Uncorrelated Random Fields Research Articles

Test of the Latent Dimension of a Spatial Blind Source Separation Model

We assume a spatial blind source separation model in which the observed multivariate spatial data is a linear mixture of latent spatially uncorrelated Gaussian random fields containing a number of pure white noise components. We propose a test on the number of white noise components and obtain the asymptotic distribution of its statistic for a general domain. We also demonstrate how computations can be facilitated in the case of gridded observation locations. Based on this test, we obtain a consistent estimator of the true dimension. Simulation studies and an environmental application demonstrate that our test is at least comparable to and often outperforms bootstrap-based techniques, which are also introduced in this paper.

Robust second-order stationary spatial blind source separation using generalized sign matrices

Consider a spatial blind source separation model in which the observed multivariate spatial data are assumed to be a linear mixture of latent stationary spatially uncorrelated random fields. The objective is to recover an unknown mixing procedure as well as the latent random fields. Recently, spatial blind source separation methods that are based on the simultaneous diagonalization of two or more scatter matrices were proposed. In cases involving uncontaminated data, such methods can solve the blind source separation problem, however, in the presence of outlying observations, these methods perform poorly. We propose a robust blind source separation method that employs robust global and local covariance matrices based on generalized spatial signs in simultaneous diagonalization. Simulation studies are employed to illustrate the robustness and efficiency of the proposed methods in various scenarios.

One-loop functional renormalization group study for the dimensional reduction and its breakdown in the long-range random field O( N ) spin model near lower critical dimension

We consider the random-field O($N$) spin model with long-range exchange interactions which decay with distance $r$ between spins as $r^{-d-\sigma}$ and/or random fields which correlate with distance $r$ as $r^{-d+\rho}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\to d-\sigma$ dimensional reduction at the leading order of the $d-2\sigma$ expansion holds only for $N>2(4+3{\sqrt{3}})\simeq 18.3923\cdots$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B {\bf{8}}, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\to d-\sigma-\rho$ dimensional reduction does not hold within the present framework, as far as $N$ is finite.

A probabilistic framework for stability assessment of existing spatial structures

In the safety assessment of existing spatial structures, sampling measurements are widely used to investigate structural geometries. The information from sampling measurements is limited and leads to the positional uncertainty of unmeasured nodes. Uncertain nodal positions have a significant influence on the stability assessment of the spatial structures with high geometric-imperfection sensitivity. To deal with the uncertainty of nodal positions, a probabilistic framework based on the randomness of nodal positional deviations was proposed. In the framework, uncertain nodal positions are modeled by random field, whose relevant parameters were inferred with the data from sampling measurements; based on the random field model, a full-probability analysis is adopted to determine the load-bearing capacity and further perform the stability assessment. According to the analysis of the correlation between nodal positions, correlated and uncorrelated random fields were respectively introduced to model the correlated and uncorrelated nodal deviations. And the detailed modeling methods were discussed, including the inference of distribution types, parameters and correlations. Finally, an existing single-layer reticulated shell was used to validate the reliability of the proposed probabilistic framework.

Visibility graphs of random scalar fields and spatial data.

We extend the family of visibility algorithms to map scalar fields of arbitrary dimension into graphs, enabling the analysis of spatially extended data structures as networks. We introduce several possible extensions and provide analytical results on the topological properties of the graphs associated to different types of real-valued matrices, which can be understood as the high and low disorder limits of real-valued scalar fields. In particular, we find a closed expression for the degree distribution of these graphs associated to uncorrelated random fields of generic dimension. This result holds independently of the field's marginal distribution and it directly yields a statistical randomness test, applicable in any dimension. We showcase its usefulness by discriminating spatial snapshots of two-dimensional white noise from snapshots of a two-dimensional lattice of diffusively coupled chaotic maps, a system that generates high dimensional spatiotemporal chaos. The range of potential applications of this combinatorial framework includes image processing in engineering, the description of surface growth in material science, soft matter or medicine, and the characterization of potential energy surfaces in chemistry, disordered systems, and high energy physics. An illustration on the applicability of this method for the classification of the different stages involved in carcinogenesis is briefly discussed.

Modeling nonstationary covariance function with convolution on sphere

The wide use of satellite-based instruments provides measurements in climatology on a global scale, which often have nonstationary covariance structure. The issue of modeling a spatial random fields on sphere which is stationary across longitudes is addressed with a kernel convolution approach. The observed random field is generated by convolving a latent uncorrelated random field with a class of Matérn type kernel functions. By allowing the parameters in the kernel functions to vary with locations, it is possible to generate a flexible class of covariance functions and capture the nonstationary properties. Since the corresponding covariance functions generally do not have a closed form, numerical evaluations are necessary and a pre-computation table is used to speed up the computation. For regular grid data on sphere, the circulant block property of the covariance matrix enables us to use Fast Fourier Transform (FFT) to get its determinant and inverse matrix efficiently. The proposed approach is applied to the Total Ozone Mapping Spectrometer (TOMS) data for illustration.

Estimation of the Effective Permeability of Heterogeneous Porous Media by Using Percolation Concepts

In this paper we present new methods to estimate the effective permeability ( $$k_\mathrm{eff}$$ ) of heterogeneous porous media with a wide distribution of permeabilities and various underlying structures, using percolation concepts. We first set a threshold permeability ( $$k_\mathrm{th}$$ ) on the permeability density function and use standard algorithms from percolation theory to check whether the high permeable grid blocks (i.e., those with permeability higher than $$k_\mathrm{th}$$ ) with occupied fraction of “p” first forms a cluster connecting two opposite sides of the system in the direction of the flow (high permeability flow pathway). Then we estimate the effective permeability of the heterogeneous porous media in different ways: a power law ( $$k_\mathrm{eff} =k_\mathrm{th} p^{m}$$ ), a weighted power average ( $$k_\mathrm{eff} =\left[ {p\cdot k_\mathrm{th}^m +\left( {1-p} \right) \cdot k_\mathrm{g}^m } \right] ^{1/m}$$ with $$k_\mathrm{g}$$ the geometric average of the permeability distribution) and a characteristic shape factor multiplied by the permeability threshold value. We found that the characteristic parameters (i.e., the exponent “m”) can be inferred either from the statistics and properties of percolation subnetworks at the threshold point (i.e., high and low permeable regions corresponding to those permeabilities above and below the threshold permeability value) or by comparing the system properties with an uncorrelated random field having the same permeability distribution. These physically based approaches do not need fitting to the experimental data of effective permeability measurements to estimate the model parameter (i.e., exponent m) as is usually necessary in empirical methods. We examine the order of accuracy of these methods on different layers of $$10{\mathrm{th}}$$ SPE model and found very good estimates as compared to the values determined from the commercial flow simulators.

Macroscopic probabilistic cracking approach for the numerical modelling of fluid leakage in concrete

The article presents a numerical finite element study of fluid leakage in concrete. Concrete cracking is numerically modelled in the framework of a macroscopic probabilistic approach. Material heterogeneity and the re- lated mechanical effects are taken into account by defining the elementary mechanical properties according to spatially uncorrelated random fields. Each finite element is consid- ered as representative of a volume of heterogeneous ma- terial, whose mechanical behaviour depends on its own volume. The parameters of the statistical distributions defining the elementary mechanical properties thus vary over the computational mesh element-by-element. A weak hydro-mechanical coupling assumption is introduced to represent the influence of cracking on the variation of transfer properties: it is assumed that the mechanical cracking of a finite element induces a loss of isotropy of its own permeability tensor. At the elementary level, an ex- perimentally enhanced parallel plates model is used to re- late the local crack permeability to the elementary crack aperture. A Monte Carlo-like approach allows to statisti- cally validate the numerical method. The self-consistency of the proposed modelling strategy is finally explored through the numerical simulation of the hydro-mechanical splitting test, recently proposed by authors to evaluate the real-time evolution of the transfer properties of a concrete sample under loading.

Computational studies of history dependence in nematic liquid crystals in random environments.

Glassy liquid crystalline systems are expected to show significant history-dependent effects. Two model glassy systems are the RAN and SSS (sprinkled silica spin) lattice models. The RAN model is a Lebwohl-Lasher lattice model with locally coupled nematic spins, together with uncorrelated random anisotropy fields at each site, while the SSS model has a finite concentration of impurity spins frozen in random directions. Here Brownian simulation is used to study the effect of different sample histories in the low temperature regime in a three-dimensional (d = 3) model intermediate between SSS and RAN, in which a finite concentration p < p(c) (p(c) the percolation threshold) of frozen spins interacts with neighboring nematic spins with coupling W. Simulations were performed at temperature T ∼ T(NI)/2 (T(NI) the bulk nematic-isotropic transition temperature) for temperature-quenched and field-quenched histories (TQH and FQH, respectively), as well as for temperature-annealed histories (AH). The first two of these limits represent extreme histories encountered in typical experimental studies. Using long-time averages for equilibrated systems, we calculate orientational order parameters and two-point correlation functions. Finite-size scaling was used to determine the range of the orientational ordering, as a function of coupling strength W,p and sample history. Sample history plays a significant role; for given concentration p, as disorder strength W is increased, TQH systems sustain quasi-long-range order (QLRO) and short-range order (SRO). The data are also consistent with a long-range order (LRO) phase at very low disorder strength. By contrast, for FQH and p ≤ 0.1, only LRO and QLRO occur within the range of parameters investigated. The crossover between regimes depends on history, but in general, the FQH phase is more ordered than the AH phase, which is more ordered than the TQH phase. However, at temperatures close to the isotropic-nematic phase transition of pure samples we observe SRO for p = 0.1 even for FQH. We detect also in the QLRO phase a domain-type structural pattern, consistent with ideas introduced by Giamarchi and Doussal [Phys. Rev. B 52, 1242 (1995)] on superconducting flux lattices. In the weak-disorder limit the orientational correlation length obeys the Larkin-Imry-Ma scaling ξ ∼ D(-2/(4-d)).

On helical multiscale characterization of homogeneous turbulence

The helical properties of five prototypical homogeneous turbulent flows are investigated: statistically steady forced isotropic turbulence, decaying isotropic turbulence, decaying rotating turbulence, growing sheared turbulence, and growing rotating sheared turbulence with a rotation ratio f/S=+0.5. The five turbulent flows were originally studied using direct numerical simulations, and well-developed flow fields are chosen for this analysis. For comparison, a solenoidal uncorrelated Gaussian random field is included in the analysis as a sixth case. An orthogonal wavelet decomposition is used to study the scale-dependent properties of the cases. It is found that flows with growing turbulent kinetic energy and turbulent motion at large scales show a maximum in the relative kinetic helicity probability distribution functions (PDFs) at zero, corresponding to a trend to local two-dimensionalization of the flow with vorticity and velocity tending to be perpendicular. Flows with decaying turbulent kinetic energy and turbulent motion at small scales, however, show maxima of the relative kinetic helicity PDFs at plus and minus one, indicating a preference for helical motion with a trend to alignment or anti-alignment of vorticity and velocity. The PDFs of relative super-helicity always assume maxima at plus and minus one for all flows. The helical properties of statistically steady forced isotropic turbulence follow those of flows with decaying turbulent kinetic energy and a small asymmetry in the relative helicity PDFs is observed. Joint PDFs of relative kinetic helicity and relative super-helicity show that the quantities tend to have the same sign for all flows, including the random field, indicating that super-helicity dissipates kinetic helicity.

Reliability of silo structure with geometric imperfections

AbstractReliability of a silo structure with geometric imperfections is estimated. As the silo cylindrical constructions can be loaded with pressure below the atmospheric any geometric discrepancies are very dangerous and can significantly reduce their capacities. A probabilistic approach was applied. Two cases were considered. The imperfections were described by a uniform and a truncated Gaussian distribution. In the first case an uncorrelated random field was considered while in the second a theoretical nonhomogeneous correlation function was formulated. The realizations of the random imperfection fields are generated using a conditional rejection method. A version of a stratified sampling method is applied. (© 2009 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Effects of evolving scales of heterogeneity on hydraulic head predictions under convergent flow conditions

Two-dimensional steady-state flow towards a well that fully penetrates a randomly heterogeneous aquifer with a deterministically prescribed constant head boundary is considered. Flow occurs over an infinite hierarchy of mutually uncorrelated, statistically homogeneous and isotropic random fields (modes) of natural log-transmissivity, Y, each of which is associated with a Gaussian variogram. A lower and upper cut-off of the hierarchy, respectively related to the length scale of the domain and data support (sample volume), is introduced. This allows the direct incorporation of the scale dependence of the integral scale of Y into groundwater flow (ensemble) moments and leads to a geostatistical description of the system in terms of a (stationary) truncated power variogram (TPV). An analytical solution for the hydraulic head mean and variance is developed based on recursive approximations of exact nonlocal moment equations. The solution allows assessment of the functional dependences of the distribution of the leading (statistical) moments of hydraulic head on parameters of the variogram associated with the hierarchy of log-transmissivity modes.

Detecting the emergence of a signal in a noisy image

We study sequential change-point detection when obser- vations form a sequence of independent Gaussian random fields, and the change-point is the time at which a sig- nal of known functional form involving a finite number of unknown parameters appears. Building on Siegmund and Yakir (2008), which identifies in a simpler problem a detec- tion procedure of Shiryayev-Roberts type that is asymptot- ically minimax up to terms that vanish as the false detec- tion rate converges to zero, we compare easily computed ap- proximations to the Shiryayev-Roberts detection procedure with similar approximations to CUSUM type procedures. Although the CUSUM type procedures are suboptimal, our studies indicate that they compare favorably to the asymp- totically optimal procedures. gradually. Generally speaking, the problem is to detect the signal as soon as possible after it appears, under the con- straint that (false) detection occurs very rarely if no signal appears. To illustrate the principles involved we will assume that the observations consist of a sequence of uncorrelated Gaussian random fields and that the signal is a parameter- ized function of a known form superimposed on the noisy observations at an unknown change-point. The discussion will be formulated in the context of image analysis. Yet the results and the principles that we introduce are meaningful in other settings as well. Following the developments in the companion to this paper (12), we measure detection delay by the expected Kullback-Leibler information accumulated between the change-point and its detection. We begin in the next sub- section with a precise description of the model. In Subsec- tion 1.2 the criterion for asymptotic minimax optimality, which is stated and proved in the companion paper, is re- formulated to fit the current context. The asymptotic minimax policy uses a randomized form of the Shiryayev-Roberts monitoring scheme. The alterna- tive Cumulative Sum (CUSUM) monitoring scheme is a bet- ter known approach. In Section 2 expressions describing the asymptotic performance of optimal CUSUM and optimal Shiryayev-Roberts rules are obtained. Suboptimal formula- tions of these procedures are also assessed. It turns out that the natural candidate for optimal Shiryayev-Roberts and CUSUM rules may require substan- tial computation. In Section 3 we propose alternative rules which are asymptotically equivalent but require less com- putational effort. A simulation study is conducted in order to investigate the finite-sample properties of these simplified rules. The paper concludes with a discussion of related open problems. The analysis of the different detection methods draws on a substantial literature for its justification. The calculations are only sketched here and emphasize new features of the present formulation. See (12) and the references cited there, and (3).

Generation of turbulent inflow and initial conditions based on multi‐correlated random fields

AbstractDirect or large eddy simulation of a turbulent flow field is strongly influenced by its initial or inflow boundary condition. This paper presents a new stochastic approach to generate an artificial turbulent velocity field for initial or inflow boundary condition based on digital filtering. Each velocity component of the artificial turbulent velocity field is generated by linear combination of individual uncorrelated random fields. These uncorrelated random fields are obtained by filtering random white‐noise fields. Using common elements in these linear combinations results in multi‐correlation among different velocity components. The generated velocity field reproduces locally desired Reynolds stress components and integral length scales including cross‐integral length scales. The method appears to be simple, flexible and more accurate in comparison with previously developed methods. The accuracy and performance of the method are demonstrated by numerical simulation of a homogeneous turbulent shear flow with high and low shear rates. To assess the accuracy and performance of the method, simulation results are compared with a reference simulation. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Bose-Einstein-condensed systems in random potentials

The properties of systems with Bose-Einstein condensate in external time-independent random potentials are investigated in the frame of a self-consistent stochastic mean-field approximation. General considerations are presented, which are valid for finite temperatures, arbitrary strengths of the interaction potential, and for arbitrarily strong disorder potentials. The special case of a spatially uncorrelated random field is then treated in more detail. It is shown that the system consists of three components, condensed particles, uncondensed particles, and a glassy density fraction, but that the pure Bose glass phase with only a glassy density does not appear. The theory predicts a first-order phase transition for increasing disorder parameter, where the condensate fraction and the superfluid fraction simultaneously jump to zero. The influence of disorder on the ground-state energy, the stability conditions, the compressibility, the structure factor, and the sound velocity are analyzed. The uniform ideal condensed gas is shown to be always stochastically unstable, in the sense that an infinitesimally weak disorder destroys the Bose-Einstein condensate, returning the system to the normal state; but the uniform Bose-condensed system with finite repulsive interactions becomes stochastically stable and exists in a finite interval of the disorder parameter.

STOCHASTIC POST-BUCKLING OF FRAMES USING KOITER'S METHOD

The objective of this paper is to explore the impact of stochastic inputs on the buckling and post-buckling response of structural frames. In particular, we examine the impact of random member stiffness on the buckling load, and the initial slope and curvature of the post-buckling response of three example frames. A finite element implementation of Koiter's perturbation method is employed to efficiently examine the post-buckling response. Monte Carlo simulations where the member stiffness is treated as a random variable, as well as correlated and uncorrelated random fields, are completed. The efficiency of Koiter's perturbation method is the key to the feasibility of applying Monte Carlo simulation techniques, which typically requires a large number of sample simulations. In an attempt to curtail the need for multiple sample calculations, an alternative first-order perturbation expansion is proposed for approximating the mean and variance of the post-buckling behavior. However, the limitations of this first-order perturbation approximation are demonstrated to be significant. The simulations indicate that deterministic characteristics of the post-buckling response can be inadequate in the face of input randomness. In one case, a frame that is stable symmetric in the deterministic case is found to be asymmetric when randomness in the input is incorporated; therefore, this frame has real potential for imperfection sensitivity. The importance of random field models for the member stiffness as opposed to random variable models is highlighted. The simulations indicate that the post-buckling response can magnify input randomness, as variability in the post-buckling parameters can be greater than the variability in the input parameters.

On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate

The connections between the random elements of a discrete random flow field and the uncertainty in the hierarchical set of its spatio-temporal scales, obtained by the symmetric version of the proper orthogonal decomposition (POD) method, are investigated. It is shown that the relevant statistics for the energy levels, the temporal modes and the spatial modes can be expressed in an explicit form as power series of the flow field standard deviation. Such expansions characterize accurately interesting phenomena of mixing between different flow scales. The basis of the present work is the assumption that the randomness is characterized by a Gaussian uncorrelated random field. Two applications of the theory developed are proposed: to the incompressible flow past a cylinder at Reynolds number . The theoretical predictions are confirmed well by Monte Carlo simulations and interesting relations between the random flows and the relevant statistics of their POD spatio-temporal scales are determined and discussed.

Free vibration analysis of rotating beams with random properties

In this paper, free vibration of rotating beam with random properties is studied. The cross-sectional area, elasticity modulus, moment of inertia, shear modulus and density are modeled as random fields and the rotational speed as a random variable. To study uncertainty, stochastic finite element method based on second order perturbation method is applied. To discretize random fields, the three methods of midpoint, interpolation and local average are applied and compared. The effects of rotational speed, setting angle, random property variances, discretization scheme, number of elements, correlation of random fields, correlation function form and correlation length on "Coefficient of Variation" (C.O.V.) of first mode eigenvalue are investigated completely. To determine the significant random properties on the variation of first mode eigenvalue the sensitivity analysis is performed. The results are studied for both Timoshenko and Bernoulli-Euler rotating beam. It is shown that the C.O.V. of first mode eigenvalue of Timoshenko and Bernoulli-Euler rotating beams are approximately identical. Also, compared to uncorrelated random fields, the correlated case has larger C.O.V. value. Another important result is, where correlation length is small, the convergence rate is lower and more number of elements are necessary for convergence of final response.

Multiple-quantum relaxation in the magic-angle-spinning NMR of [formula omitted] spin pairs

We determine the decay rate constants of zero-, double- and single-quantum coherence for 13 C spin pairs in magic-angle-spinning solid-state NMR. The double-quantum coherence is excited by a C7 pulse sequence and converted into zero-quantum coherence by a frequency-selective pair of π/2 pulses. The zero-quantum coherence is reconverted into observable magnetization by a second pair of π/2 pulses followed by a second C7 sequence. In a magnetically dilute system where the 13 C– 13 C distance is 0.296 nm, the relaxation rate constants are consistent with a model of uncorrelated random fields at the two labeled 13 C sites. In a fully-labelled system with a short 13 C– 13 C distance of 0.153 nm, the measured rate constants are inconsistent with the uncorrelated random field model.

A numerical three‐dimensional conditioned/unconditioned stochastic facies type model applied to a remediation well system

In this study a three‐dimensional stochastic facies‐based aquifer model was developed. The model can be used to numerically simulate flow and solute transport in heterogeneous groundwater aquifers. The stochastic generation process can be conditioned by using available facies information in one vertical plane or two orthogonal vertical ones. In this study the information was obtained from a facies interpretation of a vertical georadar profile in a natural gravel formation in Switzerland. In the domain outside the known profile, unconditioned lenses and layers were generated at random according to statistical information on coherent sedimentary structures based on observations in adjacent gravel pits [Jussel et al., 1994a]. The method was applied to a single extraction well designed to capture an initially block‐shaped contaminant plume. A total of 80 conditioned and unconditioned synthetic aquifers was generated. The flow and transport simulations were performed using a finite element flow model and a random walk transport model. The results are presented as the ensemble of integral solute mass recovery curves of single realizations. One would expect conditioning to reduce the bandwidth of the recovery curves representing the uncertainty, but the results show that the bandwidth even increased. This effect was attributed to a discrepancy in the mean volumetric fraction of the different facies types in the conditioned and the unconditioned cases. Moreover, a simulation using a homogeneous model with constant equivalent flow and transport parameters overestimated the remediation efficiency. The influence of a linear, reversible equilibrium sorption on the remediation well efficiency was taken into account by an uncorrelated random field of the retardation factor based on values from the literature. However, the impact of the variability in hydraulic conductivity clearly exceeded the effect of the variability in the retardation factor.