Isotropic Random Field Research Articles

An inverse random source problem for the one-dimensional Helmholtz equation with attenuation

This paper is concerned with an inverse random source problem for the one-dimensional stochastic Helmholtz equation with attenuation. The source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The random sources under consideration are equivalent to the generalized fractional Gaussian random fields which include rough fields and can be even rougher than the white noise, and hence should be interpreted as distributions. The well-posedness of the direct source problem is established in the distribution sense. The micro-correlation strength of the random source, which appears to be the strength in the principal symbol of the covariance operator, is proved to be uniquely determined by the wave field in an open measurement set. Numerical experiments are presented for the white noise model to demonstrate the validity and effectiveness of the proposed method.

Flow of Spatiotemporal Turbulentlike Random Fields

We study the Lagrangian trajectories of statistically isotropic, homogeneous, and stationary divergence-free spatiotemporal random vector fields. We design this advecting Eulerian velocity field such that it gets asymptotically rough and multifractal, both in space and time, as it is demanded by the phenomenology of turbulence at infinite Reynolds numbers. We then solve numerically the flow equations for a differentiable version of this field. We observe that trajectories get also rough, characterized by nearly the same Hurst exponent as the one of our prescribed advecting field. Moreover, even when considering the simplest situation of the advection by a fractional Gaussian field, we evidence in the Lagrangian framework additional intermittent corrections. The present approach involves properly defined random fields, and asks for a rigorous treatment that would explain our numerical findings and deepen our understanding of this long lasting problem.

Fracture in random quasibrittle media: II. Analytical model based on extremes of the averaging process

The paper delivers an analytical model for prediction of the peak force in concrete specimens loaded in bending (both notched and unnotched). The model is capable of predicting peak force statistics by computing the extreme values of sliding averages of random strength fields. The local strength of the specimen is modeled by a stationary isotropic random field with Gaussian distribution and a given autocorrelation function. The averaging operation represents the progressive loss in material integrity and the associated stress redistribution that takes place prior to reaching the peak load. Once the (linear) averaging process is performed analytically, the resulting random field of averaged strength is assumed to represent a series of representative volume elements (RVEs) and the global strength is found by solving for the minimum of such an effective strength field. All these operations can be written analytically and there are only four model parameters: the three dimensions of the averaging volume (RVE) and the length of the final weakest-link chain.The model is verified using detailed numerical computations of notched and unnotched concrete beams simulated by mesoscale discrete simulations of concrete fracture performed with probabilistic distributions of model parameters. These are presented in the companion paper Part I (Eliáš and Vořechovský, 2020). The numerical model used for verification represents material randomness both by assigning random locations to the largest aggregates and by simulating random fluctuations of material parameters via a homogeneous random field.

The turning arcs: a computationally efficient algorithm to simulate isotropic vector-valued Gaussian random fields on the d-sphere

Random fields on the sphere play a fundamental role in the natural sciences. This paper presents a simulation algorithm parenthetical to the spectral turning bands method used in Euclidean spaces, for simulating scalar- or vector-valued Gaussian random fields on the d-dimensional unit sphere. The simulated random field is obtained by a sum of Gegenbauer waves, each of which is variable along a randomly oriented arc and constant along the parallels orthogonal to the arc. Convergence criteria based on the Berry-Esseen inequality are proposed to choose suitable parameters for the implementation of the algorithm, which is illustrated through numerical experiments. A by-product of this work is a closed-form expression of the Schoenberg coefficients associated with the Chentsov and exponential covariance models on spheres of dimensions greater than or equal to 2.

Soil spatial variability impact on the behavior of a reinforced earth wall

This article presents the soil spatial variability effect on the performance of a reinforced earth wall. The serviceability limit state is considered in the analysis. Both cases of isotropic and anisotropic non-normal random fields are implemented for the soil properties. The Karhunen-Loeve expansion method is used for the discretization of the random field. Numerical finite difference models are considered as deterministic models. The Monte Carlo simulation technique is used to obtain the deformation response variability of the reinforced soil retaining wall. The influences of the spatial variability response of the geotechnical system in terms of horizontal facing displacement is presented and discussed. The results obtained show that the spatial variability has an important influence on the facing horizontal displacement as well as on the failure probability.

Influence of soil spatial variability on the response of strip footing on geocell-reinforced slope

Abstract The present study investigates the influences of spatial variability and randomness of the soil strength parameters on the load-settlement response of a strip footing placed on a geocell-reinforced soil slope by combining finite difference method with random field model. Both isotropic and anisotropic random fields of lognormally distributed soil friction angle are generated with the utilization of Cholesky-Decomposition technique. In some representative cases, cross-correlation between soil friction angle and soil cohesion, and spatial variability of soil dilation angle are considered. The mean and variation in the load carrying capacity of the footing corresponding to a particular settlement value are obtained after running several numbers of Monte-Carlo simulations. Mean load carrying capacity of the footing reduces with increasing randomness and spatial variation in soil properties, which are defined by the higher values of coefficient of variation and lower values of correlation lengths in both the horizontal and vertical directions. Influence of soil dilatancy is found significant for higher values of soil friction angle. Apart from the above-mentioned parameters, effects of footing setback distance, position of water table, and flexural rigidity of the geocell mattress on the load-settlement response of the footing are examined. Finally, the importance of performing probabilistic analysis is explained from the point of view of failure probability of the footing, calculated with respect to few design factors of safety.

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

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

Hypothesis testing for the smoothness parameter of Matérn covariance model on a regular grid

We consider the hypothesis testing problem for the smoothness parameter ν in a stationary isotropic Gaussian random field with Matérn covariance. For the data observed on a regular grid, we construct the rejection region for one-tailed tests, and starting from there, we develop a chain-like testing procedure, which can determine an interval containing the true value of ν. Such an interval can help improve the performance of various estimation methods for ν, such as restricting the parameter space or validating the assumptions for the asymptotic properties of the estimator. The test statistic is built on recursive applications of the Laplace operator to the observations. For this statistic, the fixed-domain asymptotic normality is established and the forms of asymptotic mean and variance are derived. Therefore, the proposed tests are guaranteed to have correct asymptotic size under certain conditions. Simulation studies indicate that our proposed methods are efficient for moderate sample sizes. As an application of the chain-like testing procedure, we provide a method of choosing the number of differencing for a local Whittle-likelihood type estimator of ν proposed by Wu, Lim, and Xiao, and show that it can avoid obtaining inconsistent estimates of ν via a numerical experiment.

Non-central limit theorems for functionals of random fields on hypersurfaces

This paper derives non-central asymptotic results for non-linear integral functionals of homogeneous isotropic Gaussian random fields defined on hypersurfaces in ℝd. We obtain the rate of convergence for these functionals. The results extend recent findings for solid figures. We apply the obtained results to the case of sojourn measures and demonstrate different limit situations.

Cross‐dimple in the cross‐covariance functions of bivariate isotropic random fields on spheres

Multivariate random fields allow to simultaneously model multiple spatially indexed variables, playing a fundamental role in geophysical, environmental, and climate disciplines. This paper introduces the concept of cross‐dimple for bivariate isotropic random fields on spheres and proposes an approach to build parametric models that possess this attribute. Our findings are based on the spectral representation of the matrix‐valued covariance function. We show that our construction is compatible with both the negative binomial and circular‐Matérn bivariate families of covariance functions. We illustrate through simulation experiments that the models proposed in this work allow to achieve improvements in terms of predictive performance when a dimple‐like intrinsic structure is present.

RVE Problem: Mathematical aspects and related stochastic mechanics

The paper examines (i) formulation of field problems of mechanics accounting for a random material microstructure and (ii) solution of associated boundary value problems. The adopted approach involves upscaling of constitutive properties according to the Hill--Mandel condition, as the only method yielding hierarchies of scale-dependent bounds and their statistics for a wide range of (non)linear elastic and inelastic, coupled-field, and even electromagnetic problems requiring (a) weakly homogeneous random fields and (b) corresponding variational principles. The upscaling leads to statistically homogeneous and isotropic mesoscale tensor random fields (TRFs) of constitutive properties, whose realizations are, in general, everywhere anisotropic. A summary of most general admissible correlation tensors for TRFs of ranks 1, …, 4 is given. A method of solving boundary value problems based on the TRF input is discussed in terms of the torsion of a randomly structured rod. Given that many random materials encountered in nature (e.g., in biological and geological structures) are fractal and possess long-range correlations, we also outline a method for simulating such materials, accompanied by an application to wave propagation.

Estimation of the trend function and auto-covariance for spatial models

We first establish, through a Berry–Esseen-type bound, the asymptotic normality of a local linear estimate of the regression function in a fixed design setting when the errors are stationary isotropic spatial random fields. On the other hand, we investigate the weak convergence of an empirical estimate of the variance of these errors in a general α-mixing setting.

Light Induced Inverse-Square Law Interactions between Nanoparticles: "Mock Gravity" at the Nanoscale.

The interaction forces between identical resonant molecules or nanoparticles, optically induced by a quasimonochromatic isotropic random light field, are theoretically analyzed. In general, the interaction force exhibits a far-field oscillatory behavior at separation distances larger than the light wavelength. However, we show that the oscillations disappear when the frequency of the random field is tuned to an absorption Fröhlich resonance, at which the real part of the particle's electric polarizability is zero. At the resonant condition, the interaction forces follow a long-range gravitylike inverse square distance law which holds for both near- and far-field separation distances.

Imprecise random field analysis with parametrized kernel functions

Abstract The application of isotropic random fields in engineering analysis requires the definition of their first two central moments, as well as their covariance function. In general, insufficient data are available to make a fully objective crisp estimate on these quantities, and hence, subjectivity enters implicitly into the analysis. The framework of imprecise probabilities is gaining popularity in this context, as it allows to explicitly separate the epistemic uncertainty, present due to data insufficiency, from the aleatory nature of the random parameters. However, an approach that is capable of handling imprecision in the complete definition of the imprecise random field is lacking to date. This paper proposes a framework for imprecise random field analysis with parametrized covariance functions. As such, the functional form of the covariance function is assumed to be known deterministically, whereas the governing parameters are subjected to imprecision. First, a comprehensive analysis of the effect of imprecise random fields, given imprecision on both the mean and auto-covariance structure, is presented. It is shown that the discretization of an imprecise random field, given interval-valued correlation lengths cannot be performed using interval arithmetical procedures as the resulting basis functions are in that case no longer a complete basis on L 2 . Therefore, an iterative procedure is proposed where the bounds on the imprecise random field basis are determined via an optimization procedure. This procedure provides exact bounds on the response of a system in case the system is monotonic with respect to the random field realizations. Two illustrative case studies on a mass-damper-spring system and a dynamic state-space model of a car suspension are included to illustrate the methodology. These case studies illustrate that indeed a separation between aleatory and epistemic uncertainty is possible in random field analysis, and hence, more objective results are obtained at only slightly increased computational cost.

A semiparametric class of axially symmetric random fields on the sphere

The paper provides a way to model axially symmetric random fields defined over the two-dimensional unit sphere embedded in the three-dimensional Euclidean space. Specifically, our strategy is to integrate an isotropic random field on the sphere over longitudinal arcs with a given central angle. The resulting random field is shown to be axially symmetric and to have the arc central angle as a tuning parameter that allows for isotropy as well as for longitudinal independence as limit cases. We then consider multivariate longitudinally integrated random fields, having the same properties of axial symmetry and a tuning parameter (arc central angle) proper to each random field component. This construction allows for a unified framework for vector-valued random fields that can be geodesically isotropic, axially symmetric, or longitudinally independent. Additionally, all the components of the vector random field are allowed to be cross-correlated. We finally show how to simulate the proposed axially symmetric scalar and vector random fields through a computationally efficient algorithm that exactly reproduces the desired covariance structure and provides approximately Gaussian finite-dimensional distributions.

Direct simulation of two-dimensional isotropic or anisotropic random field from sparse measurement using Bayesian compressive sampling

Random field theory has been increasingly adopted to simulate spatially varying environmental properties and hydrogeological data in recent years. In a two-dimensional (2D) stochastic analysis, variation of the environmental properties or hydrogeological data along different directions can be similar (i.e., isotropic) or quite different (i.e., anisotropic). To model the spatially isotropic or anisotropic variability in a stochastic analysis, conventional random field generators generally require a vast amount of measurement data to identify the random field parameters (e.g., mean, variance, and correlation structure and correlation length in different directions). However, measurement data available in practice are usually sparse and limited. The random field parameters estimated from sparse measurements might be unreliable, and the subsequent random field modeling or stochastic analysis might be misleading. This underscores the significance and challenge of generating 2D isotropic or anisotropic random fields from sparse measurements. This paper develops a novel 2D random field generator, which does not require a parametric form of correlation function or estimation of correlation length and other random field parameters, and directly generates 2D isotropic or anisotropic random field samples from sparse measurements. The proposed generator is highly efficient because simulation of a 2D random field is achieved by generation of a short 1D random vector. The effectiveness and applicability of the proposed generator are illustrated using isotropic and anisotropic numerical examples.

Series representations of isotropic vector random fields on balls

This paper deals with a class of second-order vector random fields in the unit ball of Rd, whose direct/cross covariances are invariant or isotropic with respect to a distance defined on the ball, and gives a series representation of such an isotropic vector random field. A necessary format of covariance matrix functions is also derived for isotropic and mean square continuous vector random fields on the ball.

Simulating isotropic vector-valued Gaussian random fields on the sphere through finite harmonics approximations

The paper tackles the problem of simulating isotropic vector-valued Gaussian random fields defined over the unit two-dimensional sphere embedded in the three-dimensional Euclidean space. Such random fields are used in different disciplines of the natural sciences to model observations located on the Earth or in the sky, or direction-dependent subsoil properties measured along borehole core samples. The simulation is obtained through a weighted sum of finitely many spherical harmonics with random degrees and orders, which allows accurately reproducing the desired multivariate covariance structure, a construction that can actually be generalized to the simulation of isotropic vector random fields on the d-dimensional sphere. The proposed algorithm is illustrated with the simulation of bivariate random fields whose covariances belong to the $${{{\mathcal{F}}}}$$ , spectral Matern and negative binomial classes of covariance functions on the two-dimensional sphere.

Multiple testing of local maxima for detection of peaks on the (celestial) sphere

We present a topological multiple testing scheme for detecting peaks on the sphere under isotropic Gaussian noise, where tests are performed at local maxima of the observed field filtered by the spherical needlet transform. Our setting is different from the standard Euclidean large domain asymptotic framework, yet highly relevant to realistic experimental circumstances for some important areas of application in astronomy, namely point-source detection in cosmic Microwave Background radiation (CMB) data. Motivated by this application, we shall focus on cases where a single realization of a smooth isotropic Gaussian random field on the sphere is observed, and a number of well-localized signals are superimposed on such background field. The proposed algorithms, combined with the Benjamini–Hochberg procedure for thresholding p-values, provide asymptotic control of the False Discovery Rate (FDR) and power consistency as the signal strength and the frequency of the needlet transform get large.

Average Linear and Angular Momentum and Power of Random Fields Near a Perfectly Conducting Boundary

The effect of a perfectly conducting planar boundary on the average linear momentum (LM), angular (momentum (AM), and power of a time-harmonic statistically isotropic random field is analyzed. These averages are purely imaginary and their magnitude decreases in a damped oscillatory manner with distance from the boundary. At discrete quasi-periodic distances and frequencies, the average LM and AM attain their free-space value. Implications for the optimal placement or tuning of power and field sensors are analyzed. Conservation of the flux of the mean LM and AM with respect to the difference of the average electric and magnetic energies and the radiation stresses via the Maxwell stress dyadic is demonstrated. The second-order spatial derivatives of differential radiation stress can be directly linked to the electromagnetic energy imbalance. Analytical results are supported by Monte Carlo simulation results. As an application, performance based estimates for the working volume of a reverberation chamber are obtained. In the context of multiphysics compatibility, mechanical self-stirred reverberation is proposed as an exploitation of electromagnetic stress.