Resource allocation in sparse graphs

Resource allocation in sparsely connected networks, a representative problem of systems with real variables, is studied using the replica and Bethe approximation methods. An efficient distributed algorithm is devised on the basis of insights gained from the analysis and is examined using numerical simulations,showing excellent performance and full agreement with the theoretical results. The physical properties of the resource allocation model are discussed.

Sparse on-line Gaussian processes

We develop an approach for sparse representations of Gaussian Process (GP) models (which are Bayesian types of kernel machines) in order to overcome their limitations for large data sets. The method is based on a combination of a Bayesian online algorithm together with a sequential construction of a relevant subsample of the data which fully specifies the prediction of the GP model. By using an appealing parametrisation and projection techniques that use the RKHS norm, recursions for the effective parameters and a sparse Gaussian approximation of the posterior process are obtained. This allows both for a propagation of predictions as well as of Bayesian error measures. The significance and robustness of our approach is demonstrated on a variety of experiments.

Inference and optimization of real edges on sparse graphs:A statistical physics perspective

Inference and optimization of real-value edge variables in sparse graphs are studied using the Bethe approximation and replica method of statistical physics. Equilibrium states of general energy functions involving a large set of real edge variables that interact at the network nodes are obtained in various cases. When applied to the representative problem of network resource allocation, efficient distributed algorithms are also devised. Scaling properties with respect to the network connectivity and the resource availability are found, and links to probabilistic Bayesian approximation methods are established. Different cost measures are considered and algorithmic solutions in the various cases are devised and examined numerically. Simulation results are in full agreement with the theory. © 2007 The American Physical Society.

Influence of memory in deterministic walks in random media: Analytical calculation within a mean-field approximation

Consider a random medium consisting of N points randomly distributed so that there is no correlation among the distances separating them. This is the random link model, which is the high dimensionality limit (mean-field approximation) for the Euclidean random point structure. In the random link model, at discrete time steps, a walker moves to the nearest point, which has not been visited in the last mu steps (memory), producing a deterministic partially self-avoiding walk (the tourist walk). We have analytically obtained the distribution of the number n of points explored by the walker with memory mu=2, as well as the transient and period joint distribution. This result enables us to explain the abrupt change in the exploratory behavior between the cases mu=1 (memoryless walker, driven by extreme value statistics) and mu=2 (walker with memory, driven by combinatorial statistics). In the mu=1 case, the mean newly visited points in the thermodynamic limit (N >> 1) is just < n >=e=2.72... while in the mu=2 case, the mean number < n > of visited points grows proportionally to N(1/2). Also, this result allows us to establish an equivalence between the random link model with mu=2 and random map (uncorrelated back and forth distances) with mu=0 and the abrupt change between the probabilities for null transient time and subsequent ones.

Approximation to density functional theory for the calculation of band gaps of semiconductors

The local-density approximation (LDA) together with the half occupation (transitionstate) is notoriously successful in the calculation of atomic ionization potentials. When it comes to extended systems, such as a semiconductor infinite system, it has been very difficult to find a way to half ionize because the hole tends to be infinitely extended (a Bloch wave). The answer to this problem lies in the LDA formalism itself. One proves that the half occupation is equivalent to introducing the hole self-energy (electrostatic and exchange correlation) into the Schrodinger equation. The argument then becomes simple: The eigenvalue minus the self-energy has to be minimized because the atom has a minimal energy. Then one simply proves that the hole is localized, not infinitely extended, because it must have maximal self-energy. Then one also arrives at an equation similar to the self- interaction correction equation, but corrected for the removal of just 1/2 electron. Applied to the calculation of band gaps and effective masses, we use the self- energy calculated in atoms and attain a precision similar to that of GW, but with the great advantage that it requires no more computational effort than standard LDA.

Nonuniversal behavior for aperiodic interactions within a mean-field approximation

We study the spin-1/2 Ising model on a Bethe lattice in the mean-field limit, with the interaction constants following one of two deterministic aperiodic sequences, the Fibonacci or period-doubling one. New algorithms of sequence generation were implemented, which were fundamental in obtaining long sequences and, therefore, precise results. We calculate the exact critical temperature for both sequences, as well as the critical exponents beta, gamma, and delta. For the Fibonacci sequence, the exponents are classical, while for the period-doubling one they depend on the ratio between the two exchange constants. The usual relations between critical exponents are satisfied, within error bars, for the period-doubling sequence. Therefore, we show that mean-field-like procedures may lead to nonclassical critical exponents.

A STUDY OF PENALTY FORMULATIONS USED IN THE NUMERICAL APPROXIMATION OF A RADIALLY SYMMETRIC ELASTICITY PROBLEM

We consider a class of two-dimensional problems in classical linear elasticity for which material overlapping occurs in the absence of singularities. Of course, material overlapping is not physically realistic, and one possible way to prevent it uses a constrained minimization theory. In this theory, a minimization problem consists of minimizing the total potential energy of a linear elastic body subject to the constraint that the deformation field must be locally invertible. Here, we use an interior and an exterior penalty formulation of the minimization problem together with both a standard finite element method and classical nonlinear programming techniques to compute the minimizers. We compare both formulations by solving a plane problem numerically in the context of the constrained minimization theory. The problem has a closed-form solution, which is used to validate the numerical results. This solution is regular everywhere, including the boundary. In particular, we show numerical results which indicate that, for a fixed finite element mesh, the sequences of numerical solutions obtained with both the interior and the exterior penalty formulations converge to the same limit function as the penalization is enforced. This limit function yields an approximate deformation field to the plane problem that is locally invertible at all points in the domain. As the mesh is refined, this field converges to the exact solution of the plane problem.

Discrete approximation to the global spectrum of the tangent operator for flow past a circular cylinder

This paper deals with the calculation of the discrete approximation to the full spectrum for the tangent operator for the stability problem of the symmetric flow past a circular cylinder. It is also concerned with the localization of the Hopf bifurcation in laminar flow past a cylinder, when the stationary solution loses stability and often becomes periodic in time. The main problem is to determine the critical Reynolds number for which a pair of eigenvalues crosses the imaginary axis. We thus present a divergence-free method, based on a decoupling of the vector of velocities in the saddle-point system from the vector of pressures, allowing the computation of eigenvalues, from which we can deduce the fundamental frequency of the time-periodic solution. The calculation showed that stability is lost through a symmetry-breaking Hopf bifurcation and that the critical Reynolds number is in agreement with the value presented in reported computations. (c) 2007 IMACS. Published by Elsevier B.V. All rights reserved.

On the convergence of successive Galerkin approximation for nonlinear output feedback H(infinity) control

Although the formulation of the nonlinear theory of H(infinity) control has been well developed, solving the Hamilton-Jacobi-Isaacs equation remains a challenge and is the major bottleneck for practical application of the theory. Several numerical methods have been proposed for its solution. In this paper, results on convergence and stability for a successive Galerkin approximation approach for nonlinear H(infinity) control via output feedback are presented. An example is presented illustrating the application of the algorithm.

Quantum mechanics as an approximation to classical mechanics in Hilbert space

Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.

A master equation model for bimolecular reaction via multi-well isomerization intermediates

A reversible linear master equation model is presented for pressure- and temperature-dependent bimolecular reactions proceeding via multiple long-lived intermediates. This kinetic treatment, which applies when the reactions are measured under pseudo-first-order conditions, facilitates accurate and efficient simulation of the time dependence of the populations of reactants, intermediate species and products. Detailed exploratory calculations have been carried out to demonstrate the capabilities of the approach, with applications to the bimolecular association reaction C3H6 + H reversible arrow C3H7 and the bimolecular chemical activation reaction C2H2 +(CH2)-C-1--> C3H3+H. The efficiency of the method can be dramatically enhanced through use of a diffusion approximation to the master equation, and a methodology for exploiting the sparse structure of the resulting rate matrix is established.

A numerical study of large sparse matrix exponentials arising in Markov chains

Krylov subspace techniques have been shown to yield robust methods for the numerical computation of large sparse matrix exponentials and especially the transient solutions of Markov Chains. The attractiveness of these methods results from the fact that they allow us to compute the action of a matrix exponential operator on an operand vector without having to compute, explicitly, the matrix exponential in isolation. In this paper we compare a Krylov-based method with some of the current approaches used for computing transient solutions of Markov chains. After a brief synthesis of the features of the methods used, wide-ranging numerical comparisons are performed on a power challenge array supercomputer on three different models. (C) 1999 Elsevier Science B.V. All rights reserved.AMS Classification: 65F99; 65L05; 65U05.

Expectation Propagation with Factorizing Distributions: A Gaussian Approximation and Performance Results for Simple Models

We discuss the expectation propagation (EP) algorithm for approximate Bayesian inference using a factorizing posterior approximation. For neural network models, we use a central limit theorem argument to make EP tractable when the number of parameters is large. For two types of models, we show that EP can achieve optimal generalization performance when data are drawn from a simple distribution.

Two-dimensional approximation for tidal dynamics in coastal aquifers: Capillarity correction

Most previous investigations on tide-induced watertable fluctuations in coastal aquifers have been based on one-dimensional models that describe the processes in the cross-shore direction alone, assuming negligible along-shore variability. A recent study proposed a two-dimensional approximation for tide-induced watertable fluctuations that took into account coastline variations. Here, we further develop this approximation in two ways, by extending the approximation to second order and by taking into account capillary effects. Our results demonstrate that both effects can markedly influence watertable fluctuations. In particular, with the first-order approximation, the local damping rate of the tidal signal could be subject to sizable errors.

Alternatives for parallel Krylov subspace basis computation

Numerical methods related to Krylov subspaces are widely used in large sparse numerical linear algebra. Vectors in these subspaces are manipulated via their representation onto orthonormal bases. Nowadays, on serial computers, the method of Arnoldi is considered as a reliable technique for constructing such bases. However, although easily parallelizable, this technique is not as scalable as expected for communications. In this work we examine alternative methods aimed at overcoming this drawback. Since they retrieve upon completion the same information as Arnoldi's algorithm does, they enable us to design a wide family of stable and scalable Krylov approximation methods for various parallel environments. We present timing results obtained from their implementation on two distributed-memory multiprocessor supercomputers: the Intel Paragon and the IBM Scalable POWERparallel SP2. (C) 1997 by John Wiley & Sons, Ltd.