Multidimensional Random Fields Research Articles

Non-Parametric Estimation of the Renewal Function for Multidimensional Random Fields

This paper addresses the almost sure convergence and the asymptotic normality of an estimator of the multidimensional renewal function based on random fields. The estimator is based on a sequence of non-negative independent and identically distributed (i.i.d.) multidimensional random fields and is expressed as infinite sums of k-folds convolutions of the empirical distribution function. It is an extension of the work from the case of the two-dimensional random fields to the case of the d-dimensional random fields where d>2. This is established by the definition of a “strict order relation”. Concrete applications are given.

A decomposed Karhunen–Loève expansion scheme for the discretization of multidimensional random fields in geotechnical variability analysis

A decomposed Karhunen–Loève expansion scheme for the discretization of multidimensional random fields in geotechnical variability analysis

Development of a model for stochastic analysis of vehicle–track interaction considering spatial variability of track parameters

In this work, a model for achieving stochastic analysis of vehicle–track interaction systems is proposed based on random field realization of track parameters and the finite element coupling of rigid-elastic bodies. Aiming at the clarification of the spatial variability and correlation of track parameters on the influence of vehicle–track dynamic performance, a general scheme based on Karhunen–Loève Expansion and cyclic calculation method is developed to realize a multi-dimensional random field that holds an arbitrary marginal distribution function and correlation structure, and to consider the inhomogeneity of system parameters along the longitudinal direction efficiently. Furthermore, the validation of the random field expansion against the vehicle–track system is implemented, and the dynamic effects of random track parameters are investigated through two special issues. Numerical results show that the spatial correlation of fastener stiffness affects the extreme response, especially the low-frequency component of rail displacement. The spatial dependence of the filling layer has little influence on the dynamic behavior of the vehicle subsystem, but it obviously changes the extreme stress and strain of the filling layer. Ignoring the spatial correlation of random variables might underestimate the performance of the track system.

Realization of multi-dimensional random field based on Jacobi–Lagrange–Galerkin method in geotechnical engineering

Geomaterial has spatial variability, which can be described by the random field theory. A powerful tool for realizing random fields is the Karhunen–Loève series expansion (K–L expansion) where the number of random variables depends on the number of K–L expansion terms rather than on the number of grids of geo-structure model. However, the K–L expansion requires the solution of an integral eigenvalue problem whose analytical form exists only in the special case. Hence, the Galerkin method is usually employed to calculate the approximate solution, which especially for multi-dimensional random field inevitably causes the huge computational cost of multi-fold integrals and the approximate error of the solution. For quickly and accurately simulating the random field with fewer random variables, a Jacobi–Lagrange–Galerkin (JLG) method is proposed where the huge amounts of multi-fold integrals are transformed into simple matrix multiplications which are implemented quickly in a MATLAB environment. Furthermore, the discretization error and its influence factors are discussed to determine the parameters of the JLG method, and the procedure of the JLG method is proposed. Finally, a two-dimensional shallow foundation and a three-dimensional slope are employed to demonstrate computational efficiency, accuracy, and applicability of the JLG method.

A new perspective on the simulation of cross-correlated random fields

Cross-correlated random fields are widely used to model multiple uncertain parameters and/or phenomena with inherent spatial/temporal variability in numerous engineering systems. The effective representation of such fields is therefore the key element in the stochastic simulation, reliability analysis and safety assessment of engineering problems with mutual correlations. However, the simulation of such fields is generally not straightforward given the complexity of correlation structure. In this paper, we develop a unified framework for simulating non-Gaussian and non-stationary cross-correlated random fields that have been specified by their correlation structure and marginal cumulative distribution functions. Our method firstly represents the cross-correlated random fields by means of a new general stochastic expansion, in which the fields are expanded in terms of a set of deterministic functions with corresponding random variables. A finite element discretization scheme is then developed to further approximate the fields, so that the sets of deterministic functions reflecting the cross-covariance structure can be straightforwardly determined from the spectral decomposition of the resulting discretized fields. For non-Gaussian random fields, an iterative mapping procedure is developed to generate random variables to fit non-Gaussian marginal distribution of the fields. By virtue of the remarkable property of the presented stochastic expansion, i.e., various random fields share an identical set of random variables, the framework we develop is conceptually simple for simulating non-Gaussian cross-correlated fields with arbitrary covariance functions, which need not be stationary. In particular, the developed method is further generalized to a consistent framework for the simulation of multi-dimensional random fields. Five illustrative examples, including a spatially varying non-Gaussian and nonstationary seismic ground motions, are used to demonstrate the application of the developed method.

Stochastic Method to Predict Effects of Roll Grinding Deviations on Sheet Flatness in Cold Rolling

Abstract Industrial measurements of the diameter profiles of work-rolls used in cold sheet rolling are applied with a stochastic roll-stack model to better understand how residual error from the roll grinding process affects the rolled sheet flatness quality. Roll diameter measurements taken via a noncontact, optical device on new, warm, and worn work-rolls show that the diameter deviations vary along the roll lengths, across roll samples, and at different operational states, suggesting a multidimensional random field problem. Studies are conducted for a 4-high rolling mill with 301 stainless steel sheet to investigate the reliability in achieving target flatness considering the work-roll diameter random field. Also investigated is the sensitivity of the flatness reliability to roll diameter deviations at different locations along the roll lengths and for the three operational states (newly machined, warm, and worn following several passes). The results lead to several key findings. Foremost, it is shown that an assumption of statistical independence among the residual grinding errors at different roll axis locations is improper. Furthermore, it is demonstrated that, for the measured grinding error correlation patterns, the roll diameter deviations external to the roll/sheet contact region play an important role in contributing to flatness defects within the sheet and that these influences vary according to the roll operational state (new, warm, worn). The presented stochastic model and applied measurement data thus provide for a new understanding into how roll grinding performance influences dimensional quality in the sheet rolling process.

A 3-D numerical study of the random wave-induced response in a spatially heterogenous seabed

The seabed failure poses a significant threat to the stability of marine structures. The actual wave-induced dynamic response in a porous seabed is a complex issue due to the randomness in hydrodynamic pressure and the spatial variability in soil properties. A probabilistic framework was proposed based on a 3-D stochastic finite element model for the investigation of the random wave-induced responses in a spatially heterogenous seabed. The framework integrates the generation of random waves as well as the random soil and the finite element analysis of poroelasticity via a Livelink code. A decomposed Karhunen-Loève expansion method was proposed, making the generation of multi-dimensional random fields with high resolution and large scale possible. Contrastive analyses of different scenarios were conducted to study the effects of the randomness of wave pressure and the spatial variability of the soil shear modulus on the seabed response. Monte Carlo simulations were used for the statistics of seabed response, and then the effects of the correlation length of random fields on the liquefaction depth were investigated. The results of the analysis highlighted the significant impacts of the randomness of ocean waves and the spatial variability in soil properties, which were often ignored in previous studies.

3rd-order Spectral Representation Method: Simulation of multi-dimensional random fields and ergodic multi-variate random processes with fast Fourier transform implementation

This paper introduces a generalized 3rd-order Spectral Representation Method for the simulation of multi-dimensional random fields and ergodic multi-variate stochastic processes with asymmetric non-linearities. The formula for the simulation of general d-dimensional random fields is presented and the method is applied to simulate 2D and 3D random fields. The differences between samples generated by the proposed methodology and the existing classical Spectral Representation Method are analysed. The formula for the simulation of multi-variate random processes is subsequently developed. An important feature of the methodologies is that they can be implemented efficiently with the Fast Fourier Transform (FFT), details of which are presented. Computational savings are shown to grow exponentially with dimensionality as a testament of the scalability of the simulation methodology. Examples highlighting the salient features of these methodologies are also presented.

Dynamic reliability analysis for structure with temporal and spatial multi-parameter

For efficiently estimating the dynamic failure probability of the structure with random variables, stochastic processes and temporal and spatial multi-parameter, an estimation strategy is presented based on the random field transformation. The random field transformation focusing on the dynamic reliability with only one time parameter is further investigated, and it is extended to temporal and spatial multi-parameter issue, which simulates the output as multi-dimensional Gaussian random field. Also, the active learning Kriging method is used to construct the surrogate models for the mean function and auto-covariance function of performance function. After that, the temporal and spatial dynamic failure probability can be obtained by the simulation method. Although it doesn’t need to call the real performance function during the process of simulation method, it is time computationally expensive. To address this issue, the optimization algorithm procedure is established to estimate the dynamic failure probability. Several examples including an aero engine turbine disk and a cylindrical pressure vessel are introduced to illustrate the significance and effectiveness of the proposed methods for analyzing the temporal and spatial multi-parameter dynamic failure probability.

Noncanonical Spectral Model of Multidimensional Uniform Random Fields

The estimates for the power spectrum of multidimensional uniform random fields in the form of models of finite mixtures of standard spectra are proposed. The learning algorithm of the models demonstrates improved convergence properties for degenerate spectra and small interclass distances in the frequency space, as well as for small volumes of the experimental data. Based on this, the noncanonical models of uniform random fields are presented as a sum of statistically independent spatial harmonics with random amplitudes and frequencies. The alternative representation of a multidimensional spectrum as a sample of random frequencies allowed us to propose computationally efficient algorithms of digital synthesis of background and underlying surface images with the topology of spectral estimates that are adequate for the experimental data. The algorithms are free from simplifying assumptions regarding the method of discretization of the field and functional form of the power spectral density.

Topological data analysis and diagnostics of compressible magnetohydrodynamic turbulence

The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical magnetohydrodynamics, it is important to verify that such simulations are in agreement with observations. One of the main challenges in this area is to identify robust quantitative measures to compare structures found in simulations with those inferred from astrophysical observations. A similar challenge is to compare quantitatively results from different simulations. Topological data analysis offers a range of techniques, including the Betti numbers and persistence diagrams, that can be used to facilitate such a comparison. After describing these tools, we first apply them to synthetic random fields and demonstrate that, when the data are standardized in a straightforward manner, some topological measures are insensitive to either large-scale trends or the resolution of the data. Focusing upon one particular astrophysical example, we apply topological data analysis to H iobservations of the turbulent interstellar medium (ISM) in the Milky Way and to recent magnetohydrodynamic simulations of the random, strongly compressible ISM. We stress that these topological techniques are generic and could be applied to any complex, multi-dimensional random field.

Simulation of multi-dimensional random fields by Karhunen–Loève expansion

The Karhunen–Loève (K–L) expansion method is a powerful tool for simulating stationary and nonstationary, Gaussian and non-Gaussian stochastic processes with explicitly known covariance functions. Since the K–L expansion requires the solution of Fredholm integral equation of the second kind, it is generally not feasible to simulate multi-dimensional random fields. This is because even the numerical solution of the Fredholm integral multi-dimensional eigenvalue problem is difficult to obtain. In order to address this problem, this paper develops a consistent generalization of K–L expansion for multi-dimensional random field simulation. The new method decomposes an n-dimensional random field into a total of n stochastic processes, each can be represented by using the traditional K–L expansion. Thus, the developed method is embedded into the well-established framework of the K–L expansion for simulating stochastic process, and obviating the need for solving the multi-dimensional integral eigenvalue problems. Four examples, including random fields with different kinds of covariance functions, are used to demonstrate the application of the proposed method.

A CLT for multi-dimensional martingale differences in a lexicographic order

We prove a central limit theorem for a square-integrable ergodic stationary multi-dimensional random field of martingale differences with respect to a lexicographic order.

Optimal representation of multi-dimensional random fields with a moderate number of samples: Application to stochastic mechanics

A significant amount of problems and applications in stochastic mechanics and engineering involve multi-dimensional random functions. The probabilistic analysis of these problems is usually computationally very expensive if a brute-force Monte Carlo method is used. Thus, a technique for the optimal selection of a moderate number of samples effectively representing the entire space of sample realizations is of paramount importance. Functional Quantization is a novel technique that has been proven to provide optimal approximations of random functions using a predetermined number of representative samples. The methodology is very easy to implement and it has been shown to work effectively for stationary and non-stationary one-dimensional random functions. This paper discusses the application of the Functional Quantization approach to the domain of multi-dimensional random functions and the applicability is demonstrated for the case of a 2D non-Gaussian field and a two-dimensional panel with uncertain Young modulus under plane stress.

Space–Time Wave Extremes: The Role of Metocean Forcings

AbstractWave observations and modeling have recently demonstrated that wave extremes of short-crested seas are poorly predicted by statistics of time records. Indeed, the highest waves pertain to wave groups at focusing that have space–time dynamics. Therefore, the statistical prediction of extremes of short-crested sea states should rely on the multidimensional random wave fields’ assumption. To adapt wave extreme statistics to the space–time domain, theoretical models using parameters of the directional wave spectrum have been recently developed. In this paper, the influence of metocean forcings (wind conditions, ambient current, and bottom depth) on these parameters and hence on wave extremes is studied with a twofold strategy. First, parametric spectral formulations [Pierson–Moskowitz and Joint North Sea Wave Project (JONSWAP) frequency spectra with cos2 directional distribution function] are considered to represent the dependence of wave extremes upon wind speed, fetch, and space domain size. Afterward, arbitrary conditions are simulated by using the SWAN numerical model adapted to store the spectral parameters, and the effects on extremes of current- and depth-induced shoaling are investigated. Preliminarily, the space–time extremes prediction model adopted is assessed by means of numerical simulations of Gaussian random seas. Compared to the significant wave height of the sea state and for a given space domain size, results show that space–time extremes are enhanced by opposite currents, whereas they are weakened by increasing wind conditions (wind speed and fetch) and by depth-induced shoaling. In this respect, the remarkable contribution to wave extremes of the size of the space domain is substantiated.

Italian seas wave extremes: a preliminary assessment

Occasional damages occurring to offshore structures and routing ships have brought attention on whether conventional analyses of stormy seas are adequate to represent wave extremes. Indeed, field and laboratory data showed that in short-crested seas (typical of stormy conditions) the maximum wave elevation is rarely retrieved by a single-point observation (a buoy, for example), which tends to underestimate the actual extreme that occurs over an area surrounding the point. Recently, stochastic models for the prediction of maxima of multidimensional Gaussian random fields (e.g. the Piterbarg’s theorem and the Adler and Taylor’s Euler Characteristics approach) have been successfully applied to ocean wave statistics, thus permitting to extend the extreme value analysis from the time to the space–time domain. Results from space–time models are here firstly compared to observations gathered by an optical stereo system, and then used to preliminarily assess wave extremes over the Italian Seas, by using a numerical model analysis covering the period 2007–2013. In particular, space–time maxima are estimated as affecting a fishing boat and a cruise ship, showing that, locally, space–time extremes may exceed time-based values up to about 50 %.

Particle picture interpretation of some Gaussian processes related to fractional Brownian motion

We construct fractional Brownian motion, sub-fractional Brownian motion and negative sub-fractional Brownian motion by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter.We employ the so-called white noise approach. Our construction is quite general, permitting to obtain also some other Gaussian processes, as well as multidimensional random fields. In particular, we generalize and presumably simplify some results by Hambly and Jones (2007). We also obtain a new class of S′-valued density processes, containing as a particular case the density process of Martin-Löf (1976).

Image variants generated by using vector models of multidimensional random fields

The author proposes vector models of multidimensional random fields to obtain the image variants for an object under observation, which differ in color and geometric distortions simulating the reflection of the interaction between the observed object and the environment.

Stochastic model for service life prediction of RC structures exposed to carbonation using random field simulation

Deterioration of Reinforced Concrete (RC) structures is related to the corrosion of reinforcement since it propagates to structural degradation. In urban or underground conditions, carbonation-initiated corrosion in reinforcement is reported. In this paper, a stochastic model for predicting service life of RC structure subjected to carbonation is proposed through Monte Carlo simulation method with the spatial variation of material, geometric properties of RC structures, and environmental factors. This model can take into account the spatial variation of uncertain parameters by generating non-Gaussian and multi-dimensional random fields using spectral representation method. To estimate the temporal variation of the probability to exceed a pre-defined serviceability, time-dependent reliability analyses are performed. Spatial reliability indicators such as the mean and standard deviation of the area are calculated through 20-sample with 4-variable full-factorial realization. It is evaluated that there exists large variation in predicted service life and quality control for concrete is required during construction for the service life. Based on these results, design and construction considerations are provided.

An implementation of matched filtering using the multi-dimensional Fourier transform

We present an implementation of matched filtering comprised three basic steps: (1) compute a fast Fourier transform (FFT) of the data, (2) map the FFT to a (data independent) one-dimensional track in a multi-dimensional space, and (3) take an inverse multi-dimensional Fourier transform. In contrast to the standard approach, this method does not require the explicit use of template waveforms. This implementation was found as a byproduct of an exploration of sampling theory in the context of the template placement problem. We show that the latter problem can be completely mapped, in all its details, onto that of sampling an analog multi-dimensional random field, including the use of anti-aliasing before the acquisition of samples. The last point is non-trivial since the analog signal in question, namely the output of a template bank, does not physically exist unless it is computed.