9,317 research outputs found
Short Cycles Connectivity
Short cycles connectivity is a generalization of ordinary connectivity.
Instead by a path (sequence of edges), two vertices have to be connected by a
sequence of short cycles, in which two adjacent cycles have at least one common
vertex. If all adjacent cycles in the sequence share at least one edge, we talk
about edge short cycles connectivity.
It is shown that the short cycles connectivity is an equivalence relation on
the set of vertices, while the edge short cycles connectivity components
determine an equivalence relation on the set of edges. Efficient algorithms for
determining equivalence classes are presented.
Short cycles connectivity can be extended to directed graphs (cyclic and
transitive connectivity). For further generalization we can also consider
connectivity by small cliques or other families of graphs
Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrodinger equations
Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are
approximated by equations of the discrete nonlinear Schrodinger type. We show
how to justify this approximation by two methods, which have been very popular
in the recent literature. The first method relies on a priori energy estimates
and multi-scale decompositions. The second method is based on a resonant normal
form theorem. We show that although the two methods are different in the
implementation, they produce equivalent results as the end product. We also
discuss applications of the discrete nonlinear Schrodinger equation in the
context of existence and stability of breathers of the Klein--Gordon lattice
Nearest-Neighbor Interaction Systems in the Tensor-Train Format
Low-rank tensor approximation approaches have become an important tool in the
scientific computing community. The aim is to enable the simulation and
analysis of high-dimensional problems which cannot be solved using conventional
methods anymore due to the so-called curse of dimensionality. This requires
techniques to handle linear operators defined on extremely large state spaces
and to solve the resulting systems of linear equations or eigenvalue problems.
In this paper, we present a systematic tensor-train decomposition for
nearest-neighbor interaction systems which is applicable to a host of different
problems. With the aid of this decomposition, it is possible to reduce the
memory consumption as well as the computational costs significantly.
Furthermore, it can be shown that in some cases the rank of the tensor
decomposition does not depend on the network size. The format is thus feasible
even for high-dimensional systems. We will illustrate the results with several
guiding examples such as the Ising model, a system of coupled oscillators, and
a CO oxidation model
Totally parallel multilevel algorithms
Four totally parallel algorithms for the solution of a sparse linear system have common characteristics which become quite apparent when they are implemented on a highly parallel hypercube such as the CM2. These four algorithms are Parallel Superconvergent Multigrid (PSMG) of Frederickson and McBryan, Robust Multigrid (RMG) of Hackbusch, the FFT based Spectral Algorithm, and Parallel Cyclic Reduction. In fact, all four can be formulated as particular cases of the same totally parallel multilevel algorithm, which are referred to as TPMA. In certain cases the spectral radius of TPMA is zero, and it is recognized to be a direct algorithm. In many other cases the spectral radius, although not zero, is small enough that a single iteration per timestep keeps the local error within the required tolerance
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