Homogeneous Random Field Research Articles

Robust topology and discrete fiber orientation optimization under principal material uncertainty

This paper introduces a formulation of the robust topology optimization problem that is tailored for designing fiber-reinforced composite structures with spatially varying principal mechanical properties. Specifically, a methodology is developed that incorporates the spatial variability in the engineering constants of the composite lamina into the concurrent topology (i.e., material distribution) and morphology (i.e., fiber orientation distribution) optimization problem for the minimization of the robust compliance function. The spatial variability in the mechanical properties of the lamina is modeled as a homogeneous random field within the design domain by means of the Karhunen-Loe´ve series expansion, and is thereafter intrusively propagated into the stochastic finite element analysis of the composite structure. To carry out the stochastic finite element analysis per iteration of the optimization cycle, the first-order perturbation method is utilized for approximating the current state variables of the physical system. The resulting robust topology and fiber orientation optimization problem is formulated step-by-step for the minimization of the robust compliance function. With the view of solving the optimization problem at hand by means of gradient-based solution algorithms, the first-order derivatives of the involved design functions w.r.t. the associated design variables are analytically derived. The present work concludes with a series of numerical examples, focusing on the benchmark academic case studies of the 2D cantilever and the half part of the Messerschmitt-Bölkow-Blohm beam, aiming to demonstrate the developed methodology as well as to explore the effect that different parameterization instances of the random field bear on the predicted topology and morphology of the beams.

Randomized limit theorems for stationary ergodic random processes and fields

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Zm and Rm,m≥1.

Cross-modality mapping using image varifolds to align tissue-scale atlases to molecular-scale measures with application to 2D brain sections

This paper explicates a solution to building correspondences between molecular-scale transcriptomics and tissue-scale atlases. This problem arises in atlas construction and cross-specimen/technology alignment where specimens per emerging technology remain sparse and conventional image representations cannot efficiently model the high dimensions from subcellular detection of thousands of genes. We address these challenges by representing spatial transcriptomics data as generalized functions encoding position and high-dimensional feature (gene, cell type) identity. We map onto low-dimensional atlas ontologies by modeling regions as homogeneous random fields with unknown transcriptomic feature distribution. We solve simultaneously for the minimizing geodesic diffeomorphism of coordinates through LDDMM and for these latent feature densities. We map tissue-scale mouse brain atlases to gene-based and cell-based transcriptomics data from MERFISH and BARseq technologies and to histopathology and cross-species atlases to illustrate integration of diverse molecular and cellular datasets into a single coordinate system as a means of comparison and further atlas construction.

On fractional spherically restricted hyperbolic diffusion random field

The paper investigates solutions of the fractional hyperbolic diffusion equation in its most general form with two fractional derivatives of distinct orders. The solutions are given as spatial–temporal homogeneous and isotropic random fields and their spherical restrictions are studied. The spectral representations of these fields are derived and the associated angular spectrum is analysed. The obtained mathematical results are illustrated by numerical examples. In addition, the numerical investigations assess the dependence of the covariance structure and other properties of these fields on the orders of fractional derivatives.

Stochastic analysis of thin beams and plates incorporating von-Kármán nonlinear strains using meshless method

The current paper proposes methods for stochastic meshless analysis of thin beams and plates, by considering von-Kármán nonlinear strains. Here, Young’s modulus is assumed to have a spatial variation over the domain and is modeled as a homogeneous random field. Further, it is discretized utilizing moving least square based shape functions. Gaussian and lognormal properties are presumed to model the randomness related to Young’s modulus. Element-free Galerkin method is used as the meshless tool. The present study suggests employing perturbation method if the quantities of interest are limited to the first or second probabilistic moments of the responses. The nonlinear equations in the proposed perturbation formulations are solved using Newton-Raphson iterative scheme for finding out the deterministic part of the response, as well as its initial two derivatives with respect to random variables. Additionally, if there is an interest in higher probabilistic moments, probability density functions, cumulative distribution functions, and so forth, the study proposes a method using high-dimensional model representation (HDMR) in the meshless framework. In HDMR, the nonlinear connection between input variables and the output response is represented in terms of hierarchical correlated function expansions. These hierarchical functions are approximated over the response values evaluated using deterministic analysis on selected number and combinations of control points of random variables involved. Lagrange interpolation method is employed for the construction of response surfaces. Unlike Monte Carlo simulation (MCS) which needs numerous deterministic nonlinear analyses to be performed during each simulation, HDMR requires only a few deterministic nonlinear analyses to construct the component functions. Hence, simulations performed on the reduced order models of high-dimensional response can be used to estimate the probabilistic moments as well as density functions and cumulative distributions of response, with high computational efficiency compared to MCS. The probabilistic moments and distributions produced for a range of coefficient of variation of input random field up to 30%, for both the normal and lognormal input random fields are compared with those provided by crude MCS on the system of equations and are observed to be matching well.

NUMERICAL-STATISTICAL INVESTIGATION OF SUPEREXPONENTIAL GROWTH OF THE MEAN PARTICLE FLUX WITH MULTIPLICATION IN A HOMOGENEOUS RANDOM MEDIUM

The new correlative-grid approximation of a homogeneous random field is introduced for the effective numerically-analytical investigation of the superexponential growth of the mean particles flux with multiplication in a random medium. A complexity of particle trajectory realization is not dependent on the correlation scale. The test computations for a critical ball with isotropic scattering showed high accuracy of the corresponding mean flux estimates. For the correlative-grid approximation the possibility of Gaussian asymptotics of the mean particles multiplication rate when the correlation scale decreases is justified.

Correlation Structure of the Solution to the Reaction-Diffusion Equation in Respond to Random Fluctuations of the Boundary Conditions

In this paper, we deal with the reaction-diffusion equation subject to Dirichlet and Neumann boundary conditions where the input function on the boundary is randomly fluctuated. First we study the fundamental case when this function is a white noise. Explicit form of the correlation function is derived for the reaction-diffusion equation in a half-plane. In this case we obtain the Karhunen–Loève expansion (KL) of the solution which is a partially homogeneous random field, i.e., it is homogeneous along the horizontal direction, and is inhomogeneous in the vertical direction. Then, based on this representation, we extend this result to the general case when the function prescribed on the boundary is an arbitrary homogeneous random field.

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.

Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes the physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp., rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp., its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For the convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.

Stationary solutions and local equations for interacting diffusions on regular trees

We study the invariant measures of infinite systems of stochastic differential equations (SDEs) indexed by the vertices of a regular tree. These invariant measures correspond to Gibbs measures associated with certain continuous specifications, and we focus specifically on those measures which are homogeneous Markov random fields. We characterize the joint law at any two adjacent vertices in terms of a new two-dimensional SDE system, called the “local equation”, which exhibits an unusual dependence on a conditional law. Exploiting an alternative characterization in terms of an eigenfunction-type fixed point problem, we derive existence and uniqueness results for invariant measures of the local equation and infinite SDE system. This machinery is put to use in two examples. First, we give a detailed analysis of the surprisingly subtle case of linear coefficients, which yields a new way to derive the famous Kesten-McKay law for the spectral measure of the regular tree. Second, we construct solutions of tree-indexed SDE systems with nearest-neighbor repulsion effects, similar to Dyson’s Brownian motion.

THE STATISTICAL SIMULATION OF RANDOM FIELDS WITH THE GAUSSIAN TYPE CORRELATION FUNCTION BY THE INVESTIGATION OF THE MAGNETOMETRY DATA

In the article, universal methods of statistical modeling (Monte Carlo methods) of geophysical data using the Gaussian correlation function have been developed, which make it possible to solve the problems of generating adequate realizations of random fields on a grid in three-dimensional space of required regularity and detail. Since in geophysics, most of the results of object research are presented in digital form, the accuracy of which depends on various random influences, the problem of the condition of the maps arises in the case when the data cannot be obtained with the specified detail in some observation areas. It is proposed to apply statistical simulation of random fields methods, to solve the problems of conditional maps, supplement the required detail of research results with additional data, to achieve the required accuracy of observations, and other similar problems in geophysics. An algorithm for numerical modeling of realizations of homogeneous isotropic random fields in three-dimensional space with a Gaussian correlation function is formulated on the basis of the theorem on estimation of the mean-square approximation of such random fields by the partial sum of the "spectral decomposition" series. Using the example of data from aeromagnetic surveying in the area of the Ovruch depression, the proposed algorithm for statistical modeling of random fields is implemented in solving the problems of map fitness by supplementing the data with simulated adequate implementations to the required level of detail. When analyzing data by profiles, they are divided into deterministic (trend) and random components. The trend is proposed to approximate by cubic splines and the homogeneous isotropic random component is proposed to modeling on the basis "spectral decomposition" of random fields on 3-D space in the Ovruch depression. According to the algorithm, authors received random component implementations on the study area with twice detail for each profile. When checking their adequacy, authors made the conclusions that the relevant random components histogram has Gaussian distribution. The built variogram of these implementations has the best approximation by theoretical variogram which is connected to the Gaussian type correlation function. As a result of superimposing the simulated array of the random component on the spline approximation of the real data, a more detailed implementation was obtained for the data of geomagnetic observations in the selected area. A comparative analysis of the results of modeling realizations random fields with the Gaussian correlation function with other correlation functions is carried out. Therefore, the method of statistical modeling of realizations of random fields in three-dimensional space with the Gaussian correlation function makes it possible to supplement the results of measurements of the full magnetic field intensity vector with data with a given detail as much as possible.

Generalization of the Glivenko–Cantelli theorem to finite-dimensional distributions of ergodic homogeneous random fields

Let X(t),t∈Zm×Rk, be an ergodic stationary random processes or an ergodic homogeneous random field (k+m≥1). We prove that the distribution function of each random vector (X(t1),…,X(ts)) can be a.s. arbitrary fine uniformly approximated by the empirical distribution functions, if, e.g., in their construction increasing bounded convex sets Tn with infinitely increasing intrinsic diameters are used. We consider also the case when X is observed on a mixing homogeneous countable random set Sm(ω)⊂Rm (e.g., on a Poisson random set). These results open a way to consistent statistical inference on finite-dimensional distribution functions of ergodic processes and fields.

Three‐stage non‐Gaussian homogeneous random field representation over manifolds

AbstractRandom fields are widely used to characterize the inherent spatial uncertainty in engineering applications. However, for widely existing manifolds, few studies focus on the simulation of Gaussian/non‐Gaussian fields over manifolds due to the geometry complexity. Recently, we proposed a two‐stage method to simulate Gaussian fields over manifolds. In this study, we extend it to a three‐stage method for non‐Gaussian fields. In the first stage, the original manifold is dimensional reduced to a Euclidean domain by the isometric feature mapping. By this step, the geodesic distance is preserved to guarantee the correlation structure unchanged. In the second stage, conventional representations can be used to simulate the underlying Gaussian field over the Euclidean domain. In the third stage, an enhanced translation procedure is proposed to translate the underlying Gaussian field to the target non‐Gaussian field over the manifold. To explain the applicability and accuracy of the three‐stage method, two kinds of non‐Gaussian fields over three different manifolds are simulated. Then, a global error is proposed to evaluate the performance of the method, and the error analysis demonstrates the method is accurate enough. Finally, two engineering structures, that is, a compressed half‐cylinder and a Canopy shell with complex geometry, are investigated to illustrate the potential applications of the method.

Polyadic random fields

The paper considers mean-square continuous, wide-sense homogeneous, and isotropic random fields taking values in a linear space of polyadics. We find a set of such fields whose values are symmetric and positive-definite dyadics, and outline a strategy for their simulation.

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 simple and efficient stochastic meshfree method for linear eigenvalue problems in structural mechanics

A novel and simple stochastic meshfree method is proposed in the present study for the stochastic eigenvalue analysis of problems in structural mechanics. Young’s modulus is modelled as homogeneous random field. Both truncated normal and lognormal distribution characteristics are used. The current work proposes to use Monte Carlo simulation (MCS) on a modified system of equations. Here, the stochastic stiffness matrix is approximated using Taylor series expansion as in perturbation methods. However, the method does not make any approximations for eigensolutions, which are also unknown functions of random variables. This enables the numerical integration of stiffness matrix and its derivatives outside the simulation loop. Unlike the direct MCS on system of equations, this simple but previously unexplored modification proposed, can result in accurate and more efficient evaluation of stochastic eigensolutions. A few numerical examples are solved using the proposed method; buckling of columns, buckling of thin plates and free vibration of beams. The probabilistic characteristics evaluated using the proposed method are found to be matching well with those obtained from direct MCS.

Active Motion Induced by Random Electromagnetic Fields

Active particles are capable of extracting energy from their environment and converting it into direct motion. This direct motion is usually attained by asymmetries, which induce a net force in the particle. In this article we show how a dimer made up of two nanoparticles can obtain this energy supply from the nonreciprocal force induced by an homogeneous and isotropic random electromagnetic field. We explicitly obtain the value of the force and find it to be different from zero only for sets of different particles in which at least one of them is absorbent. In addition, we prove that the nonreciprocity of the force is the condition for the interaction dynamics to be nonconservative. We further show how this mechanism can be used to induce active Brownian motion with persistent random walks, effective diffusion coefficients, and noticeable Péclet numbers.

Series representations and simulations of isotropic random fields in the Euclidean space

This paper introduces the series expansion for homogeneous, isotropic and mean square continuous random fields in the Euclidean space, which involves the Bessel function and the ultraspherical polynomial, but differs from the spectral representation in terms of the ordinary spherical harmonics that has more terms at each level.The series representation provides a simple and efficient approach for simulation of isotropic (non-Gaussian) random fields.

Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum

This paper presents a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.

Randomized multivariate Central Limit Theorems for ergodic homogeneous random fields

We present new versions of the CLT which are valid for each ergodic homogeneous multivariate random field (X1(⋅),…,Xd(⋅)) on Rm or Zm(m≥1) (in particular, for each ergodic stationary random process and for each ergodic stationary random sequence) such that for all l E[|Xl(0)|2+δ]<∞ for some δ>0(l=1,...,d); in some statements ergodicity is not assumed. Randomization made it possible to significantly weaken the strong mixing conditions and other restrictions of dependence, that are imposed in the conventional CLTs. These results pave the way to consistent statistical inference for homogeneous random fields and stationary processes with strong dependence.