An invariant subspace-based approach to the random eigenvalue problem of systems with clustered spectrum

The repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non-unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal-based reduced-order model of the system. Copyright (C) 2012 John Wiley & Sons, Ltd.

Self-organized criticality in dynamics without branching

We demonstrate the phenomenon of self-organized criticality (SOC) in a simple random walk model described by a random walk of a myopic ant, i.e., a walker who can see only nearest neighbors. The ant acts on the underlying lattice aiming at uniform digging, i.e., reduction of the height profile of the surface but is unaffected by the underlying lattice. In one, two, and three dimensions we have explored this model and have obtained power laws in the time intervals between consecutive events of "digging." Being a simple random walk, the power laws in space translate to power laws in time. We also study the finite size scaling of asymptotic scale invariant process as well as dynamic scaling in this system. This model differs qualitatively from the cascade models of SOC.

Speeding up AdaBoost Classifier with Random Projection

The development of techniques for scaling up classifiers so that they can be applied to problems with large datasets of training examples is one of the objectives of data mining. Recently, AdaBoost has become popular among machine learning community thanks to its promising results across a variety of applications. However, training AdaBoost on large datasets is a major problem, especially when the dimensionality of the data is very high. This paper discusses the effect of high dimensionality on the training process of AdaBoost. Two preprocessing options to reduce dimensionality, namely the principal component analysis and random projection are briefly examined. Random projection subject to a probabilistic length preserving transformation is explored further as a computationally light preprocessing step. The experimental results obtained demonstrate the effectiveness of the proposed training process for handling high dimensional large datasets.

Quantum-Ohmic Resistance Fluctuation In Disordered Conductors - An Invariant Imbedding Approach

It is now well known that in extreme quantum limit, dominated by the elastic impurity scattering and the concomitant quantum interference, the zero-temperature d.c. resistance of a strictly one-dimensional disordered system is non-additive and non-self-averaging. While these statistical fluctuations may persist in the case of a physically thin wire, they are implicitly and questionably ignored in higher dimensions. In this work, we have re-examined this question. Following an invariant imbedding formulation, we first derive a stochastic differential equation for the complex amplitude reflection coefficient and hence obtain a Fokker-Planck equation for the full probability distribution of resistance for a one-dimensional continuum with a Gaussian white-noise random potential. We then employ the Migdal-Kadanoff type bond moving procedure and derive the d-dimensional generalization of the above probability distribution, or rather the associated cumulant function –‘the free energy’. For d=3, our analysis shows that the dispersion dominates the mobilitly edge phenomena in that (i) a one-parameter B-function depending on the mean conductance only does not exist, (ii) an approximate treatment gives a diffusion-correction involving the second cumulant. It is, however, not clear whether the fluctuations can render the transition at the mobility edge ‘first-order’. We also report some analytical results for the case of the one dimensional system in the presence of a finite electric fiekl. We find a cross-over from the exponential to the power-low length dependence of resistance as the field increases from zero. Also, the distribution of resistance saturates asymptotically to a poissonian form. Most of our analytical results are supported by the recent numerical simulation work reported by some authors.

Effect of creep deformations in mechanical systems under combined deterministic and random inputs

The effect of creep on the vibrations of a single degree of freedom system subjected to combined random and deterministic excitation has been studied in this paper. The deterministic part of the excitation is assumed to be a sinusoidal function while the random part of the excitation is considered as a narrow band process with a central frequency equal to the frequency of sinusoidal part of the excitation. Creep, an energy absorbing process, introduces an equivalent damping into the system. A measure of this damping and the statistical properties of the response of the mechanical system have been derived.

Modelling of solute transport in a mild heterogeneous porous medium using stochastic finite element method: Effects of random source conditions

Randomness in the source condition other than the heterogeneity in the system parameters can also be a major source of uncertainty in the concentration field. Hence, a more general form of the problem formulation is necessary to consider randomness in both source condition and system parameters. When the source varies with time, the unsteady problem, can be solved using the unit response function. In the case of random system parameters, the response function becomes a random function and depends on the randomness in the system parameters. In the present study, the source is modelled as a random discrete process with either a fixed interval or a random interval (the Poisson process). In this study, an attempt is made to assess the relative effects of various types of source uncertainties on the probabilistic behaviour of the concentration in a porous medium while the system parameters are also modelled as random fields. Analytical expressions of mean and covariance of concentration due to random discrete source are derived in terms of mean and covariance of unit response function. The probabilistic behaviour of the random response function is obtained by using a perturbation-based stochastic finite element method (SFEM), which performs well for mild heterogeneity. The proposed method is applied for analysing both the 1-D as well as the 3-D solute transport problems. The results obtained with SFEM are compared with the Monte Carlo simulation for 1-D problems.

Stability of scheduled multi-access communication over quasi-static flat fading channels with random coding and independent decoding

The stability of scheduled multiaccess communication with random coding and independent decoding of messages is investigated. The number of messages that may be scheduled for simultaneous transmission is limited to a given maximum value, and the channels from transmitters to receiver are quasistatic, flat, and have independent fades. Requests for message transmissions are assumed to arrive according to an i.i.d. arrival process. Then, we show the following: (1) in the limit of large message alphabet size, the stability region has an interference limited information-theoretic capacity interpretation, (2) state-independent scheduling policies achieve this asymptotic stability region, and (3) in the asymptotic limit corresponding to immediate access, the stability region for non-idling scheduling policies is shown to be identical irrespective of received signal powers.

Monte-Carlo estimation of time-dependent statistical characteristics of random dynamical systems

The problem of estimating the time-dependent statistical characteristics of a random dynamical system is studied under two different settings. In the first, the system dynamics is governed by a differential equation parameterized by a random parameter, while in the second, this is governed by a differential equation with an underlying parameter sequence characterized by a continuous time Markov chain. We propose, for the first time in the literature, stochastic approximation algorithms for estimating various time-dependent process characteristics of the system. In particular, we provide efficient estimators for quantities such as the mean, variance and distribution of the process at any given time as well as the joint distribution and the autocorrelation coefficient at different times. A novel aspect of our approach is that we assume that information on the parameter model (i.e., its distribution in the first case and transition probabilities of the Markov chain in the second) is not available in either case. This is unlike most other work in the literature that assumes availability of such information. Also, most of the prior work in the literature is geared towards analyzing the steady-state system behavior of the random dynamical system while our focus is on analyzing the time-dependent statistical characteristics which are in general difficult to obtain. We prove the almost sure convergence of our stochastic approximation scheme in each case to the true value of the quantity being estimated. We provide a general class of strongly consistent estimators for the aforementioned statistical quantities with regular sample average estimators being a specific instance of these. We also present an application of the proposed scheme on a widely used model in population biology. Numerical experiments in this framework show that the time-dependent process characteristics as obtained using our algorithm in each case exhibit excellent agreement with exact results. (C) 2010 Elsevier Inc. All rights reserved.

Performance analysis of the random early detection algorithm

In this article we consider a finite queue with its arrivals controlled by the random early detection algorithm. This is one of the most prominent congestion avoidance schemes in the Internet routers. The aggregate arrival stream from the population of transmission control protocol sources is locally considered stationary renewal or Markov modulated Poisson process with general packet length distribution. We study the exact dynamics of this queue and provide the stability and the rates of convergence to the stationary distribution and obtain the packet loss probability and the waiting time distribution. Then we extend these results to a two traffic class case with each arrival stream renewal. However, computing the performance indices for this system becomes computationally prohibitive. Thus, in the latter half of the article, we approximate the dynamics of the average queue length process asymptotically via an ordinary differential equation. We estimate the error term via a diffusion approximation. We use these results to obtain approximate transient and stationary performance of the system. Finally, we provide some computational examples to show the accuracy of these approximations.

Evolving Random Geometric Graph Models for Mobile Wireless Networks

We consider evolving exponential RGGs in one dimension and characterize the time dependent behavior of some of their topological properties. We consider two evolution models and study one of them detail while providing a summary of the results for the other. In the first model, the inter-nodal gaps evolve according to an exponential AR(1) process that makes the stationary distribution of the node locations exponential. For this model we obtain the one-step conditional connectivity probabilities and extend it to the k-step case. Finite and asymptotic analysis are given. We then obtain the k-step connectivity probability conditioned on the network being disconnected. We also derive the pmf of the first passage time for a connected network to become disconnected. We then describe a random birth-death model where at each instant, the node locations evolve according to an AR(1) process. In addition, a random node is allowed to die while giving birth to a node at another location. We derive properties similar to those above.

Nonuniform random geometric graphs with location-dependent radii

We propose a distribution-free approach to the study of random geometric graphs. The distribution of vertices follows a Poisson point process with intensity function n f(center dot), where n is an element of N, and f is a probability density function on R-d. A vertex located at x connects via directed edges to other vertices that are within a cut-off distance r(n)(x). We prove strong law results for (i) the critical cut-off function so that almost surely, the graph does not contain any node with out-degree zero for sufficiently large n and (ii) the maximum and minimum vertex degrees. We also provide a characterization of the cut-off function for which the number of nodes with out-degree zero converges in distribution to a Poisson random variable. We illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero. Finally, we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually.

Asymptotics of the invariant measure in mean field models with jumps

We consider the asymptotics of the invariant measure for the process of spatial distribution of N coupled Markov chains in the limit of a large number of chains. Each chain reflects the stochastic evolution of one particle. The chains are coupled through the dependence of transition rates on the spatial distribution of particles in the various states. Our model is a caricature for medium access interactions in wireless local area networks. Our model is also applicable in the study of spread of epidemics in a network. The limiting process satisfies a deterministic ordinary differential equation called the McKean-Vlasov equation. When this differential equation has a unique globally asymptotically stable equilibrium, the spatial distribution converges weakly to this equilibrium. Using a control-theoretic approach, we examine the question of a large deviation from this equilibrium.

EFFICIENT QUANTUM RANDOM NUMBER GENERATION USING QUANTUM INDISTINGUISHABILITY

In this paper, we propose a quantum method for generation of random numbers based on bosonic stimulation. Randomness arises through the path-dependent indeterministic amplification of two competing bosonic modes. We show that the process provides an efficient method for macroscopic extraction of microscopic randomness.

Optimal sequential wireless relay placement on a random lattice path

Our work is motivated by impromptu (or ``as-you-go'') deployment of wireless relay nodes along a path, a need that arises in many situations. In this paper, the path is modeled as starting at the origin (where there is the data sink, e.g., the control center), and evolving randomly over a lattice in the positive quadrant. A person walks along the path deploying relay nodes as he goes. At each step, the path can, randomly, either continue in the same direction or take a turn, or come to an end, at which point a data source (e.g., a sensor) has to be placed, that will send packets to the data sink. A decision has to be made at each step whether or not to place a wireless relay node. Assuming that the packet generation rate by the source is very low, and simple link-by-link scheduling, we consider the problem of sequential relay placement so as to minimize the expectation of an end-to-end cost metric (a linear combination of the sum of convex hop costs and the number of relays placed). This impromptu relay placement problem is formulated as a total cost Markov decision process. First, we derive the optimal policy in terms of an optimal placement set and show that this set is characterized by a boundary (with respect to the position of the last placed relay) beyond which it is optimal to place the next relay. Next, based on a simpler one-step-look-ahead characterization of the optimal policy, we propose an algorithm which is proved to converge to the optimal placement set in a finite number of steps and which is faster than value iteration. We show by simulations that the distance threshold based heuristic, usually assumed in the literature, is close to the optimal, provided that the threshold distance is carefully chosen. (C) 2014 Elsevier B.V. All rights reserved.