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The present work proposes a new high-order simulation framework based on statistical learning. The training data consist of the sample data together with a training image, and the learning target is the underlying random field model of spatial attributes of interest. The learning process attempts to find a model with expected high-order spatial statistics that coincide with those observed in the available data, while the learning problem is approached within the statistical learning framework in a reproducing kernel Hilbert space (RKHS). More specifically, the required RKHS is constructed via a spatial Legendre moment (SLM) reproducing kernel that systematically incorporates the high-order spatial statistics. The target distributions of the random field are mapped into the SLM-RKHS to start the learning process, where solutions of the random field model amount to solving a quadratic programming problem. Case studies with a known data set in different initial settings show that sequential simulation under the new framework reproduces the high-order spatial statistics of the available data and resolves the potential conflicts between the training image and the sample data. This is due to the characteristics of the spatial Legendre moment kernel and the generalization capability of the proposed statistical learning framework. A three-dimensional case study at a gold deposit shows practical aspects of the proposed method in real-life applications.

The first step to simulate random fields in practice is usually to obtain or estimate random field parameters, such as mean, standard deviation, correlation function, among others. However, it is difficult to estimate these parameters, particularly the correlation length and correlation functions, in the presence of sparse measurement data. In such cases, assumptions are often made to define the probabilistic distribution and correlation structure (e.g. Gaussian distribution and stationarity), and the sparse measurement data are only used to estimate the parameters tailored by these assumptions. However, uncertainty associated with the degree of imprecision in this estimation process is not taken into account in random field simulations. This paper aims to address the challenge of properly simulating non-stationary non-Gaussian random fields, when only sparse data are available. A novel method is proposed to simulate non-stationary and non-Gaussian random field samples directly from sparse measurement data, bypassing the difficulty in random field parameter estimation from sparse measurement data. It is based on Bayesian compressive sampling and Karhunen–Loeve expansion. First, the formulation of the proposed generator is described. Then, it is illustrated through simulated examples, and tested with wind speed time series data. The results show that the proposed method is able to accurately depict the underlying spatial correlation from sparse measurement data for both non-Gaussian and non-stationary random fields. In addition, the proposed method is able to quantify the uncertainty related to random field parameter estimation from the sparse measurement data and propagate it to the generated random field.

Stochastic simulation has a prominent position in a variety of scientific domains including those of environmental and water resources sciences. This is due to the numerous applications that can benefit from it, such as risk-related studies. In such domains, stochastic models are typically used to generate synthetic weather data with the desired properties, often resembling those of hydrometeorological observations, which are then used to drive deterministic models of the understudy system. However, generating synthetic weather data with the desired properties is not an easy task. This is due to the peculiarities of such processes, i.e., non-Gaussianity, intermittency, dependence, and periodicity, and the limited availability of open-source software for such purposes. This work aims to simplify the synthetic data generation procedure by providing an R-package called anySim, specifically designed for the simulation of non-Gaussian correlated random variables, stochastic processes at single and multiple temporal scales, and random fields. The functionality of the package is demonstrated through seven simulation studies, accompanied by code snippets, which resemble real-world cases of stochastic simulation (i.e., generation of synthetic weather data) of hydrometeorological processes and fields (e.g., rainfall, streamflow, temperature, etc.), across several spatial and temporal scales (ranging from annual down to 10-min simulations).

This paper presents a general decoupled method for reliability-based geotechnical design that takes into account the spatial variability of soil properties. In this method, reliability analyses that require a lot of computational resources are decoupled from the optimization procedure by approximating the failure probability function globally. Failure samples are iteratively generated over the entire design space so that their global distribution information can be extracted to construct the failure probability function. The method is computationally efficient, is flexible to implement, and is well suited for geotechnical problems that may involve sophisticated models. A design example of two-dimensional deep excavation against basal heave is discussed for Singapore marine clay where the density and normalized undrained shear strength of soil mass are modeled as random fields. Results demonstrate that the proposed method works well in practice and is advantageous over the coupled or locally decoupled reliability-based geotechnical design methods. Nanyang Technological University This work was supported by the start-up grant (M4080123.030), NTU Internal Seed Grant (M020030110) and research student scholarship from Nanyang Technological University, Singapore. The authors gratefully acknowledge the support of the Institute of Catastrophe Risk Management and the School of Civil and Environmental Engineering at Nanyang Technological University.

This paper deals with numerical simulation of random fields corresponding to a stochastic structure of atmospheric clouds. We construct numerical models to simulate the optical thickness of stratus clouds as well as indicator fields of broken clouds. To simulate the random fields we use spectral and autoregressive models, as well as a method of nonlinear transformations of Gaussian functions.

This paper marks the debut of a Galerkin isogeometric method for solving a Fredholm integral eigenvalue problem, enabling random field discretization by means of the Karhunen-Loeve expansion. The method involves a Galerkin projection onto a finite-dimensional subspace of a Hilbert space, basis splines (B-splines) and non-uniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Compared with the existing Galerkin methods, such as the finite-element and mesh-free methods, the NURBS-based isogeometric method upholds exact geometrical representation of the physical or computational domain and exploits regularity of basis functions delivering globally smooth eigensolutions. Therefore, the introduction of the isogeometric method for random field discretization is not only new; it also offers a few computational advantages over existing methods. In the big picture, the use of NURBS for random field discretization enriches the isogeometric paradigm. As a result, an uncertainty quantification pipeline of the future can be envisioned where geometric modeling, stress analysis, and stochastic simulation are all integrated using the same building blocks of NURBS. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the isogeometric method for obtaining eigensolutions., Comment: 31 pages, 12 figures, five tables; accepted by Computer Methods in Applied Mechanics and Engineering

Sequential Gaussian Simulation (SGS) is a stochastic simulation technique commonly employed for generating realizations of Gaussian random fields. Arguably, the main limitation of this technique is the high computational cost associated with determining the kriging weights. This problem is compounded by the fact that often many realizations are required to allow for an adequate uncertainty assessment. A seemingly simple way to address this problem is to keep the same simulation path for all realizations. This results in identical neighbourhood configurations and hence the kriging weights only need to be determined once and can then be re-used in all subsequent realizations. This approach is generally not recommended because it is expected to result in correlation between the realizations. Here, we challenge this common preconception and make the case for the use of a constant path approach in SGS by systematically evaluating the associated benefits and limitations. We present a detailed implementation, particularly regarding parallelization and memory requirements. Extensive numerical tests demonstrate that using a constant path allows for substantial computational gains with very limited loss of simulation accuracy. This is especially the case for a constant multi-grid path. The computational savings can be used to increase the neighbourhood size, thus allowing for a better reproduction of the spatial statistics. The outcome of this study is a recommendation for an optimal implementation of SGS that maximizes accurate reproduction of the covariance structure as well as computational efficiency.

This work addresses the challenge of efficiently determining the representative volume size in computational homogenization by exploiting the requirement that mechanical response quantities are statistically homogeneous and ergodic random fields. The proposed computational homogenization approach focuses on empirically determining the autocorrelation functions of output response quantities through post processing finite element analyses. Once the autocorrelation functions are known, only trivial computations are needed to determine the variance of apparent properties as a function of domain size without any additional finite element computation by utilizing a simple formula relating the autocorrelation function to the variance of spatially averaged quantities. This approach improves upon the current established approach by circumventing the need to analyze numerous successively larger domains in order to determine convergence of apparent properties, which can be computationally prohibitive. Furthermore, similar previous analytical expressions for the variance of apparent properties are asymptotic with respect to domain size while the expression in the proposed approach is exact. After presenting the formulation, the method is first demonstrated for a one-dimensional bar model where analytical expressions can be derived. Then the approach is demonstrated on two numerical examples: (1) a plane strain, linear elastic domain with a lognormal random field describing its compliance, and (2) a stochastic polycrystalline microstructure of a Nickel super alloy having a crystal plasticity model for its constitutive law.

In this paper, a random field model based on nodal integration domain is presented to solve stochastic heat transfer problems. In the proposed model, the uncertainty of the inputs are considered as random field, which is discretized into a number of node-based subdomains and the properties of the uncertainties under random field can be considered at the nodes. The proposed method is efficient to model non-uniform material under random field with constitutive equation, meanwhile, the random field with arbitrary geometry can be simulated conveniently and efficiently by using the Karhunen-Loeve expansion truncated in this work. The statistical moments of the structural responses using the perturbation method is also performed and compared with the solutions of Monte Carlo simulation. The proposed method is successfully applied to the steady-state heat transfer problem with spatially varying random material parameter introduced in the thermal conductivity in this work. Finally, we demonstrate the accuracy and performance of the proposed method through a series of numerical examples both in 2D and 3D steady-state heat transfer problems under different random fields.

Numerous translation approaches have been developed to generate non-Gaussian random fields; however, non-translation or direct simulation methods are much rare. The translation approaches invoke the memoryless transformation from an underlying Gaussian random field to a target non-Gaussian random field. The correlation structure is changed by the memoryless transformation from a Gaussian to a non-Gaussian field and as a result the non-Gaussian correlation structure is obtained after amending the Gaussian correlation structure. In addition, the underlying Gaussian random field is often generated by series expansion approaches (e.g. spectral representation method), which require a sum of infinite terms. Consequently, the computational efforts involved in generating the Gaussian random fields can be significant. After considering the difficulties in simulating a translation random field, a direct approach without utilizing the memoryless transformation is proposed to generate a beta random field. Though the marginal distribution is restricted to the beta distribution, the proposed approach is simple and efficient. This would make it attractive in simulating large-scale random fields for material properties. This is exemplified with an engineering application.

A mathematical form for the response of the stochastic finite element analysis of elliptical partial differential equations has been established through summing products of random scalars and random vectors. The method is based upon the eigendecomposition of a system’s stiffness matrix. The computational reduction is achieved by only summing the dominant terms and by approximating the random eigenvalues and the random eigenvectors. An error analysis has been conducted to investigate the effect of the truncation and the approximations. Consequently, a novel error minimisation technique has been applied through the Galerkin error minimisation approach. This has been implemented by utilising the orthogonal nature of the random eigenvectors. The proposed method is used to solve three numerical examples: the bending of a stochastic beam, the flow through a porous media with stochastic permeability and the bending of a stochastic plate. The results obtained through the proposed random eigenfunction expansion approach are compared with those obtained by using direct Monte Carlo Simulations and by using polynomial chaos.

This work is concerned with the discussion of the numerical approximation for random Cauchy transport model in one dimension. The random (forward time, backward space) finite difference scheme is used to find the stochastic solution. The impression of the consistency and the random von-Neumann stability technique under the mean square sense are studied. Using some examples, we can support our main objective of this model statistically.

Contemporary real-time problems like CAPTCHA generation and optical character recognition can be solved effectively using correlated random fields. These random fields should be produced on a graph in order that problems of any dimension and shape can be handled. However, traditional solutions are often too slow, inaccurate or both. Herein, the Quick Simulation Random Field algorithm to produce correlated random fields on general undirected graphs is introduced. It differs from prior algorithms by completing the graph and setting the unspecified covariances to zero, which facilitates analytic study. The Quick Simulation Random Field graph distribution is derived within and the following questions are studied: (1) For which marginal pmfs and covariances will this algorithm work? (2) When does the marginal property hold, where the sub-graph distribution of an algorithm-simulated field matches the distribution of the algorithm-simulated field on the subgraph? (3) When does the permutation property hold, where the vertex simulation order does not affect the joint distribution?

Anomalous diffusion is often simulated by random walks on random fractal structures. As existing simulation methods either lack a high degree of parallelism or impose restrictions on the choice of fractal structures, a new approach is proposed here. We present a parallel algorithm for simulating random walks on fractal structures that is suitable for a wide variety of hardware architectures. The degree of parallelism of the algorithm equals the number of random walkers, which is achieved by its communication-avoiding design. In contrast to other approaches, the random fractal structure is not pre-computed at whole. Instead, only the surrounding of each random walker is calculated by the parallel threads while the random walker moves around on the fractal structure.

Geometric parameters such as the thickness and width of a beam are random for various reasons including manufacturing tolerance and operation wear. Due to these random parameter properties, the vibration characteristics of the structure are also random. In this paper, we derive equations of motion to conduct stochastic vibration analysis of a rotating beam using the assumed mode method and stochastic spectral method. The accuracy of the proposed method is first verified by comparing analysis results to those obtained with Monte-Carlo simulation (MCS). The efficiency of the proposed method is then compared to that of MCS. Finally, probability densities of various modal and transient response characteristics of rotating beams are obtained with the proposed method.

Existing approaches for generating non-Gaussian random fields typically utilize translation process theory that applies a memoryless nonlinear transformation to an underlying Gaussian random field. In the current study, a direct non-translation approach is proposed to generate random fields with a marginal gamma distribution. The proposed approach is based on the additive reproductive property of the gamma distribution; it results in a conceptually simple algorithm that is straightforward to implement in Monte-Carlo simulations. It is demonstrated that an arbitrary marginal gamma distribution is achievable. The resulting auto-correlation functions are non-negative and decreasing functions with a prescribed scale of fluctuation. These characteristics make the proposed non-translation approach suitable for modelling the spatial variability in material properties. The engineering implications of the proposed approach are illustrated through an application example wherein the proposed approach is utilized to generate a spatially varying undrained shear strength field for a two-dimensional plane strain slope, and the stability of the slope is analyzed by the finite element method with Monte-Carlo simulations. Since many material properties have a non-zero lower bound, the three-parameter gamma distribution is also discussed, and an asymptotically unbiased and consistent estimate of the lower bound is proposed.

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

We study asymptotic behavior of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix-valued random step-size sequence, and a dynamically changing random regression function. In particular, we show that under quitemild conditions, stochastic approximation procedures are asymptotically linear in the statistical sense, that is, they can be represented as weighted sums of random variables. Therefore, a suitable formof the central limit theoremcan be applied to derive asymptotic distribution of the corresponding processes. The theory is illustrated by various examples and special cases.

In this article, a random and a stochastic version of a SIR nonautonomous model previously introduced in [ 19 ] is considered. In particular, the existence of a random attractor is proved for the random model and the persistence of the disease is analyzed as well. In the stochastic case, we consider some environmental effect on the model, in fact, we assume that one of the coefficients of the system is affected by some stochastic perturbation, and analyze the asymptotic behavior of the solutions. The paper is concluded with a comparison between the two different modeling strategies.

The use of Markov random fields (MRFs) is a common approach for performing image segmentation, where the problem is modeled using MRFs that incorporate priors on neighborhood nodes to allow for efficient Maximum a Posteriori inference. These local MRF models often result in smoothed segmentation boundaries, since they penalize the assignment of different labels to neighboring pixels and are limited in the use of long-range interactions. Although recent work on fully connected random fields and deep random fields has shown to be very promising in addressing these issues, both streams of approaches face certain limitations, which could affect inference performance and computational tractability. In this paper, we introduce the concept of deep randomly connected conditional random fields DRCRF, which fuse fully-connected random fields and deep random fields together to obtain benefits from long-range interactions while allowing for efficient inference using arbitrary potential functions. Leveraging random graph theory, the concept of stochastic cliques is incorporated into a deep CRF structure to take better advantage of long-range interactions while maintaining computational tractability. The experimental results demonstrate that the proposed DRCRF framework outperforms existing fully connected CRF frameworks and provides results comparable to the principled deep random field framework, which is among the state of the art in random field frameworks for image segmentation.

Designing a random dynamical system requires the prediction of the statistics of the response, knowing the random model of the uncertain parameters. Direct Monte Carlo simulation (MCS) is the reference method for propagating uncertainties but its main drawback is the high numerical cost. A surrogate model based on a polynomial chaos expansion (PCE) can be built as an alternative to MCS. However, some previous studies have shown poor convergence properties around the deterministic eigenfrequencies. In this study, an extended Pade approximant approach is proposed not only to accelerate the convergence of the PCE but also to have a better representation of the exact frequency response, which is a rational function of the uncertain parameters. A second approach is based on the random mode expansion of the response, which is widely used for deterministic dynamical systems. A PCE approach is used to calculate the random modes. Both approaches are tested on an example to check their efficiency.

In this paper, we consider the stochastic higher-order Kirchhoff-type equation with nonlinear strongly dissipation and white noise. We first deal with random term by using Ornstein-Uhlenbeck process and establish the wellness of the solution, then the existence of global random attractor are proved.

The main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called thevariation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.

This paper presents a learning-based stochastic simulation method that incorporates high-order spatial statistics at multiple scales from sources with different resolutions. Regarding the simulation of a certain spatial attribute, the high-order spatial information from different sources is encapsulated as aggregated kernel statistics in a spatial Legendre moment kernel space, and the probability distribution of the underlying random field model is derived by a statistical learning algorithm, which matches the high-order spatial statistics of the target model to the observed ones. In addition, a related software is developed as the SGeMS plugin. Case studies are conducted with a known data set and a gold deposit, demonstrating reproduction of high-order spatial statistics from the available data, as well as practical aspects in mining applications.

In this paper, we consider the asymptotic behavior of solutions for stochastic non-autonomous FitzHugh–Nagumo system with multiplicative white noise. First we prove the existence of random attractor of the random dynamical system generated by the solutions of considered system. Then we present some conditions for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable Banach space and apply these conditions to prove the finiteness of fractal dimension of random attractor.

Uncertainty in geological structural modeling, especially geological corrosion(a kind of karst cave), is a bottleneck that restricts the development and application of geological computer modeling and effect estimation. To solve this issue, a stochastic modeling method based on the random field theory is proposed in comparison with the deterministic geometric modeling method. Then the constraint random field modeling method and the random field modeling method without constrained parameters are compared and analyzed. A case study shows that the novel stochastic simulation method is an effective tool to describe the distribution characteristics of corrosion parameters and reflect the updated geological prospecting information. The influence of geological corrosion on the dam behavior can also be better analyzed by using the stochastic simulation method. At the same time, the unconfined random field ignores the sample location information and may lead to higher variability. Therefore, the constraint random field modeling method can provide a useful reference for the numerical analysis under complex geological conditions.

The parameters in a structure such as geometric and material properties are generally uncertain due to manufacturing tolerance, wear, fatigue and material irregularity. Such parameters are random fields because the uncertain properties vary along the spatial domain of a structure. Since the parameter uncertainties in a structure result in the uncertainty of the structural dynamic behavior, they need to be identified accurately for structural analysis or design. In order to identify the random fields of geometric parameters, the parameters can be measured directly using a 3-dimensional coordinate measuring machine. However, it is often very expensive to measure them directly. It is even impossible to directly measure some parameters such as density and Young's modulus. For that case, the parameter random fields should be identified from measurable response data samples. In this paper, a stochastic inverse method to identify parameter random fields in a structure using modal data is proposed. The proposed method consists of the following three steps: (i) obtaining realizations of the parameter random field from modal data samples by solving an optimization problem, (ii) obtaining the deterministic terms in the Karhunen-Loeve expansion by solving an eigenvalue problem and (iii) estimating the distributions of random variables in the Karhunen-Loeve expansion using a maximum likelihood estimation method with kernel density.

For modeling material properties having a bounded range, the beta distribution may be adopted as the marginal distribution of a second-order non-Gaussian random field. Three aspects related to the simulation of such random field are discussed in this study. First, an unbiased and consistent estimator for the lower (and upper) bound of the beta distribution based on sample data is proposed. This estimator is shown to be generally more efficient than that given by the method of moments. Second, a simple explicit function relating the auto-correlation function of the non-Gaussian random field to that of the underlying Gaussian field is proposed. The relationship facilitates control on the scale of fluctuation of the non-Gaussian field. Third, an algorithm is proposed for generating random fields with an approximate marginal beta distribution and a prescribed cross-correlation, where the latter can range from −1 to 1. Numerical examples are given to illustrate the effectiveness and efficiency of each of the three aspects. The estimation of the lower bound of material property is exemplified through field data from a real project.

In conjunction with the formulation of random functions, a family of renewed spectral representation schemes is proposed. The selected random function serves as a random constraint correlating the random variables included in the spectral representation schemes. The objective stochastic process can thus be completely represented by a dimension-reduced spectral model with just few elementary random variables, through defining the high-dimensional random variables of conventional spectral representation schemes (usually hundreds of random variables) into the low-dimensional orthogonal random functions. To highlight the advantages of this scheme, orthogonal trigonometric functions with one and two random variables are constructed. Representative-point set of the dimension-reduced spectral model is derived by employing the probability-space partition techniques. The complete set with assigned probabilities of points gains a low-number-sample stochastic process. For illustrative purposes, the stochastic modeling of seismic acceleration processes is proceeded, of which the stationary and non-stationary cases are investigated. It is shown that the spectral acceleration of simulated processes matches well with the target spectrum. Stochastic seismic response analysis, moreover, and reliability assessment of a framed structure with Bouc-Wen behaviors are carried out using the probability density evolution method. Numerical results reveal the applicability and efficiency of the proposed simulation technique.

Based on the Monte Carlo simulation and probabilistic analysis, stochastic radiative models are effectively averaged; that is, deterministic models that reproduce the mean probabilities of particle passage through a stochastic medium are constructed. For this purpose, special algorithms for the double randomization and conjugate walk methods are developed. For the numerical simulation of stochastic media, homogeneous isotropic Voronoi and Poisson mosaic models are used. The parameters of the averaged models are estimated based on the properties of the exponential distribution and the renewal theory.

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.

Markov random evolutions and their approximations are analyzed. The main object of study is generators of random processes with independent increments. These processes are considered in Poisson approximation and Levy approximation schemes. Generators of random processes are normalized by parameters that are nonlinear functions. The explicit form of such normalization parameters is shown. The asymptotic representation of generators in both approximation schemes is shown. Normalizing factors of random evolution are presented.

The research work presents the study on non-deterministic problems in the presence of the multiple imprecise-random-field uncertainties by extending the spectral stochastic finite element framework. The standard random field is expanded to characterize the behaviour of physical model with imprecise randomness more appropriately for real engineering problems based on available sources of uncertainty. The Young’s modulus and body force of structures are considered as imprecise random fields with bounded statistical moments. Mathematical expressions and solution procedure are developed to produce the uncertain-but-bounded statistical characteristics of responses, namely interval mean value, interval standard deviation and bounding distribution functions. The bounding probability density and cumulative distribution of the imprecise random response are then effectively visualized by means of Polynomial Chaos Expansion. The feasibility and effectiveness of the presented method are illustrated by three numerical examples.

This paper addresses the problem of simulating multivariate random fields with stationary Gaussian increments in a d-dimensional Euclidean space. To this end, one considers a spectral turning-bands algorithm, in which the simulated field is a mixture of basic random fields made of weighted cosine waves associated with random frequencies and random phases. The weights depend on the spectral density of the direct and cross variogram matrices of the desired random field for the specified frequencies. The algorithm is applied to synthetic examples corresponding to different spatial correlation models. The properties of these models and of the algorithm are discussed, highlighting its computational efficiency, accuracy and versatility.

We study n independent stochastic processes(xi (t),tiЄ[o,t1 ],i=1,......n) defined by a stochastic differential equation with diffusion coefficients depending nonlinearly on a random variables and (the random effects).The distributions of the random effects фi,and,μi and depends on unknown parameters which are to be estimated from the continuous observations of the processes xi (t) . When the distributions of the random effects ф ,μ, are Gaussian and exponential respectively, we obtained an explicit formula for the likelihood function and the asymptotic properties (consistency and asymptotic normality) of the maximum likelihood estimator (MLE) are derived when tend to infinity.

In this paper, we consider the existence of a pullback attractor for the random dynamical system generated by stochastic two-compartment Gray-Scott equation for a multiplicative noise with the homogeneous Neumann boundary condition on a bounded domain of space dimension n ≤ 3. We first show that the stochastic Gray-Scott equation generates a random dynamical system by transforming this stochastic equation into a random one. We also show that the existence of a random attractor for the stochastic equation follows from the conjugation relation between systems. Then, we prove pullback asymptotical compactness of solutions through the uniform estimate on the solutions. Finally, we obtain the existence of a pullback attractor.

[EN] This paper deals with the numerical solution of the random Cauchy one-dimensional heat model. We propose a random finite difference numerical scheme to construct numerical approximations to the solution stochastic process. We establish sufficient conditions in order to guarantee the consistency and stability of the proposed random numerical scheme.The theoretical results are illustrated by means of an example where reliable approximations of the mean and standard deviation to the solution stochastic process are given., This work has been partially supported by the Ministerio de Economía y Competitividad Grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. M. A. Sohaly is also indebted to Egypt Ministry of Higher Education, Cultural Affairs, for its financial support [mohe-casem (2016)].

With the ever-growing computational power of personal computers over the past few decades, stochastic simulation of spatially varying three-dimensional (3D) quantities has been coupled with numerical analysis models (e.g., 3D finite element model) to consider explicitly the influence of spatial variability of engineering quantities when carrying out design or analysis for mechanical systems and civil- or geo-structures. Random field theory is often adopted in stochastic simulations, where random field samples (RFSs) are generated for representing the spatially varying engineering quantities encountered in practice. Nevertheless, simulation of 3D RFSs is not trivial, especially when measurements are sparse and limited, a case often encountered in engineering practice of a specific project. This is because random field parameters (e.g., the type of auto-correlation structure, correlation length, marginal probability distribution) are difficult to determine when measurements from a project are sparse and limited. The problem becomes more challenging when the quantity of interest is non-stationary and/or non-Gaussian. This renders a great challenge for proper simulation of non-stationary non-Gaussian 3D RFSs from sparse measurements. To address this challenge, this paper proposes a method which integrates the concept of signal decomposition in digital signal processing with Markov Chain Monte Carlo (MCMC) simulation. The proposed method is non-parametric and data-driven. It takes sparse measurements and their corresponding 3D spatial coordinates as input and returns many high-resolution non-stationary non-Gaussian 3D RFSs as output. The method is illustrated using a series of numerical examples. The results show that the proposed method performs reasonably well.

In recent years, stochastic simulation has obtained increasing attention in geomechanics as it allows to consider uncertainties in geomaterials’ properties. Spatial uncertainties may influence simulation results and lead to either unsafe or uneconomic designs. By implementing stochastic random field algorithms in Finite-Element simulations, the impact of such uncertainty can be integrated in the simulation results. However, the application of such algorithms implicates new requirements in understanding the relationships between correlation length, mesh coarseness, and required number of Finite-Element simulations. The present study intends to provide a deeper understanding of such aspects. In the presented approach, stochastic random fields analysis is applied in a Finite-Element code by remote scripting. Considering a slope stability analysis and using Monte-Carlo Simulations, the impact of varying mesh coarseness and the correlation length on the obtained safety factor is investigated. As for large numbers of mesh elements, computational costs increase rapidly, the study expands its investigations by employing two further advanced methods, namely mesh adaptivity and subset simulation. These state-of-the-art methods allow to reduce calculation efforts, once by iterative remeshing, once by selective sampling according to failure probability. The results show promising improvements in the concept of computational efforts.

If the physical properties of a structural component are sufficiently random then the statistical distribution of the natural frequencies and mode shapes tends to a universal distribution associated with the Gaussian Orthogonal Ensemble (GOE) of random matrices. Previous work has exploited this result to yield expressions for the relative variance of the energy of the response of a random system to harmonic excitation. The derivation of these expressions employed random point process theory, and in the theoretical development it was assumed that the modal generalised forces were uncorrelated. Although this assumption is often valid, there are cases in which correlations between the generalised forces can significantly affect the response variance, and in the present work the existing theory is extended to include correlations of this type. The extended theory is applicable to both single frequency responses and to band average responses, and the developed closed form expressions are validated by comparison with direct simulations for a random plate structure. ispartof: Journal of Sound and Vibration vol:413 pages:456-466 status: published

It is well known that geotechnical parameters impact dramatically soil-tunneling mechanics. Random field theory has been widely used for a single geotechnical parameter, including aleatory uncertainty, spatial variability and local singularity. Current challenge is focused on analyzing the spatially correlated soil layers. Coefficient of cohesion, internal friction angle, and Young’s modulus are accounted in reliability analysis of ground surface settlement due to shallow-buried shield tunneling. At first, convert multiple non-stationary soil layers into stationary field using a local detrending method, and define the geostatistical parameters. Then, assume spatial variability of geotechnical parameters into aleatory randomness, and classical response surface method is utilized into reliability analysis. Furthermore, Co-sequential Gaussian discretization is conceived for multivariate spatial random field, in which failure probability of ground surface settlement is calculated directly by subset Monte-Carlo simulation. This approach is applied into the paralleling zone of four shield tunnels of the 5th and 6th metro lines linking to Huanhu W Rd station, Tianjin China. Results prove that reliability index of considering geotechnical parameters as random variables is lower than the assumption of multivariate spatial random field, which would support substantially to construction control and design optimization in complex shallow-buried shield tunneling projects.

In this paper, we consider the asymptotic behaviour of solutions to second-order non-autonomous stochastic lattice equations with dispersive term and additive white noises in the space of infinite sequences. We first transfer the stochastic lattice equations into random lattice equations, and prove the existence and uniqueness of solutions that generate a random dynamical system. Second, we prove the existence of a tempered random absorbing set and a random attractor for the system. Finally, we establish the upper semi-continuity of the random attractors as the coefficient of the white noise term tends to zero.