Non-parametric simulation of random field samples from incomplete measurements using generative adversarial networks

The paper deals with the theory and methods of statistical simulation of random processes and fields based on their spectral decomposition and Kotelnikov-Shennon modified interpolation sums, as well as applying these methods for environmental geophysical monitoring. Statistical simulation of multivariate random fields (those homogeneous in time and homogeneous isotropic in n other variables) are considered to be essential for seismological research into frequency characteristics of geological media. A statistical model and a numerical algorithm of simulating random fields are built on the basis of Kotelnikov-Shennon modified interpolation decomposition to generate adequate realizations of seismic noise. The paper examines real-valued random fields ξ(t,x),tϵÎR,xÎRn, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. It also considers approximation of random fields by the random fields with a bounded spectrum. There is made an analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum. Besides, there are obtained estimates of the mean-square approximation of random fields in the space R´Rn by a model constructed with the help of spectral decomposition and Kotelnikov–Shannon interpolation formula. The paper provides a mechanism for statistical simulation of Gaussian random fields with a bounded spectrum; namely, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. Proved have been the theorems of the mean-square approximation of random fields (those homogeneous in time and homogeneous isotropic ones relative to n- other variables) by special partial sums. A simulation method was used to formulate an algorithm of numerical simulation by means of these theorems. There are also considered ways to carry out spectral analysis of generated seismic noise realizations. Finally, there have been developed universal methods of statistical simulation (Monte Carlo methods) of multi-parameter seismology data for generating seismic noise on 2D and 3D grids of the required detail and regularity.

The lifespan of a reinforced concrete (RC) structure can be greatly influenced by the spatial variability of its material characteristics which, in particular, explains the observed or measured reduction of the tensile strength at first crack when the volume under tension increases. This paper discusses the ability of accounting for the spatial variability of the tensile strength of concrete in RC structures using a stochastic finite element (SFE) method based on random field simulations. In this work, the generation of random fields on the concrete tensile strength aims at computing the force corresponding to the first crack occurrence, and the reduced tensile strength of the structure. The method can be applied in particular to large‐sized structures, which show a pronounced size effect, for different types of loading. The method consists of, first, estimating the mean of the random field, using the analytical approach of the weakest link and localization method (WL ). Then, the discretized random field is defined on a particular 2D or 3D grid, and it is finally projected on the finite element mesh of the studied structure. The study of the parameters that influence the prediction of the cumulative density functions (CDFs) of the rupture force or the tensile strength is highlighted using experimental series of concrete beams having different volumes and subjected to 4‐point bending loading. Moreover, the SFE method is applied to a RC tie‐beam under tensile loading, characterized by a weak stress gradient, which complicates the prediction of crack positions.

Uncertainty Quantification of Random Fields Based on Spatially Sparse Data by Synthesizing Bayesian Compressive Sensing and Stochastic Harmonic Function

The three-condition version of the uniform conditional state combustion model makes use of the mixture fraction, progress variable, and normalized total enthalpy as conditioning variables to build a three-dimensional conditional manifold for chemistry. In order to map the solution in conditional space into the flow domain, the joint Probability Density Function (PDF) of the conditioning variables needs to be modeled. In simulations, presumed functions (i.e., β-PDF for the mixture fraction and progress variable and δ-PDF for total enthalpy) are often used for modeling the marginal PDFs. In this work, the measurements from the Cambridge/Sandia burner are employed to obtain the marginal PDFs for the conditioning variables at various points in the reacting domain. The measurements are then combined from all positions in space to form conditional PDFs of the normalized total enthalpy for various values of the other two variables. In the vicinity of the flame brush, the marginal PDF of the normalized total enthalpy resembles a bimodal Gaussian distribution; nonetheless, the conditional PDFs for this variable are nearly Gaussian distributions. The correlation coefficients between the conditioning variables are also investigated, and the assumption of their statistical independence is examined. To consider the association between the conditioning variables for modeling, the copula concept is introduced, and the performances of three different copulas are tested. Furthermore, the statistical moments of the conditioning variables are computed from the experimental data at different points and are utilized for modeling the joint PDF of the conditioning variables from two different approaches that are compared.

Conditioned simulation of local fields of earthquake ground motion

Medical image reconstruction is typically an ill-posed inverse problem. In order to address such ill-posed problems, the prior distribution of the sought after object property is usually incorporated by means of some sparsity-promoting regularization. Recently, prior distributions for images estimated using generative adversarial networks (GANs) have shown great promise in regularizing some of these image reconstruction problems. In this work, we apply an image-adaptive GAN-based reconstruction method (IAGAN) to reconstruct high fidelity images from incomplete medical imaging data. It is observed that the IAGAN method can potentially recover fine structures in the object that are relevant for medical diagnosis but may be oversmoothed in reconstructions with traditional sparsity-promoting regularization.

A rigorous multiscale random field approach to generate large scale rough rock surfaces

3rd-order Spectral Representation Method: Simulation of multi-dimensional random fields and ergodic multi-variate random processes with fast Fourier transform implementation

A new perspective on the simulation of cross-correlated random fields

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

Simulation of simply cross correlated random fields by series expansion methods

Simulation of cross-correlated non-Gaussian random fields for layered rock mass mechanical parameters

The effect of random porosity field on supercritical carbonation of cement-based materials

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from “best” to “worst” conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.

In this work, the theory of random fields is used to account for the influence of spatial variability on slope reliability. Within this framework the friction coefficient along a discontinuity is treated as a Gaussian random field which is fully described by its mean value, standard deviation and spatial correlation length. The random field is simulated using the Local Average Subdivision (LAS) method. As shown by the examples presented herein, the spatial correlation of shear strength along a failure plane can have an important influence on slope performance, as expressed by the failure probability. This is a significant observation since ignoring the influence of spatial correlation in design may lead to non-conservative estimations of slope reliability. The planar mode of failure is considered.