79,300 research outputs found
Multibody Multipole Methods
A three-body potential function can account for interactions among triples of
particles which are uncaptured by pairwise interaction functions such as
Coulombic or Lennard-Jones potentials. Likewise, a multibody potential of order
can account for interactions among -tuples of particles uncaptured by
interaction functions of lower orders. To date, the computation of multibody
potential functions for a large number of particles has not been possible due
to its scaling cost. In this paper we describe a fast tree-code for
efficiently approximating multibody potentials that can be factorized as
products of functions of pairwise distances. For the first time, we show how to
derive a Barnes-Hut type algorithm for handling interactions among more than
two particles. Our algorithm uses two approximation schemes: 1) a deterministic
series expansion-based method; 2) a Monte Carlo-based approximation based on
the central limit theorem. Our approach guarantees a user-specified bound on
the absolute or relative error in the computed potential with an asymptotic
probability guarantee. We provide speedup results on a three-body dispersion
potential, the Axilrod-Teller potential.Comment: To appear in Journal of Computational Physic
Perspectives on the Neuroscience of Cognition and Consciousness
The origin and current use of the concepts of computation, representation and information in Neuroscience are examined and conceptual flaws are identified which vitiate their usefulness for addressing problems of the neural basis of Cognition and Consciousness. In contrast, a convergence of views is presented to support the characterization of the Nervous System as a complex dynamical system operating in the metastable regime, and capable of evolving to configurations and transitions in phase space with potential relevance for Cognition and Consciousness
Polynomial Linear Programming with Gaussian Belief Propagation
Interior-point methods are state-of-the-art algorithms for solving linear
programming (LP) problems with polynomial complexity. Specifically, the
Karmarkar algorithm typically solves LP problems in time O(n^{3.5}), where
is the number of unknown variables. Karmarkar's celebrated algorithm is known
to be an instance of the log-barrier method using the Newton iteration. The
main computational overhead of this method is in inverting the Hessian matrix
of the Newton iteration. In this contribution, we propose the application of
the Gaussian belief propagation (GaBP) algorithm as part of an efficient and
distributed LP solver that exploits the sparse and symmetric structure of the
Hessian matrix and avoids the need for direct matrix inversion. This approach
shifts the computation from realm of linear algebra to that of probabilistic
inference on graphical models, thus applying GaBP as an efficient inference
engine. Our construction is general and can be used for any interior-point
algorithm which uses the Newton method, including non-linear program solvers.Comment: 7 pages, 1 figure, appeared in the 46th Annual Allerton Conference on
Communication, Control and Computing, Allerton House, Illinois, Sept. 200
Cortical region interactions and the functional role of apical dendrites
The basal and distal apical dendrites of pyramidal cells occupy distinct
cortical layers and are targeted by axons originating in different cortical
regions. Hence, apical and basal dendrites receive information from distinct
sources. Physiological evidence suggests that this anatomically observed
segregation of input sources may have functional significance. This possibility
has been explored in various connectionist models that employ neurons with
functionally distinct apical and basal compartments. A neuron in which separate
sets of inputs can be integrated independently has the potential to operate in a
variety of ways which are not possible for the conventional model of a neuron in
which all inputs are treated equally. This article thus considers how
functionally distinct apical and basal dendrites can contribute to the
information processing capacities of single neurons and, in particular, how
information from different cortical regions could have disparate affects on
neural activity and learning
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