Homogeneous Gaussian Random Field Research Articles

NURBS-based Isogeometric Analysis and Refined Plate Theory Application on a Functionally Graded Plate Subjected to Random Loads

In structural design standards, loads are often assumed to be random quantities to give load factors. This study deals with the Stochastic Isogeometric Analysis (SIGA) for a Functionally Graded Plate (FGP) subjected to random distribution loads. The spatial random variation of distribution loads is modeled as a homogeneous Gaussian random field in the plane of the functionally graded plate. The governing equation of the functional grade plate is derived using the NURBS-based isogeometric analysis and the refined plate theory. SIGA is developed based on standard NURBS-based isogeometric analysis in conjunction with the first-order perturbation expansions of random loads. This approach was verified with Monte Carlo simulation, and the numerical results showed the effect of random loads on the variation of displacements and stresses of the functionally graded plate.

Anomalies in the topology of the temperature fluctuations in the cosmic microwave background: An analysis of the NPIPE and FFP10 data releases

We present a topological analysis of the temperature fluctuation maps from the Planck 2020 Data Release 4 NPIPE dataset and the Planck 2018 Data Release 3 FFP10 dataset. We performed a multiscale analysis in terms of the homology characteristics of the maps, invoking relative homology to account for the analysis in the presence of masks. We performed our analysis for a range of smoothing scales spanning sub- and super-horizon scales corresponding to a full width at half maximum (FWHM) of 5′,10′,20′,40′,80′,160′,320′, and 640′, and employed simulations based on the standard model for comparison, which assumes the initial fluctuation field to be an isotropic and homogeneous Gaussian random field. Examining the behavior of topological components, represented by the 0D homology group, we find the observations to be approximately 2σ or less deviant from the simulations for all resolutions and scales for the NPIPE dataset. For the FFP10 dataset, we detect a 2.96σ deviation between the observations and simulations at N = 128, FWHM = 80′. For the topological loops, represented by the first homology group, the simulations and observations are consistent within 2σ for most resolutions and scales for both the datasets. However, for the NPIPE dataset, we observe a high deviation between the observation and simulations in the number of loops at FWHM = 320′, but at a low dimensionless threshold ν = −2.5. Under a Gaussian assumption, this would amount to a deviation of ∼4σ. However, the distribution in this bin is manifestly non-Gaussian and does not obey Poisson statistics either. In the absence of a true theoretical understanding, we simply note that the significance is higher than what may be resolved by 600 simulations, yielding an empirical p-value of at most 0.0016. Specifically in this case, our tests indicate that the numbers arise from a statistically stable regime, despite being based on small numbers. For the FFP10 dataset, the differences are not as strong as for the NPIPE dataset, indicating a 2.77σ deviation at this resolution and threshold. The Euler characteristic, which is the alternating sum of the ranks of relative homology groups, reflects the deviations in the components and loops. To assess the significance of combined levels for a given scale, we employed the empirical and theoretical versions of the χ2 test as well as the nonparametric Tukey depth test. Although all statistics exhibit a stable distribution, we favor the empirical version of the χ2 test in the final interpretation, as it indicates the most conservative differences. For the NPIPE dataset, we find that the components and loops differ at more than 95%, but agree within the 99% confidence level with respect to the base model at N = 32, FWHM = 320′. The Euler characteristic at this resolution displays a per mil deviation. In contrast, the FFP10 dataset shows that the observations are consistent with the base model within the 95% confidence level, at this and smaller scales. This is consistent with the observations of the Planck analysis pipeline via Minkowski functionals. For the largest smoothing scale, N = 16, FWHM = 640′, both datasets exhibit an anomalous behavior of the loops, where FFP10 data exhibit a deviation that is larger by an order of magnitude than that of the NPIPE dataset. In contrast, the values for the topological components and the Euler characteristic agree between observations and model to within a confidence level of 99%. However, for the largest scales, the statistics are based on low numbers and may have to be regarded with caution. Even though both datasets exhibit mild to significant discrepancies, they also exhibit contrasting behaviors at various instances. Therefore, we do not find it feasible to convincingly accept or reject the null hypothesis. Disregarding the large-scale anomalies that persist at similar scales in WMAP and Planck, observations of the cosmic microwave background are largely consistent with the standard cosmological model within 2σ.

A random field model of external metal-loss corrosion on buried pipelines

This paper proposes a random field model to characterize the corrosion depth on the external surface of buried steel oil and gas pipelines. The model addresses the intermingling of corroded and corrosion-free areas on the pipe surface, similar to the spatial intermittency of rainfall accumulation within a geographic area, by using a latent homogeneous Gaussian random field and a spatial position-dependent threshold associated with the latent Gaussian field. High-resolution corrosion measurement data obtained from corroded pipe segments removed from in-service pipelines are used to estimate parameters of the proposed model, including the probability of corrosion at a given point, marginal distribution of the nonzero corrosion depth and correlation structure of the latent Gaussian field. The results indicate that the nonzero corrosion depth follows a shifted lognormal distribution. The isotropic γ-exponential correlation function is adequate to characterize the correlation structure of the latent Gaussian field with the correlation length varying from 17 to 50 mm and γ from 0.69 to 0.82. A comparison of simulated and measured corrosion fields suggests that the proposed model is able to capture the characteristics of naturally-occurring corrosion field on the pipe surface. The proposed model provides a useful tool for developing fitness-for-service assessment methodologies for corroded pipelines.

Space and spectral domain relative entropy for homogeneous random fields

A classical result in spectral estimation establishes that the relative entropy rate between two zero-mean stationary Gaussian processes can be computed explicitly in terms of their spectral densities, hence inducing a pseudo-distance in the cone of positive definite spectra. Relying on the properties of multi-level circulant and multi-level Toeplitz matrices, we establish a similar theory for homogeneous Gaussian random fields, both periodic and non-periodic. As a consequence we derive a natural entropic pseudo-distance on the cone of positive definite multi-dimensional spectra. We also define the spectral relative entropy rate as an entropic divergence index between the spectral measures associated to two homogeneous Gaussian random fields and we show that the relative entropy rate and the spectral relative entropy rate are in fact equal.

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.

Fracture of beams with random field properties: Fractal and Hurst effects

The classical problem of peeling a beam off a substrate is studied through a re-examination of Griffith's fracture criterion in the presence of multiscale random properties. Four types of wide-sense homogeneous Gaussian random fields of the vector {Young's modulus E, surface energy density γ}, parametrized by the beam axis, are considered: Ornstein–Uhlenbeck, Matérn, Cauchy, and Dagum. The latter two are multiscale and allow decoupling of the fractal dimension and Hurst effects. Also calculated is the variance of the crack driving force G with any given type of random field in terms of the covariances of E and γ, under either fixed-grip or dead-load conditions. This investigation is complemented by a study of the stochastic crack stability which involves a stochastic competition between potential and surface energies. Overall, we find that, for Cauchy and Dagum models, the introduction of fractal-and-Hurst effects strongly influences the fracture mechanics results. Notably, while the Cauchy and Dagum models represent a more realistic scenario of random fields, given the same covariance on input, the response on output is strongest for Matérn, then Ornstein–Uhlenbeck, then Cauchy and, finally, Dagum model.

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

ABSTRACTIn this paper, for centred homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands' grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.

Accuracy and reliability of a model of an isotropic and homogeneous Gaussian random field in the space $C(\mathbb {T})$

Accuracy and reliability of a model of an isotropic and homogeneous Gaussian random field in the space $C(\mathbb {T})$

Increasing domain asymptotics for the first Minkowski functional of spherical random fields

The restriction to the sphere of a homogeneous and isotropic random field defines a spherical isotropic random field. This paper derives central and noncentral limit results for the first Minkowski functional subordinated to homogeneous and isotropic Gaussian and chi-squared random fields, restricted to the sphere in R3. Both scenarios are motivated by their interesting applications in the analysis of the Cosmic Microwave Background (CMB) radiation.

Selection of a covariance function for a Gaussian random field aimed for modeling global optimization problems

Bayesian approach is actively used to develop global optimization algorithms aimed at expensive black box functions. One of the challenges in this approach is the selection of an appropriate model for the objective function. Normally, a Gaussian random field is chosen as a theoretical model. However, the problem of estimation of parameters, using objective function values, is not thoroughly researched. In this paper, we consider the behavior of the maximum likelihood estimators of parameters of the homogeneous isotropic Gaussian random field with squared exponential covariance function. We also compare properties of exponential covariance function models.

Probability Characteristics of the Absolute Maximum of the Discontinuous Homogeneous Gaussian Random Field

A new method is introduced in order to determine the limiting characteristics of the homogeneous Gaussian random field with the non-differentiable correlation function at zero. This method is based on the representation of a field in the small neighborhood of the maximum point by the superposition of the statistically independent locally Markov Gaussian random processes. Then, using the expressions for the distribution function of the Markov Gaussian process, it is possible to obtain the asymptotic approximations for both the probability of the threshold crossing by the absolute maximum of the homogeneous Gaussian random field and the distribution function of the absolute maximum of the field. It is shown that the proposed approximations are monotonic and valid in a much wider range of both threshold values and the area of the definitional domain of the random field, unlike the commonly known approximations. The applicability borders of the introduced theoretical formulas are established by means of the statistical computer simulation.

Density correlations induced by temperature fluctuations in a photon gas

The impact of angular temperature variations on the thermodynamic variables and real-space correlation functions of black-body radiation are analyzed. In particular, the effect of temperature fluctuations on the number density and energy density correlations of the cosmic microwave background (CMB) is studied. The angular temperature fluctuations are modeled by an isotropic and homogeneous Gaussian random field, whose autocorrelation function is defined on the unit sphere in momentum space. This temperature correlation function admits an angular Fourier transform which determines the density correlations in real space induced by temperature fluctuations. In the case of the CMB radiation, the multipole coefficients of the angular power spectrum defining the temperature correlation function have been measured by the Planck satellite. The fluctuation-induced perturbation of the equilibrium variables (internal energy, entropy, heat capacity and compressibility) can be quantified in terms of the measured multipole coefficients by expanding the partition function around the equilibrium state in powers of the temperature random field. The real-space density correlations can also be extracted from the measured temperature power spectrum. Both the number density and energy density correlations of the electromagnetic field are long-range, admitting power-law decay; in the case of the energy density correlation, the fluctuation-induced correlation overpowers the isotropic equilibrium correlation in the long-distance limit.

Consistency of the Least Squares Estimate of Textured Surface Parameters

Background. For sinusoidal observation model of textured surface, i.e. model where regression function is a two-parameter harmonic oscillation and noise is a homogeneous isotropic Gaussian random field on the plane with zero mean and covariance function of the special kind, the problem of statistical estimation of sinusoidal model unknown amplitudes and angular frequencies is considered.Objective. The aim of the paper is to study asymptotic behavior of the least squares estimate (LSE) of unknown amplitudes and angular frequencies of textured surface sinusoidal model.Methods. The results were obtained on the application of methodology developed in the papers by A.V. Ivanov et al. (2009, 2015) and monograph by A.V. Ivanov, N.N. Leonenko (1989).Results. Sufficient conditions on random noise covariance function that ensure strong consistency of sinusoidal model parameters LSE are obtained.Conclusions. The obtained results allow extending them on the models where regression function is a sum of several two-parameter harmonic oscillations. Besides LSE consistency property will allow proving asymptotic normality of these estimates in further works.

Acoustic Waveform Inversion for Ocean Turbulence

AbstractThe seismic oceanography method is based on extracting and stacking the low-frequency acoustic energy scattered by the ocean heterogeneity. However, a good understanding on how this acoustic wavefield is affected by physical processes in the ocean is still lacking. In this work an acoustic waveform modeling and inversion method is developed and applied to both synthetic and real data. In the synthetic example, the temperature field is simulated as a homogeneous Gaussian isotropic random field with the Kolmogorov–Obukhov spectrum superimposed on a background stratified ocean structure. The presented full waveform inversion method is based on the ray-Born approximation. The synthetic seismograms computed using the ray-Born scattering method closely match the seismograms produced with a more computationally expensive finite-difference method. The efficient solution to the inverse problem is provided by the multiscale nonlinear inversion approach that is specifically stable with respect to noise. Full waveform inversion tests are performed using both the stationary and time-dependent sound speed models. These tests show that the method provides a reliable reconstruction of both the spatial sound speed variation and the theoretical spectrum due to fully developed turbulence. Finally, the inversion approach is applied to real seismic reflection data to determine the heterogeneous sound speed structure at the west Barents Sea continental margin in the northeast Atlantic. The obtained model illustrates in more detail the processes of diapycnal mixing near the continental slope.

Stochastic isogeometric analysis of free vibration of functionally graded plates considering material randomness

The paper is to develop stochastic isogeometric analysis for the free vibration of functionally graded plates with spatially varying random material properties. The isogeometric analysis method is employed in the new stochastic analysis scheme which is called Stochastic IsoGeometric Analysis (SIGA). The elastic modulus and mass density are modeled as homogeneous Gaussian random fields along the plane of structure. The governing equation of stochastic isogeometric analysis for free vibration of functionally graded plates is derived in conjunction with perturbation expansions to predict the first and second moments of eigenvalues. In order to verify proposed method, the brute force Monte Carlo simulation is employed. The mean, variance, and COV (Coefficient of Variation) predicted by SIGA and those predicted by Monte Carlo simulation show good agreement. The numerical examples demonstrate that the randomness of material properties affects significantly the structural responses of the functionally graded plates. The correlation between elastic modulus and mass density is also observed to have significant effect on the response COV of eigenvalue.

On the maxima and sums of homogeneous Gaussian random fields

Let {Xn}n≥1 be a sequence of homogeneous Gaussian random fields with mean 0 and variance 1. Under some conditions related to the correlation function of the field, the joint limiting distribution of the maximum and the sum of the Gaussian random field is obtained. The result is also extended to continuous time Gaussian random field.

Extremes of a class of nonhomogeneous Gaussian random fields

This contribution establishes exact tail asymptotics of $\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous Gaussian random fields $X$ on a bounded convex set $\mathbf{E}\subset\mathbb{R}^2$, with variance function that attains its maximum on a segment on $\mathbf{E}$. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.

Extremes of Gaussian fields with a smooth random variance

In this paper we investigate the probabilities of large extremes P(supt∈Tξ(t)η(t)>u), as u→∞, where ξ(t) is a centered homogeneous Gaussian random field, η(t) is particular smooth field independent of ξ(t) and T⊂Rn is a closed Jordan set.

Hitting probabilities and the Hausdorff dimension of the inverse images of a class of anisotropic random fields

Let X = {X(t): t ∈ RN} be an anisotropic random field with values in Rd. Under certain conditions on X, we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel–Riesz capacity. We also obtain the Hausdorff dimension of its inverse image, and the Hausdorff and packing dimensions of its level sets. These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.

A new criterion for testing hypothesis about the covariance function of the homogeneous and isotropic random field

In this paper, we consider a continuous in mean square homogeneous and isotropic Gaussian random field. A criterion for testing hypotheses about the covariance function of such field using estimates for its norm in the space $L_p(\mathbb{T}), p\geq 1$ is constructed.