non-Gaussian Fields Research Articles

Solving inverse problems using conditional invertible neural networks

Inverse modeling for computing a high-dimensional spatially-varying property field from indirect sparse and noisy observations is a challenging problem. This is due to the complex physical system of interest often expressed in the form of multiscale PDEs, the high-dimensionality of the spatial property of interest, and the incomplete and noisy nature of observations. To address these challenges, we develop a model that maps the given observations to the unknown input field in the form of a surrogate model. This inverse surrogate model will then allow us to estimate the unknown input field for any given sparse and noisy output observations. Here, the inverse mapping is limited to a broad prior distribution of the input field with which the surrogate model is trained. In this work, we construct a two- and three-dimensional inverse surrogate models consisting of an invertible and a conditional neural network trained in an end-to-end fashion with limited training data. The invertible network is developed using a flow-based generative model. The developed inverse surrogate model is then applied for an inversion task of a multiphase flow problem where given the pressure and saturation observations the aim is to recover a high-dimensional non-Gaussian log-permeability field where the two facies consist of heterogeneous log-permeability and varying length-scales. For both the two- and three-dimensional surrogate models, the predicted sample realizations of the non-Gaussian log-permeability field are diverse with the predictive mean being close to the ground truth even when the model is trained with limited data.

Reflection on inactive and energetic survey of sandwich synthesized glazed layers

Right now, study the material and geometric vulnerability consequences for the static, free vibration and dynamic conduct of sandwich shaft structures. A higher-request sandwich board hypothesis is considered for the examination. The flexible properties of the sandwich pillar are considered as 1-D non -Gaussian irregular field which causes neighborhood variety in mass and firmness frameworks of the shaft. The discretization of the non-Gaussian irregular fields is performed utilizing the development ideal direct estimation. To play out the numerical investigation, Monte-Carlo reproduction alongside the computationally proficient time-space ghastly component technique is proposed. Numerical outcomes are acquired for various limit conditions just as for various materials in the sandwich face sheet and center. Results got right now the impacts of material and geometric vulnerability in the reaction conduct of a sandwich shaft. The individual impact of center thickness and Poisson's proportion of the center on the static, free vibration and dynamic reaction is measured. It is seen that the vulnerabilities in material and geometric properties alongside stacking and limit conditions impact the static, free vibration and dynamic reaction.

Accurate analytic model for the weak lensing convergence one-point probability distribution function and its autocovariance

The one-point probability distribution function (PDF) is a powerful summary statistic for non-Gaussian cosmological fields, such as the weak lensing (WL) convergence reconstructed from galaxy shapes or cosmic microwave background (CMB) maps. Thus far, no analytic model has been developed that successfully describes the high-convergence tail of the WL convergence PDF for small smoothing scales from first principles. Here, we present a halo-model formalism to compute the WL convergence PDF, building upon our previous results for the thermal Sunyaev-Zel'dovich field. Furthermore, we extend our formalism to analytically compute the covariance matrix of the convergence PDF. Comparisons to numerical simulations generally confirm the validity of our formalism in the non-Gaussian, positive tail of the WL convergence PDF, but also reveal the convergence PDF's strong sensitivity to small-scale systematic effects in the simulations (e.g., due to finite resolution). Finally, we present a simple Fisher forecast for a Rubin-Observatory-like survey, based on our new analytic model. Considering the $\{A_s, \Omega_m, \Sigma m_\nu\}$ parameter space and assuming a Planck CMB prior on $A_s$ only, we forecast a marginalized constraint $\sigma(\Sigma m_\nu) \approx 0.08$ eV from the WL convergence PDF alone, even after marginalizing over parameters describing the halo concentration-mass relation. This error bar on the neutrino mass sum is comparable to the minimum value allowed in the normal hierarchy, illustrating the strong constraining power of the WL convergence PDF. We make our code publicly available at https://github.com/leanderthiele/hmpdf.

New interpretable statistics for large-scale structure analysis and generation

We introduce Wavelet Phase Harmonics (WPH) statistics: interpretable low-dimensional statistics that describe 2D non-Gaussian fields. These statistics are built from WPH moments, which were recently introduced in the data science and machine learning community. We apply WPH statistics to projected 2D matter density fields from the Quijote N-body simulations of the large-scale structure of the Universe. By computing Fisher information matrices, we find that the WPH statistics place more stringent constraints on four of five cosmological parameters when compared to statistics based on the combination of the power spectrum and bispectrum. We also use the WPH statistics with a maximum entropy model to statistically generate new 2D density fields that accurately reproduce the probability density function, the mean and standard deviation of the power spectrum, the bispectrum, and Minkowski functionals of the input density fields. Although other methods are efficient for either parameter estimates or statistical syntheses of the large-scale structure, WPH statistics are the first statistics that achieve state-of-the-art results for both tasks as well as being interpretable.

Stochastic simulation of the spatio-temporal field of the average daily heat index in Southern Russia

Numerical models of the heat index time series and spatio-temporal fields can be used for a variety of purposes, from the study of the dynamics of heat waves to projections of the influence of future climate on humans. To conduct these studies one must have efficient numerical models that successfully reproduce key features of the real weather processes. In this study, 2 numerical stochastic models of the spatio-temporal non-Gaussian field of the average daily heat index (ADHI) are considered. The field is simulated on an irregular grid determined by the location of weather stations. The first model is based on the method of the inverse distribution function. The second model is constructed using the normalization method. Real data collected at weather stations located in southern Russia are used to both determine the input parameters and to verify the proposed models. It is shown that the first model reproduces the properties of the real field of the ADHI more precisely compared to the second one, but the numerical implementation of the first model is significantly more time consuming. In the future, it is intended to transform the models presented to a numerical model of the conditional spatio-temporal field of the ADHI defined on a dense spatio-temporal grid and to use the model constructed for the stochastic forecasting of the heat index.

A new approach to observational cosmology using the scattering transform

ABSTRACT Parameter estimation with non-Gaussian stochastic fields is a common challenge in astrophysics and cosmology. In this paper, we advocate performing this task using the scattering transform, a statistical tool sharing ideas with convolutional neural networks (CNNs) but requiring neither training nor tuning. It generates a compact set of coefficients, which can be used as robust summary statistics for non-Gaussian information. It is especially suited for fields presenting localized structures and hierarchical clustering, such as the cosmological density field. To demonstrate its power, we apply this estimator to a cosmological parameter inference problem in the context of weak lensing. On simulated convergence maps with realistic noise, the scattering transform outperforms classic estimators and is on a par with the state-of-the-art CNN. It retains advantages of traditional statistical descriptors, has provable stability properties, allows to check for systematics, and importantly, the scattering coefficients are interpretable. It is a powerful and attractive estimator for observational cosmology and the study of physical fields in general.

Influence of non-Gaussian characteristics of wind load on fatigue damage of wind turbine

Based on translation models, both Gaussian and non-Gaussian wind fields are generated using spectral representation method for investigating the influence of non-Gaussian characteristics and directivity effect of wind load on fatigue damage of wind turbine. Using the blade aerodynamic model and multi-body dynamics, dynamic responses are calculated. Using linear damage accumulation theory and linear crack propagation theory, crack initiation life and crack propagation life are discussed with consideration of the joint probability density distribution of the wind direction and mean wind speed in detail. The result shows that non-Gaussian characteristics of wind load have less influence on fatigue life of wind turbine in the area with smaller annual mean wind speeds. Whereas, the influence becomes significant with the increase of the annual mean wind speed. When the annual mean wind speeds are 7 m/s and 9 m/s at hub height of 90 m, the crack initiation lives under softening non-Gaussian wind decrease by 10% compared with Gaussian wind fields or at higher hub height. The study indicates that the consideration of the influence of softening non-Gaussian characteristics of wind inflows can significantly decrease the fatigue life, and, if neglected, it can result in non-conservative fatigue life estimates for the areas with higher annual mean wind speeds.

Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements.

The aim of this work is to efficiently and robustly solve the statistical inverse problem related to the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneous anisotropic materials with a complex microstructure that usually cannot be properly described in terms of their mechanical constituents at microscale. Within the context of linear elasticity theory, the apparent elasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field. A general methodology using multiscale displacement field measurements simultaneously made at both macroscale and mesoscale has been recently proposed for the identification the hyperparameters of such a prior stochastic model by solving a multiscale statistical inverse problem using a stochastic computational model and some information from displacement fields at both macroscale and mesoscale. This paper contributes to the improvement of the computational efficiency, accuracy and robustness of such a method by introducing (i) a mesoscopic numerical indicator related to the spatial correlation length(s) of kinematic fields, allowing the time-consuming global optimization algorithm (genetic algorithm) used in a previous work to be replaced with a more efficient algorithm and (ii) an ad hoc stochastic representation of the hyperparameters involved in the prior stochastic model in order to enhance both the robustness and the precision of the statistical inverse identification method. Finally, the proposed improved method is first validated on in silico materials within the framework of 2D plane stress and 3D linear elasticity (using multiscale simulated data obtained through numerical computations) and then exemplified on a real heterogeneous biological material (beef cortical bone) within the framework of 2D plane stress linear elasticity (using multiscale experimental data obtained through mechanical testing monitored by digital image correlation).

Non-parametric simulation of non-stationary non-gaussian 3D random field samples directly from sparse measurements using signal decomposition and Markov Chain Monte Carlo (MCMC) simulation

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

Sampling of Bayesian posteriors with a non-Gaussian probabilistic learning on manifolds from a small dataset

This paper tackles the challenge presented by small-data to the task of Bayesian inference. A novel methodology, based on manifold learning and manifold sampling, is proposed for solving this computational statistics problem under the following assumptions: (1) neither the prior model nor the likelihood function are Gaussian and neither can be approximated by a Gaussian measure; (2) the number of functional input (system parameters) and functional output (quantity of interest) can be large; (3) the number of available realizations of the prior model is small, leading to the small-data challenge typically associated with expensive numerical simulations; the number of experimental realizations is also small; (4) the number of the posterior realizations required for decision is much larger than the available initial dataset. The method and its mathematical aspects are detailed. Three applications are presented for validation: The first two involve mathematical constructions aimed to develop intuition around the method and to explore its performance. The third example aims to demonstrate the operational value of the method using a more complex application related to the statistical inverse identification of the non-Gaussian matrix-valued random elasticity field of a damaged biological tissue (osteoporosis in a cortical bone) using ultrasonic waves.

Tensor train-Karhunen-Loève expansion: New theoretical and algorithmic frameworks for representing general non-Gaussian random fields

The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional physical domain in the tensor train format; secondly, to develop a new algorithmic framework for accurately computing the modes and describing the correlation structure of the latent factors beyond second order within moderate time. Central to the algorithmic framework is the tensor train decomposition of cumulant functions. This decomposition is accurately computed with a novel rank-revealing algorithm. For computing the modes, compared with existing Galerkin-type and collocation-type methods, the proposed computational procedure totally removes the need of selecting the basis functions or collocation points and the quadrature points, which not only greatly enhances adaptivity, but also avoids solving large-scale eigenvalue problems. Moreover, by computing with higher-order cumulant functions, the new theoretical and algorithmic frameworks show great potential for representing general non-Gaussian non-homogeneous random fields. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the efficiency and accuracy of the proposed algorithmic framework.

Kernel-based two-dimensional principal component analysis applied for parameterization in history matching

The process of minimizing the difference between observed and simulated data by adjusting the input model parameters is typically called history matching. Geological parameterization approaches allow high-dimensional geological models to be replaced by relatively low-dimensional parameters in history matching problem. It is important because of the incompetence of optimization algorithms in problems with numerous decision variables. The principal component analysis (PCA) method is commonly used for the representation of geological models in terms of a few parameters. Though, it can only preserve two-point statistics of a random field which is inadequate for regenerating the complicated structures. Kernel-based PCA (KPCA) has been developed to allow the conservation of higher-order statistics in random fields. The most important deficiency of these methods is the consideration of geological realizations in vector forms that causes parameterization methods can only survey the structure of realizations in one direction. As a major disadvantage for fields with anisotropic covariance, spatial correlations in different directions are not properly preserved. We employ two-dimensional principal component analysis (2DPCA) rather than PCA for surveying the structure of two-dimensional realizations in both directions. We consider the realizations as two-dimensional matrices and perform original KPCA in two directions. This method is applied for regenerating Gaussian and non-Gaussian geological random fields with smaller normal random fields and as a parameterization method in history matching. In the Gaussian case, the superiority of 2DPCA over PCA is demonstrated. For non-Gaussian case, both methods lack adequate precision. Hence, a kernel-based 2DPCA (K2DPCA) is developed that exhibits better performance in the regeneration of channelized geological structures compared with KPCA. It is demonstrated that K2DPCA allows preserving the higher-order statistics behind the complex channelized structures when used as a parameterization method for history matching problem in a channelized reservoir.

Efficient analytical upscaling method for elliptic equations in three-dimensional heterogeneous anisotropic media

We propose an efficient analytical upscaling method to compute the equivalent conductivity tensor for elliptic equations in three-dimensional space. Our approach uses perturbation expansion and Fourier analysis, and considers heterogeneity, anisotropy and geometry of coarse gridblocks. Through low-order approximation, the derived analytical solution accurately approximates the central-difference numerical solution with periodic boundary conditions. Numerical tests are performed to demonstrate the capability and efficiency of this analytical approach in upscaling fluid flow in heterogeneous formations. We test the method in synthetic examples and benchmark cases with both Gaussian random fields and channelized non-Gaussian fields. In addition, we examine the impact of each parameter on the upscaled conductivity, and investigate the sensitivity of the variance and correlation lengths to the coefficients. We also indicate how to extend this approach to multiphase flow problems.

High-precision Monte Carlo modelling of galaxy distribution

We revisit the case of fast Monte Carlo simulations of galaxy positions for a non-Gaussian field. More precisely, we address the question of generating a 3D field with a given one-point function (e.g. log-normal) and some power spectrum fixed by cosmology. We highlight and investigate a problem that occurs in the log-normal case when the field is filtered, and we identify a regime where this approximation still holds. However, we show that the filtering is unnecessary if aliasing effects are taken into account and the discrete sampling step is carefully controlled. In this way we demonstrate a sub-percent precision of all our spectra up to the Nyquist frequency. We extend the method to generate a full light cone evolution, comparing two methods for this process, and validate our method with a tomographic analysis. We analytically and numerically investigate the structure of the covariance matrices obtained with such simulations which may be useful for future large and deep surveys.

Non-Gaussian Lagrangian Stochastic Model for Wind Field Simulation in the Surface Layer

Wind field simulation in the surface layer is often used to manage natural resources in terms of air quality, gene flow (through pollen drift), and plant disease transmission (spore dispersion). Although Lagrangian stochastic (LS) models describe stochastic wind behaviors, such models assume that wind velocities follow Gaussian distributions. However, measured surface-layer wind velocities show a strong skewness and kurtosis. This paper presents an improved model, a non-Gaussian LS model, which incorporates controllable non-Gaussian random variables to simulate the targeted non-Gaussian velocity distribution with more accurate skewness and kurtosis. Wind velocity statistics generated by the non-Gaussian model are evaluated by using the field data from the Cooperative Atmospheric Surface Exchange Study, October 1999 experimental dataset and comparing the data with statistics from the original Gaussian model. Results show that the non-Gaussian model improves the wind trajectory simulation by stably producing precise skewness and kurtosis in simulated wind velocities without sacrificing other features of the traditional Gaussian LS model, such as the accuracy in the mean and variance of simulated velocities. This improvement also leads to better accuracy in friction velocity (i.e., a coupling of three-dimensional velocities). The model can also accommodate various non-Gaussian wind fields and a wide range of skewness–kurtosis combinations. Moreover, improved skewness and kurtosis in the simulated velocity will result in a significantly different dispersion for wind/particle simulations. Thus, the non-Gaussian model is worth applying to wind field simulation in the surface layer.

Wind turbine load dynamics in the context of turbulence intermittency

Abstract. The importance of a high-order statistical feature of wind, which is neglected in common wind models, is investigated: non-Gaussian distributed wind velocity increments related to the intermittency of turbulence and their impact on wind turbine dynamics and fatigue loads are the focus. Gaussian and non-Gaussian synthetic wind fields obtained from a continuous-time random walk model are compared and fed to a common aero-servo-elastic model of a wind turbine employing blade element momentum (BEM) aerodynamics. It is discussed why and how the effect of the non-Gaussian increment statistics has to be isolated. This is achieved by assuring that both types feature equivalent probability density functions, spectral properties and coherence, which makes them indistinguishable based on wind characterizations of common design guidelines. Due to limitations in the wind field genesis, idealized spatial correlations are considered. Three examples with idealized; differently sized wind structures are presented. A comparison between the resulting wind turbine loads is made. For the largest wind structure sizes, differences in the fatigue loads between intermittent and Gaussian are observed. These are potentially relevant in a wind turbine certification context. Subsequently, the dependency of this intermittency effect on the field's spatial variation is discussed. Towards very small structured fields, the effect vanishes.

Coupled hydrogeophysical inversion to identify non-Gaussian hydraulic conductivity field by jointly assimilating geochemical and time-lapse geophysical data

Reliable inversion of spatial heterogeneity of hydraulic conductivity is crucial to understand subsurface fluids migration. The Ensemble Smoother – Direct Sampling method (ES-DS) has proven to be an effective method to identify non-Gaussian hydraulic conductivity distributions by incorporating a variety of traditional hydrodynamic measurements, e.g., piezometric head. However, inversion problems for non-Gaussian parameters often suffer from a sparsity of the available data from direct sampling in boreholes. As a non-intrusive, cost-effective, and high sampling density method, time-lapse geophysical technique has not yet drawn much attention as a useful source of information for delineating the underlying non-Gaussian heterogeneity. In this study, we integrated coupled hydrogeophysical modeling and the ES-DS algorithm to estimate non-Gaussian hydraulic conductivity field by assimilating both geochemical and time-lapse geophysical datasets. Four synthetic Cases for a salt injection experiment, monitored by both sampling analysis and electrical resistivity tomography, are conducted to assess the ability of the proposed approach to characterize hydraulic properties by assimilating different types of data. Results show that using geochemical or geophysical data alone only allow a rough reconstruction of subsurface heterogeneity of aquifers but might lose the fine structure. By incorporating multi-source datasets, the main patterns of the non-Gaussian reference fields can be reflected with an improved resolution. The time-lapse geophysical methods open up new opportunities to accurately characterize non-Gaussian aquifers and monitor the dynamic processes of subsurface fluids.

Efficient Parallel Algorithm for Estimating Higher-order Polyspectra

Abstract Nonlinearities in the gravitational evolution, galaxy bias, and redshift-space distortion drive the observed galaxy density fields away from the initial near-Gaussian states. Exploiting such a non-Gaussian galaxy density field requires measuring higher-order correlation functions, or, its Fourier counterpart, polyspectra. Here, we present an efficient parallel algorithm for estimating higher-order polyspectra. Based upon the Scoccimarro estimator, the estimator avoids direct sampling of polygons using the fast Fourier transform, and the parallelization overcomes the large memory requirement of the original estimator. In particular, we design the memory layout to minimize the inter-CPU communications, which excels in the code performance.

A non-Gaussian random field model for earthquake slip

The present study proposes a methodology to generate stochastic slip field with desired probability structure and spectral characteristics. The characterization utilizes 100 tsunamigenic and 130 non-tsunamigenic slip models available in the SRCMOD database. First, the cumulative distribution function (CDF) of marginal slip distributions is obtained. Then, it compares the estimated CDFs with nine standard distribution functions based on model fitting criteria like Akaike Information Criterion, mean squared error, correlation coefficient, and QQ plots. The analysis showed that the majority of slip distributions follow generalized Pareto or truncated exponential distribution functions. For the parameters of CDF estimated from slip fields, an empirical relation is proposed as a function of magnitude (Mw) to predict the corresponding values. As the next step, to understand the correlation structure of the slip field, semi-variograms are estimated along strike and along downdip for all rupture models considered in the study. A total of nine theoretical variogram models is compared with the semi-variogram of each slip field. From the analysis based on model fitting criteria, most of the slip field is observed to follow a stable variogram model. An empirical equation is developed to predict the sill and range of stable variogram function for a given Mw. Then, an iterative method based on translational field theory is proposed to obtain an ensemble of non-Gaussian slip fields for a given Mw and type of source. The generated sample fields from the proposed method match with the prescribed non-Gaussian distribution and variogram structure. In the sequel, the method estimates the variogram and the corresponding spectral density of the underlying Gaussian field according to translation field theory. The scaling relations and the proposed iterative methodology can be used to obtain an ensemble of possible earthquake slip fields.

Simulations of Non-Gaussian Property Fields Based on the Apparent Properties of Statistical Volume Elements

The properties of composite materials with random microstructures are often defined by homogenizing the properties of a representative volume element (RVE). This results in the effective properties of an equivalent homogeneous material. This approach is useful for predicting a global response but smooths the underlying variability of the composite's properties resulting from the random microstructure. Statistical volume elements (SVEs) are partitions of an RVE. Homogenization of individual SVEs produces a population of apparent properties. While not as rigorously defined as RVEs, SVEs can still provide a repeatable framework to characterize mesoscale variability in composite properties. In particular, their statistical properties can be used as the basis for simulation studies. For this work, Voronoi tessellation was used to partition RVEs into SVEs and apparent properties developed for each SVE. The resulting field of properties is characterized with respect to its spatial autocorrelation and distribution. These autocorrelation and distribution functions (PDFs) are then used as target fields to simulate additional property fields, with the same probabilistic characteristics. Simulations based on SVEs may provide a method of further exploring the uncertainty within the underlying approximations or of highlighting effects that might be experimentally measurable or used to validate the use of an SVE mesoscale analysis in a specific predictive model. This work presents an update to an existing simulation technique developed by Joshi (1975, “A Class of Stochastic Models for Porous Media,” Ph.D. thesis, University of Kansas, Lawrence, KS) and initially extended by Adler et al. (1990, “Flow in Simulated Porous Media,” Int. J. Multiphase Flow, 16(4), pp. 691–712). The simulation methodology is illustrated for three random microstructures and two SVE partitioning sizes.