Nonhomogeneous Random Fields Research Articles

Approximation of Nonhomogeneous Random Field from Local Averages

In this article, we consider the extension of Shannon sampling series reconstruction theorem for nonhomogeneous random fields using local averages sampling, which helps improve certain earlier results. The upper bound of mean square truncation sampling approximation error is more precise, and we establish one approximation result in the almost sure sense.

Spatial modeling of concrete strength based on data

Structural verification of concrete structures relies on an underlying probabilistic model of the concrete strength. This concrete strength exhibits a spatial variability, which is of particular relevance in existing concrete structures, for which the strength is assessed based on samples. To accurately account for the spatial variability of the concrete material, a random field modeling approach can be adopted, which includes a spatial correlation function. Unfortunately, the available literature on spatial variability of concrete strength is not sufficient to make an educated choice of this correlation function. In this paper, we propose a hierarchical Bayesian random field model, that enables learning the parameters of a selected correlation function with in-situ spatially distributed measurements of the concrete strength. We propose a correlation function that accounts for the composite nature of the material through distinguishing micro-scale and meso-scale variability. The predictive spatial distribution of the proposed random field model given the spatial data is then obtained through an analytical random field update, resulting in a non-homogeneous random field model with log-Student’s t-marginal distribution. The proposed approach provides an effective means to employ in-situ measurements for updating verification predictions of concrete structures. We apply our approach to two case studies on chamber walls of ship locks, where measurements of the concrete strength are available from core samples.

A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields

In many instances, conditions in the subsurface are highly variable, while site investigations only provide sparse measurements. Consequently, subsurface models are usually inaccurate. These characteristics reflect uncertainty and mean significant engineering and environmental hazards. Such uncertainty should be quantified, in order, ultimately, to be reduced. To this end, in this paper, a new two-dimensional approach to quantify stratigraphic uncertainty is proposed and described. The approach is based on non-homogeneous random fields and considers categorical quantities to represent geological structures. Unlike other reported approaches in related literature, we provide evidence on the usefulness of the approach to: (i) comprehensively explore the many diverse and potential geological structures' configurations to exhaust uncertainty quantification; (ii) facilitate the representation of non-homogeneous fields in a computational inexpensive fashion; and (iii) ameliorate the complication of producing spatial Markov chains or Bayesian models to quantify uncertainty using finite difference equations. The proposed approach is demonstrated using a case analysis.

The spatial averaging method for non-homogeneous random fields with application to reliability analysis

In probabilistic assessments, inputs with significant spatial variability should be modeled with random fields. Random fields can be non-homogeneous with location-specific marginal distributions, for example, due to site-specific information incorporated through Bayesian analysis or due to spatial trends in the mean or variance of the uncertain quantity. This paper investigates the spatial averaging method for the discretization of non-homogeneous random fields. In this approach, the random field is reduced to a set of random variables representing its local averages over a corresponding set of elemental domains. This is of particular benefit when coupling the random field model with finite elements for structural analysis. We extend the application of the method to non-homogeneous Gaussian and non-Gaussian translation random fields with lognormal, Student’s t- and log-Student’s t-marginal distribution. The latter two distributions are particularly relevant if spatial data is used in a hierarchical Bayesian random field modeling. Two numerical investigations assess the ability of the method to efficiently represent the response variability and probability of failure of structural systems with spatially variable inputs. The investigations include the effect of different element sizes for the spatial averaging on the system response and applicability of the spatial averaging method to assessing local and global failure modes.

Generation of Nonhomogeneous Turbulent Velocity Fields by Modified Randomized Spectral Method

A new version of the synthetic turbulent velocity generator is proposed for simulation of turbulent flows. The method is fully stochastic and generates a statistically anisotropic and nonhomogeneous random field, which provides the initial and boundary conditions for the deterministic eddy-resolving model of turbulence. The simulation method has been tested on a three-dimensional problem of developed turbulent channel flow.

Tensor train-Karhunen-Loève expansion: New theoretical and algorithmic frameworks for representing general non-Gaussian random fields

The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional physical domain in the tensor train format; secondly, to develop a new algorithmic framework for accurately computing the modes and describing the correlation structure of the latent factors beyond second order within moderate time. Central to the algorithmic framework is the tensor train decomposition of cumulant functions. This decomposition is accurately computed with a novel rank-revealing algorithm. For computing the modes, compared with existing Galerkin-type and collocation-type methods, the proposed computational procedure totally removes the need of selecting the basis functions or collocation points and the quadrature points, which not only greatly enhances adaptivity, but also avoids solving large-scale eigenvalue problems. Moreover, by computing with higher-order cumulant functions, the new theoretical and algorithmic frameworks show great potential for representing general non-Gaussian non-homogeneous random fields. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the efficiency and accuracy of the proposed algorithmic framework.

Asymptotic Properties of the Method of Observead Means for Nonstationary Random Fields

The author considers a stochastic programming problem where the estimator is approximated by its empirical estimate based on observations of a non-homogeneous random field with continuous time and strong mixing. The strong consistency of this estimate is investigated and its asymptotic distribution is found under the constraint imposed on the unknown parameter in the form of systems of inequalities.

Limit laws on extremes of nonhomogeneous Gaussian random fields

Abstract In this paper, by using the tail asymptotics derived by Dębicki et al. (2016), we prove the Gumbel limit laws for the maximum of a class of nonhomogeneous Gaussian random fields. As an application of the main results, we derive the Gumbel limit law for Shepp statistics of fractional Brownian motion and Gaussian integrated processes.

Probability-based damage detection using model updating with efficient uncertainty propagation

Model updating method has received increasing attention in damage detection of structures based on measured modal parameters. In this article, a probability-based damage detection procedure is presented, in which the random factor method for non-homogeneous random field is developed and used as the forward propagation to analytically evaluate covariance matrices in each iteration step of stochastic model updating. An improved optimization algorithm is introduced to guarantee the convergence and reduce the computational effort, in which the design variables are restricted in search region by region truncation of each iteration step. The developed algorithm is illustrated by a simulated 25-bar planar truss structure and the results have been compared and verified with those obtained from Monte Carlo simulation. In order to assess the influences of uncertainty sources on the results of model updating and damage detection of structures, a comparative study is also given under different cases of uncertainties, that is, structural uncertainty only, measurement uncertainty only and combination of the two. The simulation results show the proposed method can perform well in stochastic model updating and probability-based damage detection of structures with less computational effort.

The Sequential Generation of Gaussian Random Fields for Applications in the Geospatial Sciences

This paper presents practical methods for the sequential generation or simulation of a Gaussian two-dimensional random field. The specific realizations typically correspond to geospatial errors or perturbations over a horizontal plane or grid. The errors are either scalar, such as vertical errors, or multivariate, such as , , and errors. These realizations enable simulation-based performance assessment and tuning of various geospatial applications. Both homogeneous and non-homogeneous random fields are addressed. The sequential generation is very fast and compared to methods based on Cholesky decomposition of an a priori covariance matrix and Sequential Gaussian Simulation. The multi-grid point covariance matrix is also developed for all the above random fields, essential for the optimal performance of many geospatial applications ingesting data with these types of errors.

A meshfree-Galerkin method in modelling and synthesizing spatially varying soil properties

Physical properties of soil vary from point to point in space and exhibit great uncertainty, suggesting random field as a natural approach in modelling and synthesizing these properties. The significance of considering spatial variability and uncertainty of soil properties is greatly manifested in the probabilistic seismic risk analysis of soil–structural system (nonlinear dynamic analysis under earthquake loading), where modelling and synthesis of the spatial variability and uncertainty of soil properties are necessary. This paper introduces a meshfree-Galerkin approach within the Karhunen–Loève (K–L) expansion scheme for representation of spatial soil properties modelled as the random fields. The meshfree shape functions are introduced and employed as a set of basis functions in the Galerkin scheme to obtain the eigen-solutions of integral equation of K–L expansion. An optimization scheme is proposed for the resulting eigenvectors in treating the compatibility between the target and analytical covariance models. Assessments of the meshfree-Galerkin method are conducted for the resulting eigen-solutions and the representation of covariance models for various homogeneous and nonhomogeneous random fields. The accuracy and validity of the proposed approach are demonstrated through the modelling and synthesis of the spatial field models inferred from the field measurements.

Sparsified Randomization algorithms for low rank approximations and applications to integral equations and inhomogeneous random field simulation

Abstract Sparsified Randomization Monte Carlo (SRMC) algorithms introduced in our recent paper [60] for solving systems of linear algebraic equations are extended to construct the SVD-based randomized low rank approximations for large matrices. We suggest some efficient implementations of SRMC based on low rank approximations, and give different applications. In particular, an important application we present in this paper is a fast simulation algorithm for a randomized approximation of non-homogeneous random fields based on a discrete version of the Karhunen-Loéve expansion. We present two examples of non-homogeneous random field simulation which include a long-correlated Lorenzian random field and the fractional Wiener process. Another application we deal in this paper concerns the randomized solvers for large linear systems. We suggest a hybrid method which combines SRMC with an algorithm for solving boundary integral equations based on a separation representation of the kernel. This method is illustrated in this paper by solving a 2D boundary integral equation from potential theory governing the Dirichlet problem for the Laplace equation.

Accurate estimation of evolutionary power spectra for strongly narrow-band random fields

One of the most widely used techniques for the simulation of non-homogeneous random fields is the spectral representation method. Its key quantity is the power spectrum, which characterizes the random field in terms of frequency content and spatial evolution in a mean square sense. The paper at hand proposes a method for the estimation of separable power spectra from a series of samples, which combines accurate spectrum resolution in space with an optimum localization in frequency. For non-separable power spectra, it can be complemented by a joint strategy, which is based on the partitioning of the space-frequency domain into several sub-spectra that have to be separable only within themselves. Characteristics and accuracy of the proposed method are demonstrated for analytical benchmark spectra, whose estimates are compared to corresponding results of established techniques based on the short-time Fourier, the harmonic wavelet and the Wigner–Ville transforms. It is then shown by a practical example from stochastic imperfection modeling in structures that in the presence of strong narrow-bandedness in frequency, the proposed method for separable random fields leads to a considerable improvement of estimation results in comparison to the established techniques.

Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients

The mean and variance of the temperature are analytically obtained in a functionally graded annular disc with spatially random heat transfer coefficients (HTCs) on the upper and lower surfaces. This annular disc has arbitrary variations in the HTCs (i.e., arbitrary thermal interaction with the surroundings) and gradient material composition only along the radial direction and is subjected to deterministic axisymmetrical heating at the lateral surfaces. The stochastic temperature field is analysed by considering the annular disc to be multilayered with spatially constant material properties and spatially constant but random HTCs in each layer. A type of integral transform method and a perturbation method are employed in order to obtain the analytical solutions for the statistics. The correlation coefficients of the random HTCs are expressed in the form of a linear function with respect to the radial distance as a non-homogeneous random field of discrete space. Numerical calculations are performed for functionally graded annular discs composed of stainless steel and ceramic, which comprise two types of material composition distributions. The effects of the magnitude of the means of HTCs, volume fraction distributions of the constitutive materials and correlation strengths of the HTCs on the standard deviation of the temperature are discussed.

Heat and mass transport in nonhomogeneous random velocity fields.

The effective equation describing the transport of passive tracers in nonsolenoidal velocity fields is determined, assuming that the velocity field U(r,t) is a function of both position r and time t, albeit remaining locally random. Assuming a strong separation of scales and applying the method of homogenization, we find a Fickian constitutive relation for the coarse-grained particle flux, as the sum of a convective part, V(E)c, and a diffusive term, -D(s). Inverted Delta c, where V(E) is the Eulerian mean tracer velocity, c the average particle concentration, and D(s) the effective diffusivity. The latter can be written as D(s)(r,t)=D(0)I+D(r,r,t), where D0 is the molecular diffusivity, I the unit dyadic and D(r(1),r(2),t) the cross diffusion dyadic. Conversely, the Eulerian mean velocity V(E)(r,t) is the sum of the microscale mean tracer velocity V(r,t) and a particle drift velocity, V(d)(r,t)=-[(delta/delta r(2)).D(T)(r,r(2),t)](r(2)=r), which depends on the nonhomogeneity of the velocity field at the macroscale. The microscale mean particle velocity, in turn, is the sum of the mean fluid velocity and the ballistic tracer velocity, which is due to the local nonuniformity of the concentration field and is therefore structurally different from the tracer drift velocity. In the limit of large Peclet numbers, D(s) coincides with the self-diffusion dyadic, as it measures the local temporal growth of the mean square displacement of a tracer particle from its average position. In this case, the motion of a tracer particle is a random process in the manner of Stratonovich, where the smoothly varying mean tracer velocity equals the microscale mean tracer velocity and the fluctuating term is described through the cross diffusion dyadic D(r(1),r(2),t).

Evaluating Nonwoven Fabric Irregularity on the Basis of Linnik Functionals

New statistical characteristics are suggested for analyzing nonwoven fabric irregularity. These characteristics are based on the concept of outliers of random field irregularity beyond (or beneath) a fixed level, and allow the area and volume of these outliers to be determined. Outlier statistics are studied and results are presented for calculating their confidence intervals and the dimensions of the sample. The results of the simulation are presented, and the influence of nonhomogeneous random field irregularity on the outliers is depicted. The experimental part of the paper presents the results of a study in micro- and macroscale of some nonwoven fabrics. Some practical aspects of the new characteristics of nonwoven irregularity are also given. Typical application problems are set up, and examples of solutions are given.

Direct Approximation of the Extreme Value Distribution of Nonhomogeneous Gaussian Random Fields

In many reliability problems, particularly in the area of offshore and marine safety analysis, the maxima of stochastic processes and random fields, describing random load responses and resistances, are critically important. In the case of stationary differentiable Gaussian processes, approximations for the distribution of the maxima using upcrossing rates are well known. In this paper, we consider nonstationary differentiable Gaussian random fields. A new technique first applied by Sun (1993) can be used to derive direct approximations for the tail probability. This is an alternative approach based on an expansion of the covariance operator and on more or less geometric concepts. The method does not rely on assumptions regarding the point process of upcrossings. It provides, directly, an asymptotic approximation for the distribution of the maximum of the random field.

Simulation of nonhomogeneous random fields for structural applications

In this paper, an original approach to the simulation of discretized, nonhomogeneous, spatial random fields is presented. To solve the simulation problem, an effective version of the rejection method and conditional probability distributions is implemented. A superior role in the method plays a propagation scheme consisting of a growing number of points, covering sequentially the nodes of a regular or irregular mesh. The results of the simulation of different types of second-order field (Brown, Wiener, Shinozuka) are presented. Estimators of local and global characteristics of these fields show agreement with theoretical Wishart (gamma) distributions. The proposed method can be used for the solution of two-dimensional stochastic nonlinear boundary value problems by computer simulation.

Simulation of stochastic waves in a non-homogeneous random field : J. Kiyono, K. Toki, T. Sato & H. Mizutani, Soil Dynamics & Earthquake Engineering, 14(5), 1995, pp 387–396

Simulation of stochastic waves in a non-homogeneous random field : J. Kiyono, K. Toki, T. Sato & H. Mizutani, Soil Dynamics & Earthquake Engineering, 14(5), 1995, pp 387–396

Orthogonal decompositions of 2-D nonhomogeneous discrete random fields

Imposing atotal-order on a two-dimensional (2-D) discrete random field induces an orthogonal decomposition of the random field into two components: Apurely-indeterministic field and adeterministic one. The purely-indeterministic component is shown to have a 2-D white-innovations driven moving-average representation. The 2-D deterministic random field can be perfectly predicted from the field's “past” with respect to the imposed total-order definition. The deterministic field is further orthogonally decomposed into anevanescent field, and aremote past field. The evanescent field is generated by the columnto-column innovations of the deterministic field with respect to the imposed nonsymmetrical-half-plane total-ordering definition. The presented decomposition can be obtained with respect to any nonsymmetrical-half-plane total-ordering definition, for which the nonsymmetrical-half-plane boundary line has rational slope.