Gaussian Random Field Model Research Articles

Rotation in Event Horizon Telescope Movies

The Event Horizon Telescope (EHT) has produced images of M87* and Sagittarius A*, and will soon produce time sequences of images, or movies. In anticipation of this, we describe a technique to measure the rotation rate, or pattern speed Ω p , from movies using an autocorrelation technique. We validate the technique on Gaussian random field models with a known rotation rate and apply it to a library of synthetic images of Sgr A* based on general relativistic magnetohydrodynamics simulations. We predict that EHT movies will have Ω p ≈ 1° per GMc −3, which is of order 15% of the Keplerian orbital frequency in the emitting region. We can plausibly attribute the slow rotation seen in our models to the pattern speed of inward-propagating spiral shocks. We also find that Ω p depends strongly on inclination. Application of this technique will enable us to compare future EHT movies with the clockwise rotation of Sgr A* seen in near-infrared flares by GRAVITY. Pattern speed analysis of future EHT observations of M87* and Sgr A* may also provide novel constraints on black hole inclination and spin, as well as an independent measurement of black hole mass.

3-D Adaptive AUV Sampling for Classification of Water Masses

Autonomous underwater vehicles with onboard computing units foster innovative approaches for sampling oceanographic phenomena. Feedback of observations via the onboard model for planning algorithms enable adaptive sampling for such robotic units. In this work, we develop, implement, and test an adaptive sampling algorithm for efficient sampling of water masses in a 3-D frontal system. Focusing on a river plume, salinity variations are used to characterize the water masses. A threshold in salinity is assumed to distinguish the ocean and river waters, so that excursions below the threshold define river waters. The onboard model builds on a Gaussian random field representation of the salinity variations in (north, east, depth) coordinates. This model is initially trained from numerical ocean model data, and then updated with data gathered by the vehicle sensor. The Gaussian random field model further allows closed-form expressions of the expected spatially integrated Bernoulli variance of the salinity excursion set, which is used to reward sampling efforts. Combining these results with forward-looking planning algorithms, we suggest a workflow for 3-D adaptive sampling to map river plume systems. Simulation studies are used to compare the suggested approach with others. Results of field trials in the Nidelva river plume in Norway are presented and discussed.

Nonsteady dynamics at the dynamic depinning transition in the two-dimensional Gaussian random-field Ising model.

With large-scale Monte Carlo simulations, we investigate the nonsteady relaxation at the dynamic depinning transition in the two-dimensional Gaussian random-field Ising model. The dynamic scaling behavior is carefully analyzed, and the transition fields as well as static and dynamic exponents are accurately determined based on the short-time dynamic scaling form. Different from the usual assumption, two distinguished growth processes of spatial correlation lengths for the velocity and height of the domain wall are found. Thus, the universality class of the depinning transition is established, which significantly differs from that of the quenched disorder equationbut agrees with that of the recent experiment as well as other simulations works. Under the influence of the mesoscopic time regime, the crossover from the second-order phase transition to the first-order one is confirmed in the weak-disorder regime, yielding an abnormal disorder-dependent nature of the criticality.

Tile low-rank approximations of non-Gaussian space and space-time Tukey g-and-h random field likelihoods and predictions on large-scale systems

Large-scale statistical modeling has become necessary with the vast flood of geospace data coming from various sources. In space statistics, the Maximum Likelihood Estimation (MLE) is widely considered for modeling geospace data by estimating a set of statistical parameters related to a predefined covariance function. This covariance function describes the correlation between a set of geospace locations where the main goal is to model given data samples and impute missing data. Climate/weather modeling is a prevalent application for the MLE operation where data interpolation and forecasting are highly required. In the literature, the Gaussian random field is often used to describe geospace data as one of the most popular models for MLE. However, real-life datasets are often skewed and/or have extreme values, and non-Gaussian random field models are more appropriate for capturing such features. In this work, we provide an exact and approximate parallel implementation of the well-known Tukey g-and-h (TGH) non-Gaussian random field in the context of climate/weather applications. The proposed implementation alleviates the computation complexity of the log-likelihood function, which requires O(n2) storage and O(n3) operations, where N is the number of geospace locations, M is the number of time slots, and n=N×M. Based on tile low-rank (TLR) approximations, our implementation of the TGH model can tackle large-scale problems. Furthermore, we rely on task-based programming models and dynamic runtime systems to provide fast execution for the MLE operation in space and space-time cases. We assess the performance and accuracy of the proposed implementations using synthetic space and space-time datasets up to 800K. We also consider a 12-month precipitation dataset in Germany to demonstrate the advantage of using non-Gaussian over Gaussian random field models. We evaluate the prediction accuracy of the TGH model on the precipitation dataset using the Probability Integral Transformation (PIT) tool showing that the TGH model outperforms the Gaussian modeling in the real dataset. Moreover, our performance assessment indicates that TLR computations allow solving larger matrix sizes while preserving the required accuracy for prediction. The TLR-based approximation shows a speedup up to 7.29X and 2.96X over the exact solution.

Data-Drive Site Characterization for Benchmark Examples: Sparse Bayesian Learning versus Gaussian Process Regression

In this paper, two data-drive site characterization methods, the sparse Bayesian learning (SBL) method and the Gaussian process regression (GPR) method, are benchmarked by a set of virtual ground examples and a real ground example of cone penetration test (CPT) data. The two methods both assume a zero-mean prior Gaussian random field model for the spatial trend, but the strategies of maintaining model simplicity are different. The SBL method produces a simple trend model by adopting sparse basis functions, whereas the GPR method produces a simple trend model by adopting a kernel function governed by few hyperparameters. The accuracy of the two methods in predicting the cone tip resistance (qt) of CPT was quantified by the root-mean square prediction error (RMSE), whereas the accuracy in identifying soil layers was quantified by the identification rate (IR). It was found that the GPR method in general outperforms the SBL method. Further accuracy improvement for the GPR method can be obtained if a clustering analysis based on the Robertson’s soil behavior index (Ic) is conducted.

Structured prediction of sparse dependent variables for traffic state estimation in large-scale networks

Currently, one of the biggest challenges in modern traffic engineering is related to traffic state estimation (TSE). Although many machine learning and domain models can be used for TSE, they do not consider the sparsity and spatial dependence of traffic state variables. In this paper, we propose a hybrid soft computing model of two Gaussian conditional random field (GCRF) models for the inference of traffic speed, which is a relevant variable for TSE and travel information systems. The proposed model can infer the traffic state variables in large-scale networks whose nodes are geographically dispersed. Moreover, by combining a Gaussian conditional random field binary classification model (GCRFBC), which classifies traffic regimes as free-flow or potentially congested, and a regression GCRF model for the prediction of traffic speed in potentially congested traffic regimes, the model addresses two specifics of the problem: sparsity in traffic data, and the fact that observations are not independent. The proposed model was tested on two large-scale real-world networks in Serbia, namely an arterial E70-E75 335 km long highway stretch and the major ski resort Kopaonik with 55 km of ski slopes. In addition, the proposed model showed better prediction performance than several other unstructured and structured models.

A Perspective on Methods to Computationally Design the Morphology of Aerogels

Reconstructing aerogel morphology presents significant challenges, in particular, if 3D visualizations of their mesoporous network are desired. Available microscopic and tomographic tools find it difficult to probe into all types of aerogels for the purposes of reconstructing their 3D nanoporous morphology. This is where computational approaches have shown promising efforts. Herein, diverse models that can be applied to describing different aerogels are explored. To begin with, cluster–cluster aggregation models are examined for simulating the sol–gel process and the resulting morphologies in fractal aerogels, e.g., silica‐based. Gaussian random field models and polymerization‐induced phase separation models are explored for modeling organic non‐fractal aerogels, e.g., resorcinol‐formaldehyde (RF) ones. This is followed by Langevin‐dynamics‐based discrete element models that are explored for simulating gelation in fibrillar aerogels, e.g., those from biopolymer sources. Lastly, modified Voronoi approaches are investigated for describing the 3D fibrillar morphology, also of fibrillar aerogels, like those from biopolymers. A perspective is presented highlighting the strengths as well as shortcomings in each of the model approaches. Possibilities to either extend available approaches or explore new ones are briefly discussed at every interval.

Superstatistical Wind Fields from Pointwise Atmospheric Turbulence Measurements

Accurate models of turbulent wind fields have become increasingly important in the atmospheric sciences, e.g., for the determination of spatiotemporal correlations in wind parks, the estimation of individual loads on turbine rotor and blades, or the modeling of particle-turbulence interaction in atmospheric clouds or pollutant distributions in urban settings. Because of the difficult task of resolving the fields across a broad range of scales, one oftentimes has to invoke stochastic wind field models that fulfill specific, empirically observed, properties. Whereas commonly used Gaussian random field models solely control second-order statistics (i.e., velocity correlation tensors or kinetic energy spectra), we explicitly show that our extended model emulates the effects of higher-order statistics as well. Most importantly, the empirically observed phenomenon of small-scale intermittency, which can be regarded as one of the key features of atmospheric turbulent flows, is reproduced with a very high level of accuracy and at considerably low computational cost. Our method is based on a multipoint statistical description of turbulent velocity fields that consists of a superposition of multivariate Gaussian statistics with fluctuating covariances. We propose a new and efficient sampling algorithm for this Gaussian scale mixture and demonstrate how such “superstatistical” wind fields can be constrained on a certain number of real-world measurement data points from a meteorological mast array.Received 12 April 2022Revised 26 July 2022Accepted 22 August 2022DOI:https://doi.org/10.1103/PRXEnergy.1.023006Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAtmospheric scienceBoundary layersShear flowsStochastic processesStructure & turbulence of boundary layersSustainabilityTurbulenceWind energyEnergy Science & TechnologyNonlinear DynamicsFluid DynamicsStatistical Physics

Gaussian conditional random fields for classification

Gaussian conditional random fields (GCRF) are a well-known structured model for continuous outputs that uses multiple unstructured predictors to form its features and at the same time exploits dependence structure among outputs, which is provided by a similarity measure. In this paper, a Gaussian conditional random field model for structured binary classification (GCRFBC) is proposed. The model is applicable to classification problems with undirected graphs, intractable for standard classification CRFs. The model representation of GCRFBC is extended by latent variables which yield some appealing properties. Thanks to the GCRF latent structure, the model becomes tractable, efficient and open to improvements previously applied to GCRF regression models. In addition, the model allows for reduction of noise, that might appear if structures were defined directly between discrete outputs. Two different forms of the algorithm are presented: GCRFBCb (GCRGBC — Bayesian) and GCRFBCnb (GCRFBC — non-Bayesian). The extended method of local variational approximation of sigmoid function is used for solving empirical Bayes in Bayesian GCRFBCb variant, whereas MAP value of latent variables is the basis for learning and inference in the GCRFBCnb variant. The inference in GCRFBCb is solved by Newton–Cotes formulas for one-dimensional integration. Both models are evaluated on synthetic data and real-world data. We show that both models achieve better prediction performance than unstructured predictors. Furthermore, computational and memory complexity is evaluated. Advantages and disadvantages of the proposed GCRFBCb and GCRFBCnb are discussed in detail.

Surrogate modelling of wind fields from point-wise atmospheric turbulence measurements

We present an advanced model for the generation of synthetic wind fields that can be understood as an extension of the well-known Mann model. In contrast to such Gaussian random field models which control second-order statistics (i.e., velocity correlation tensors or spectra), we demonstrate that our extended model incorporates the effects of higherorder statistics as well. In particular, the empirically observed phenomenon of small-scale intermittency, a key feature of atmospheric turbulent flows, can be reproduced with high accuracy and at considerably low computational cost. Our method is based on a recently developed multipoint statistical description of a turbulent velocity field [J. Friedrich et al., J. Phys. Complex. 2 045006 (2021)] and consists of a superposition of multivariate Gaussian statistics with fluctuating covariances. Furthermore, we explicitly show how such superstatistical Mann fields can be constraint on a certain number of point-wise measurement data. We give an outlook on the relevance of such surrogate wind fields in the context of fatigue loads on wind turbines.

How Do Magnetic Field Models Affect Astrophysical Limits on Light Axion-like Particles? An X-Ray Case Study with NGC 1275

Axion-like particles (ALPs) are a well-motivated extension to the standard model of particle physics, and X-ray observations of cluster-hosted AGN currently place the most stringent constraints on the ALP coupling to electromagnetism, g a γ , for very light ALPs (m a ≲ 10−11 eV). We revisit limits obtained by Reynolds et al. using Chandra X-ray grating spectroscopy of NGC 1275, the central AGN in the Perseus cluster, examining the impact of the X-ray spectral model and magnetic field model. We also present a new publicly available code, ALPro, which we use to solve the ALP propagation problem. We discuss evidence for turbulent magnetic fields in Perseus and show that it can be important to resolve the magnetic field structure on scales below the coherence length. We reanalyze the NGC 1275 X-ray spectra using an improved data reduction and baseline spectral model. We find the limits are insensitive to whether a partially covering absorber is used in the fits. At low m a (m a ≲ 10−13 eV), we find marginally weaker limits on g a γ (by 0.1–0.3 dex) with different magnetic field models, compared to Model B from Reynolds et al. (2020). A Gaussian random field (GRF) model designed to mimic ∼50 kpc scale coherent structures also results in only slightly weaker limits. We conclude that the existing Model B limits are robust assuming that β pl ≈ 100, and are insensitive to whether cell-based or GRF methods are used. However, astrophysical uncertainties regarding the strength and structure of cluster magnetic fields persist, motivating high-sensitivity RM observations and tighter constraints on the radial profile of β pl.

Stochastic aspects of crack deflection and crack path prediction in short fiber reinforced polymer matrix composites

Owing to the production process, short fiber reinforced composites exhibit a pronounced anisotropy of both elastic properties and crack growth resistance. In particular the latter issue has a major impact on crack deflection and inevitably has to be taken into account for the sake of an accurate prediction of crack paths. The perpendicular axes of transverse isotropy are associated with the fiber orientations, whereupon a crack in transverse direction encounters the largest fracture toughness. While the local mean fiber orientations in polymer matrix composites are determined by the injection molding process, their statistical fluctuations can approximately be described in terms of Gaussian random field models. Furthermore, statistical variations of the volume fraction of fibers and the fiber–matrix adhesion give rise to stochasticity of the local ratio of fracture toughness anisotropy. Influencing factors of prediction regions of crack paths obtained from finite element simulations with an adapted J-integral deflection criterion are investigated, just as stochastic aspects of bifurcation phenomena of crack deflection observed under mode-I loading.

X-Ray Tomography-Based Microstructure Representation in the Snow Microwave Radiative Transfer Model

The modular Snow Microwave Radiative Transfer (SMRT) model simulates microwave scattering behavior in snow via different selectable theories and snow microstructure representations, which is well suited to intercomparisons analyses. Here, five microstructure models were parameterized from X-ray tomography and thin-section images of snow samples and evaluated with SMRT. Three field experiments provided observations of scattering and absorption coefficients, brightness temperature, and/or backscatter with the increasing complexity of snowpack. These took place in Sodankylä, Finland, and Weissfluhjoch, Switzerland. Simulations of scattering and absorption coefficients agreed well with observations, with higher errors for snow with predominantly vertical structures. For simulation of brightness temperature, difficulty in retrieving stickiness with the Sticky Hard Sphere microstructure model resulted in relatively poor performance for two experiments, but good agreement for the third. Exponential microstructure gave generally good results, near to the best performing models for two field experiments. The Independent Sphere model gave intermediate results. New Teubner–Strey and Gaussian Random Field models demonstrated the advantages of SMRT over microwave models with restricted microstructural geometry. Relative model performance is assessed by the quality of the microstructure model fit to micro-computed tomography (CT) data and further improvements may be possible with different fitting techniques. Careful consideration of simulation stratigraphy is required in this new era of high-resolution microstructure measurement as layers thinner than the wavelength introduce artificial scattering boundaries not seen by the instrument.

Coupled dynamics of axially functionally graded graphene nanoplatelets-reinforced viscoelastic shear deformable beams with material and geometric imperfections

This paper is the first to explore the coupled dynamics of geometrically and material-wise imperfect axially functionally graded (AFG) graphene nanoplatelets-reinforced viscoelastic third-order shear deformable beams. Four AFG graphene nanoplatelets distribution patterns are considered. Porosity, as the material imperfection, is modelled using a Gaussian Random Field model. Four thickness-wise functionally graded porosity distribution patterns are modelled. Effects of geometric imperfection are included by assigning an initial curvature to the beam. To consider the influences associated with energy dissipation caused by internal friction, the Kelvin-Voigt model for viscosity is used. External dissipative energy is modelled using a transverse damper. The effective material properties of the AFG beams are calculated using a modified Halpin-Tsai micromechanics model, together with a rule of mixture. Coupled axial, transverse, and rotational motion equations are obtained by employing a Hamiltonian approach and third-order shear deformation. The natural frequencies are obtained using a modal decomposition method. A simplified version of the graphene nanoplatelets-reinforced AFG structure is verified by a computer code-based finite element method. This novel study on the effects of geometrical imperfection on the sensitivity of beams towards porosity imperfections demonstrates the significance of scrutinising the effects of one imperfection on another.

Mechanical Enhancement of Graded Nanoporous Structure

Abstract Inspired by the development of strong and ductile composite and gradient materials over the past decade, here, we report the investigation of a graded nanoporous (NP) structure utilizing molecular dynamics simulations. The structure is generated by inducing a gradient scaling parameter in a Gaussian random field model. It has a large ligament/pore size toward the core and a small ligament/pore size toward the surface. The redistribution of stress and strain under tensile loading is then scrutinized and compared between the functional graded NP structure and two conventional NP structures with identical relative density but constant ligament size. During loading, the thick ligaments in the gradient structure yield at high stress, leading to the entire structure's high mechanical strength. The thin ligaments help the structure accommodate significant plastic strain by promoting uniform deformation. Both parts of the gradient structure worked collectively and resulted in the structure exhibiting a synergy of excellent strength and good deformability.

Shape-constrained Gaussian process regression for surface reconstruction and multimodal, non-rigid image registration

We present a new statistical framework for landmark ?>curve-based image registration and surface reconstruction. The proposed method first elastically aligns geometric features (continuous, parameterized curves) to compute local deformations, and then uses a Gaussian random field model to estimate the full deformation vector field as a spatial stochastic process on the entire surface or image domain. The statistical estimation is performed using two different methods: maximum likelihood and Bayesian inference via Markov Chain Monte Carlo sampling. The resulting deformations accurately match corresponding curve regions while also being sufficiently smooth over the entire domain. We present several qualitative and quantitative evaluations of the proposed method on both synthetic and real data. We apply our approach to two different tasks on real data: (1) multimodal medical image registration, and (2) anatomical and pottery surface reconstruction.

Thermo-elasto-plastic phase-field modelling of mechanical behaviours of sintered nano-silver with randomly distributed micro-pores

Nano-silver paste is an emerging lead-free bonding material in power electronics, and has excellent mechanical properties, thermal conductivity and long-term reliability. However, it is extremely challenging to model the mechanical and failure behaviours of sintered nano-silver paste due to its random micro-porous structures and the coupled thermomechanical loading conditions. In this study, a novel computational framework was proposed to generate the random micro-porous structures and simulate their effects on mechanical properties and fracture behaviour based on the one-cut gaussian random field model and the thermo-elasto-plastic phase-field model. The elastic modulus, ultimate tensile strength and strain to failure are computed statistically, showing good agreement with the experimental results. Further, the framework was applied to model the fracture of sintered nano-silver paste under thermal cyclic conditions, demonstrating the formation of distinctive crack patterns and complex crack networks. The cracking behaviours observed in the experiments and simulations are remarkably similar to each other. The framework was implemented within Abaqus via a combination of subroutines and Python scripts, automating the process of model generation and subsequent computation. This study provides an efficient and reliable approach to simulate the mechanical and failure behaviours of sintered nano-silver paste with random micro-porous structures.

Reduced-Dimensional Monte Carlo Maximum Likelihood for Latent Gaussian Random Field Models

Monte Carlo maximum likelihood (MCML) provides an elegant approach to find maximum likelihood estimators (MLEs) for latent variable models. However, MCML algorithms are computationally expensive when the latent variables are high-dimensional and correlated, as is the case for latent Gaussian random field models. Latent Gaussian random field models are widely used, for example, in building flexible regression models and in the interpolation of spatially dependent data in many research areas such as analyzing count data in disease modeling and presence-absence satellite images of ice sheets. We propose a computationally efficient MCML algorithm by using a projection-based approach to reduce the dimensions of the random effects. We develop an iterative method for finding an effective importance function; this is generally a challenging problem and is crucial for the MCML algorithm to be computationally feasible. We find that our method is applicable to both continuous (latent Gaussian process) and discrete domain (latent Gaussian Markov random field) models. We illustrate the application of our methods to challenging simulated and real data examples for which maximum likelihood estimation would otherwise be very challenging. Furthermore, we study an often overlooked challenge in MCML approaches to latent variable models: practical issues in calculating standard errors of the resulting estimates, and assessing whether resulting confidence intervals provide nominal coverage. Our study therefore provides useful insights into the details of implementing MCML algorithms for high-dimensional latent variable models. Supplementary materials for this article are available online.

Thermal vibration of functionally graded porous nanocomposite beams reinforced by graphene platelets

The thermal vibration of functionally graded (FG) porous nanocomposite beams reinforced by graphene platelets (GPLs) is studied. The beams are exposed to the thermal gradient with a multilayer structure. The temperature varies linearly across the thickness direction. Three different types of dispersion patterns of GPLs as well as porosity distributions are presented. The material properties vary along the thickness direction. By using the mechanical parameters of closed-cell cellular solid, the variation of Poisson’s ratio and the relation between the porosity coefficient and the mass density under the Gaussian random field (GRF) model are obtained. By using the Halpin-Tsai micromechanics model, the elastic modulus of the nanocomposite is achieved. The equations of motion based on the Timoshenko beam theory are obtained by using Hamilton’s principle. These equations are discretized and solved by using the generalized differential quadrature method (GDQM) to obtain the fundamental frequencies. The effects of the weight fraction, the dispersion model, the geometry, and the size of GPLs, as well as the porosity distribution, the porosity coefficient, the boundary condition, the metal matrix, the slenderness ratio, and the thermal gradient are presented.

Adjusting for Spatial Effects in Genomic Prediction

This paper investigates the problem of adjusting for spatial effects in genomic prediction. Despite being seldomly considered in genomic prediction, spatial effects often affect phenotypic measurements of plants. We consider a Gaussian random field model with an additive covariance structure that incorporates genotype effects, spatial effects and subpopulation effects. An empirical study shows the existence of spatial effects and heterogeneity across different subpopulation families, while simulations illustrate the improvement in selecting genotypically superior plants by adjusting for spatial effects in genomic prediction.