range, connecting near-neighbours, and relatively few long-range links, or shortcuts, are present. Given a network as a set of observed links (interactions), the task of finding an ordering of the nodes that reveals such a range-dependent structure is closely related to some sparse matrix reordering problems arising in scientific computation. The spectral, or Fiedler vector, approach for sparse matrix reordering has successfully been applied to biological data sets, revealing useful structures and subpatterns. In this work we argue that a periodic analogue of the standard reordering task is also highly relevant. Here, rather than encouraging nonzeros only to lie close to the diagonal of a suitably ordered adjacency matrix, we also allow them to inhabit the off-diagonal corners. Indeed, for the classic small-world model of Watts & Strogatz (1998, Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442) this type of periodic structure is inherent. We therefore devise and test a new spectral algorithm for periodic reordering. By generalizing the range-dependent random graph class of Grindrod (2002, Range-dependent random graphs and their application to modeling large small-world proteome datasets. Phys. Rev. E, 66, 066702-1–066702-7) to the periodic case, we can also construct a computable likelihood ratio that suggests whether a given network is inherently linear or periodic. Tests on synthetic data show that the new algorithm can detect periodic structure, even in the presence of noise. Further experiments on real biological data sets then show that some networks are better regarded as periodic than linear. Hence, we find both qualitative (reordered networks plots) and quantitative (likelihood ratios) evidence of periodicity in biological networks.

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Original source : http://imajna.oxfordjournals.org/cgi/content/short...