Stochastic dynamo in critical situations

Based on the functional method of consecutive approximations, we consider the problem of magnetic field excitation (stochastic dynamo) by a random velocity field with a finite temporal correlation radius. In critical situations, in the first (diffusion) approximation, the Lyapunov characteristic parameter of the magnetic field energy vanishes. This implies the absence of structure formation (clustering) in realizations of the magnetic field in that approximation. Critical situations occur in problems of magnetic field diffusion in an equilibrium thermal and random pseudoequilibrium and acoustic (in the absence of dissipation) velocity fields. The sign of the Lyapunov characteristic parameter in the second-order approximation determines the possibility of clustering of the magnetic field energy. We show that energy clustering does not occur in a thermal velocity field. In the cases of pseudoequilibrium and acoustic velocity fields, clustering occurs with probability one, i.e., in almost every realization. We evaluate the characteristic time for clustering to be established.     

Stratified Monte Carlo Quadrature for Continuous Random Fields

We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behaviour is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the Hölder class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.     

Random walks systems on complete graphs

We study two versions of random walks systems on complete graphs. In the first one, the random walks have geometrically distributed lifetimes so we define and identify a non-trivial critical parameter related to the proportion of visited vertices before the process dies out. In the second version, the lifetimes depend on the past of the process in a non-Markovian setup. For that version, we present results obtained from computational analysis, simulations and a mean field approximation. These three approaches match.     

Transfer of a Passive Vector Admixture by a Two-Dimensional Turbulent Flow

We consider a model of a passive vector field transfer by a random two-dimensional transverse velocity field that is uncorrelated in time and has Gaussian spatial statistics given by a powerlike correlator. We use the renormalization group and the operator product expansion techniques to show that the asymptotic approximation of the structure functions of a vector field in the inertial range is determined by the energy dissipation fluctuations. The dependence of the asymptotic approximation on the external scale of turbulence is essential and has a powerlike form (the case of an anomalous scaling). The corresponding exponents are calculated in the one-loop approximation for structure functions of an arbitrary order.     

Estimation of intrinsic processes affected by additive fractal noise

Fractal Gaussian models have been widely used to represent the singular behavior of phenomena arising in different applied fields; for example, fractional Brownian motion and fractional Gaussian noise are considered as monofractal models in subsurface hydrology and geophysical studies Mandelbrot [The Fractal Geometry of Nature, Freeman Press, San Francisco, 1982 [13]]. In this paper, we address the problem of least-squares linear estimation of an intrinsic fractal input random field from the observation of an output random field affected by fractal noise (see Angulo et al. [Estimation and filtering of fractional generalised random fields, J. Austral. Math. Soc. A 69 (2000) 1-26 [2]], Ruiz-Medina et al. [Fractional generalized random fields on bounded domains, Stochastic Anal. Appl. 21 (2003a) 465-492], Ruiz-Medina et al. [Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields, J. Multivariate Anal. 85 (2003b) 192-216]. Conditions on the fractality order of the additive noise are studied to obtain a bounded inversion of the associated Wiener-Hopf equation. A stable solution is then obtained in terms of orthogonal bases of the reproducing kernel Hilbert spaces associated with the random fields involved. Such bases are constructed from orthonormal wavelet bases (see Angulo and Ruiz-Medina [Multiresolution approximation to the stochastic inverse problem, Adv. in Appl. Probab. 31 (1999) 1039-1057], Angulo et al. [Wavelet-based orthogonal expansions of fractional generalized random fields on bounded domains, Theoret. Probab. Math. Stat. (2004), in press]). A simulation study is carried out to illustrate the influence of the fractality orders of the output random field and the fractal additive noise on the stability of the solution derived.     

关于齐次树上随机场的一类强偏差定理

通过引进渐近对数似然比作为齐次树上任意Markov随机场逼近的一种度量,通过构造鞅的方法,建立了关于随机场的一类强偏差(也称小偏差)定理.所得结论推广了一个已知的结果.     

New Approximation Algorithms for Weighted Maximin Dispersion Problem with Box or Ball Constraints

In this paper, we propose new approximation algorithms for a NP-hard problem, i.e., weighted maximin dispersion problem. By using a uniformly distributed random sample method, we first propose a new random approximation algorithm for box constrained or ball constrained weighted maximin dispersion problems and analyze its approximation bound respectively. Moreover, we propose two improved approximation algorithms by combining our technique with an existing binary sample technique for both cases. To the best of our knowledge, they are the best approximation bounds for both box constrained and ball constrained weighted maximin dispersion problems respectively.

     

Variational Bayes and the Reduced Dependence Approximation for the Autologistic Model on an Irregular Grid With Applications

Discrete Markov random field models provide a natural framework for representing images or spatial datasets. They model the spatial association present while providing a convenient Markovian dependency structure and strong edge-preservation properties. However, parameter estimation for discrete Markov random field models is difficult due to the complex form of the associated normalizing constant for the likelihood function. For large lattices, the reduced dependence approximation to the normalizing constant is based on the concept of performing computationally efficient and feasible forward recursions on smaller sublattices, which are then suitably combined to estimate the constant for the entire lattice. We present an efficient computational extension of the forward recursion approach for the autologistic model to lattices that have an irregularly shaped boundary and that may contain regions with no data; these lattices are typical in applications. Consequently, we also extend the reduced dependence approximation to these scenarios, enabling us to implement a practical and efficient nonsimulation-based approach for spatial data analysis within the variational Bayesian framework. The methodology is illustrated through application to simulated data and example images. The online supplementary materials include our C++ source code for computing the approximate normalizing constant and simulation studies.     

Approximating an expectation related to cooperative robots

In this paper we develop an approximation to the expectation of a random variable implied in cooperation stability, presented in a previous work. This approximation is obtained by means of a continuous monotonous function that upper bounds the expectation. Finally, we analyze the quality of this approximation.     

A random flight process associated to a Lorentz gas with variable density in a gravitational field

We investigate a random flight process approximation to a random scatterer Lorentz gas with variable scatterer density in a gravitational field. For power function densities we show how the parameters of the model determine recurrence or transience of the vertical component of the trajectory. Finally, our methods show that, with appropriate scaling of space, time and the density of obstacles, the trajectory of the particle converges to a diffusion with explicitly given parameters.     

Some equilibrium problems under uncertainty and random variational inequalities

In this paper we describe some nonlinear equilibrium problems under uncertainty arising from economics and operations research. In particular we treat Wardrop equilibria in traffic networks. We show how the theory of monotone random variational inequalities, where random variables occur both in the operator and the constraint set, can be applied to model these problems. Therefore in this contribution we introduce the topic of random variational inequalities and present some of our recent results in this field. In particular, we treat the more structured case where a finite Karhunen-Loève expansion leads to a separation of the random and the deterministic variables. Here we describe a norm convergent approximation procedure based on averaging and truncation. We illustrate this procedure by means of some small sized numerical examples.     

The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables

In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks Λ statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution.     

Efficient approximation of random fields for numerical applications

This article is dedicated to the rapid computation of separable expansions for the approximation of random fields. We consider approaches based on techniques from the approximation of non‐local operators on the one hand and based on the pivoted Cholesky decomposition on the other hand. Especially, we provide an a posteriori error estimate for the pivoted Cholesky decomposition in terms of the trace. Numerical examples are provided to validate and quantify the presented methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Mittag-Leffler vector random fields with Mittag-Leffler direct and cross covariance functions

In terms of the two-parameter Mittag-Leffler function with specified parameters, this paper introduces the Mittag-Leffler vector random field through its finite-dimensional characteristic functions, which is essentially an elliptically contoured one and reduces to a Gaussian one when the two parameters of the Mittag-Leffler function equal 1. Having second-order moments, a Mittag-Leffler vector random field is characterized by its mean function and its covariance matrix function, just like a Gaussian one. In particular, we construct direct and cross covariances of Mittag-Leffler type for such vector random fields.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

On the Lattice Approximation for Certain Generalized Vector Markov Fields in Four Spacetime Dimensions

We extend previous results by Albeverio, Iwata and Schmidt on the construction of a convergent lattice approximation for invariant scalar 3-vector generalized random fields F of an infinitely divisible type and apply them to the construction of convergent lattice approximation for the generalized random vector field A determined by the stochastic quaternionic Cauchy–Riemann equation A = F.     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

One of the open problems in the field of forward uncertainty quantification(UQ) is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs. Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems, particularly with high dimensional random parameters. We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems. The first task in this two-step process is to employ the procedure developed in [1] to construct an "arbitrary" polynomial chaos expansion basis using a finite number of statistical moments of the random inputs. The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty. To enhance the performance of the preconditioned l1 minimization problem, we sample from the so-called induced distribution, instead of using Monte Carlo (MC) sampling from the original, unknown probability measure. We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure) when we have incomplete information about the distribution. We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions, and on a Kirchoff plating bending problem with random Young's modulus.     

Regularity,continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces

Gaussian random fields defined over compact two-point homogeneous spaces are considered and Sobolev regularity and Hölder continuity are explored through spectral representations. It is shown how spectral properties of the covariance function associated to a given Gaussian random field are crucial to determine such regularities and geometric properties. Furthermore, fast approximations of random fields on compact two-point homogeneous spaces are derived by truncation of the series expansion, and a suitable bound for the error involved in such an approximation is provided.