Lognormal Random Field Research Articles

C3NN: Cosmological Correlator Convolutional Neural Network an Interpretable Machine-learning Framework for Cosmological Analyses

Modern cosmological research in large-scale structure has witnessed an increasing number of machine-learning applications. Among them, convolutional neural networks (CNNs) have received substantial attention due to their outstanding performance in image classification, cosmological parameter inference, and various other tasks. However, many models based on CNNs are criticized as “black boxes” due to the difficulties in relating their outputs intuitively and quantitatively to the cosmological fields under investigation. To overcome this challenge, we present the Cosmological Correlator Convolutional Neural Network (C3NN)—a fusion of CNN architecture and cosmological N-point correlation functions (NPCFs). We demonstrate that its output can be expressed explicitly in terms of the analytically tractable NPCFs. Together with other auxiliary algorithms, we can open the “black box” by quantitatively ranking different orders of the interpretable outputs based on their contribution to classification tasks. As a proof of concept, we demonstrate this by applying our framework to a series of binary classification tasks using Gaussian and log-normal random fields and relating its outputs to the NPCFs describing the two fields. Furthermore, we exhibit the model’s ability to distinguish different dark energy scenarios (w 0 = −0.95 and −1.05) using N-body simulated weak-lensing convergence maps and discuss the physical implications coming from their interpretability. With these tests, we show that C3NN combines advanced aspects of machine learning architectures with the framework of cosmological NPCFs, thereby making it an exciting tool to extract physical insights in a robust and explainable way from observational data.

Uncertainty modeling and propagation for groundwater flow: a comparative study of surrogates

We compare sparse grid stochastic collocation and Gaussian process emulation as surrogates for the parameter-to-observation map of a groundwater flow problem related to the Waste Isolation Pilot Plant in Carlsbad, NM. The goal is the computation of the probability distribution of a contaminant particle travel time resulting from uncertain knowledge about the transmissivity field. The latter is modelled as a lognormal random field which is fitted by restricted maximum likelihood estimation and universal kriging to observational data as well as geological information including site-specific trend regression functions obtained from technical documentation. The resulting random transmissivity field leads to a random groundwater flow and particle transport problem which is solved realization-wise using a mixed finite element discretization. Computational surrogates, once constructed, allow sampling the quantities of interest in the uncertainty analysis at substantially reduced computational cost. Special emphasis is placed on explaining the differences between the two surrogates in terms of computational realization and interpretation of the results. Numerical experiments are given for illustration.

Stochastic analysis for bearing capacity computation of twin strip footings on geogrid-reinforced sands

ABSTRACT The interaction of closely-spaced footings on soils is of concern for recent decades. The inherent variability of soil makes the topic more challenging. This study investigates the behaviour of twin strip foundations on both unreinforced and geogrid-reinforced spatially random sands. The Random Finite Difference is performed by coupling Matlab and FLAC2D in each Monte Carlo simulation. Two types of sands, dense and loose, are assumed as the spatially correlated log-normal random fields and the friction angle is considered a random parameter. This study discusses how much the bearing capacity of twin footings is impressed by the heterogeneity of random sands. The unreinforced soil results show that when the uncertainty of φ is high, the homogeneous soil assumption could overestimate the interference effect by about 18% and 9% in the dense and loose sands, respectively. While soil reinforcement reduces the difference between results obtained from probabilistic analysis with those calculated with deterministic analysis. Moreover, the isotropic random fields with rapid fluctuation of φ yield the greatest interference influence in the unreinforced and reinforced dense sands. The combined impact of interference and reinforcement is greater in loose sand than in dense sand, regardless of whether the soil is heterogeneous or homogeneous. HIGHLIGHT Random heterogeneous variability of φ on the interaction of the twin foundations on both unreinforced and geogrid-reinforced sand was investigated. For twin footings on unreinforced dense sand, Scr/B changes from 1.25 to somewhere between 1.25 and 1.5 by increasing COV φ , while on the unreinforced loose sand, Scr/B = 1 remains constant, irrespective of COV φ values. For high variability of φ, the assumption of soil without variability results in overestimation of twin footings interference effect by about 18% and 9%, which means the risk acceptance yields irrevocable damages. Increasing the number of geogrid layers, N, declines the risk of estimating the analysis with homogeneous soil, especially for high variability of φ. By all means, the footing interference and geogrid effects become lower when N is added. The interference effect is usually higher for twin footings on unreinforced sand than on the same reinforced sand. Overall, the integrated influence of interference and reinforcement in loose sand is more than that in dense sand The combined influence of interference and the first geogrid layer is the most in the isotropic random field (θ x = θ y = 1 m) for both sand since the rapid variation of φ in both directions increases the influence and reinforcement effects. The deterministic analysis underestimates the interaction of twin foundations on the reinforced loose sand compared to the anisotropic random fields.

Impact of observation thresholds in the assessment of field-scale soil saturated hydraulic conductivity

The soil saturated hydraulic conductivity (Ks) plays a key role in the partitioning of rainfall into runoff and infiltration. Field-scale variability of Ks is often represented as a lognormal random field, and its parameters are frequently estimated by calibrating areal-average infiltration models with infiltration observations obtained under natural/artificial rainfall conditions. Such observations are obtained in terms of censored moments as infiltration is controlled by only a fraction of the Ks random field during a rainfall event. In this study, the impact of such observations on the maximum likelihood estimates (MLE) of the Ks distribution parameters is evaluated. Based on the data from several rainfall-runoff events observed under natural conditions over an experimental field, the results demonstrate the role of temporal variation of rainfall in resolving the Ks field for a rainfall event. Validation results show that the performance of the MLEs is impacted by both the fraction of the Ks field resolved by a rainfall event and its capacity to capture the variability of Ks. Finally, recommendations are provided as to what constitutes a good calibration event. The impact of incorporating reliability of the calibration events in consolidating information was tested using an information-theoretic measure.

Inverse reliability analysis and design for tunnel face stability considering soil spatial variability

The traditional deterministic analysis for tunnel face stability neglects the uncertainties of geotechnical parameters, while the simplified reliability analysis which models the potential uncertainties by means of random variables usually fails to account for soil spatial variability. To overcome these limitations, this study proposes an efficient framework for conducting reliability analysis and reliability-based design (RBD) of tunnel face stability in spatially variable soil strata. The three-dimensional (3D) rotational failure mechanism of the tunnel face is extended to account for the soil spatial variability, and a probabilistic framework is established by coupling the extended mechanism with the improved Hasofer-Lind-Rackwits-Fiessler recursive algorithm (iHLRF) as well as its inverse analysis formulation. The proposed framework allows for rapid and precise reliability analysis and RBD of tunnel face stability. To demonstrate the feasibility and efficacy of the proposed framework, an illustrative case of tunnelling in frictional soils is presented, where the soil's cohesion and friction angle are modelled as two anisotropic cross-correlated lognormal random fields. The results show that the proposed method can accurately estimate the failure probability (or reliability index) regarding the tunnel face stability and can efficiently determine the required supporting pressure for a target reliability index with soil spatial variability being taken into account. Furthermore, this study reveals the impact of various factors on the support pressure, including coefficient of variation, cross-correlation between cohesion and friction angle, as well as autocorrelation distance of spatially variable soil strata. The results also demonstrate the feasibility of using the forward and/or inverse first-order reliability method (FORM) in high-dimensional stochastic problems. It is hoped that this study may provide a practical and reliable framework for determining the stability of tunnels in complex soil strata.

Dynamic reliability analysis of three-dimensional slopes considering the spatial variability in soil parameters

Slopes are commonly a significant aspect of geotechnical engineering projects. The incorrect evaluation and prediction of slope stability can cause serious disasters when it experiences seismic activity, and reliability is the key index a slope’s dynamic state. Spatial variability is an inherent soil property and has an essential influence on the dynamic response of a slope body. However, in previous studies, soil was often described deterministically, and the interaction between the uncertainty and spatial dimensions was ignored. To fill this gap, in this paper, practical expressions for the cohesion and friction angle in each working condition are given to simulate spatial variability by a lognormal random field. Then, the finite element method and effective dynamic reliability calculation method are used to obtain the safety factor of a 3D slope. Finally, the generalized probability density evolution method is performed to obtained the probability density information of the dynamic slope response and judge the influence of the soil spatial variability probabilistically. Due to the spatial variability in soil, the probability density distribution under each condition is distributed in a large range and thus may not provide results that are much more dangerous than those of the deterministic simulation. Meanwhile, the friction angle is more sensitive than cohesion and shows stronger discreteness in the spatial variability. The change in the cross-correlation coefficient has a greater influence on reliability than the autocorrelation length. Consequently, the simplification of model slopes in dimension and variability could deviate the reliability results from actual dynamic responses, either overestimating or underestimating, resulting in incorrect designs and evaluations.

Reliability analysis of pile-reinforced slopes in width-limited failure mode considering three-dimensional spatial variation of soil strength

Stabilization pile is a common measurement for landslide mitigation. In previous literatures, the deterministic safety factor Fs method has been well established for pile-reinforced slope stability analysis, for example by limit analysis. However, the Fs method by itself cannot reflect the inherent uncertainty and spatial variability of mechanical properties of soil. This paper presents a probabilistic analysis procedure for width-constrained pile-reinforced slope where the failure pattern and soil spatial variability are both accounted for under three-dimensional (3D) condition. A modified discretization kinematic analysis-based mechanism is employed as the deterministic model. The spatial variability of cohesion c and friction angle φ is modeled as lognormal random fields by using the Karhunen–Loève expansion. The sparse polynomial chaos expansion (SPCE) is used to construct the meta-model to reduce the computational cost in high dimensional stochastic problems. The failure probability, probability density function (PDF), and other useful reliability results can be provided by performing Monte Carlo simulations of the proposed SPCE meta-model. Finally, the influences of soil spatial variability, slope geometry, and pile parameters on the slope reliability are analyzed. The combination of the modified discretization mechanism and SPCE provides a useful tool for the probabilistic analysis of pile-reinforced earth slopes under 3D condition. The probabilistic simulation results are also helpful for reliability design of pile-reinforced slope in practice.

Influences of spatial variability of hydrothermal properties on the freezing process in artificial ground freezing technique

Artificial ground freezing (AGF) has been extensively applied in underground engineering construction as a popular ground-support technique. In recent decades, a series of investigations on AGF have been conducted through analytical, experimental and numerical methods. However, as critical parameters in AGF, the inherent spatial variabilities of hydrothermal properties of soils have been merely considered in previous studies. Therefore, this work aims to explore the influences of spatial variability of the hydrothermal properties (i.e., thermal conductivity and permeability) on AGF by stochastic finite element method combined with Monte Carlo simulation (MCS). Two-dimensional lognormal random fields are generated to account for the spatial variability of hydrothermal properties. The closure time is adopted as an indicator to evaluate the impact of uncertainty in hydrothermal properties on the freezing efficiency of AGF. In addition, the effect of MCS number on the statistics of the closure time in AGF is determined. Results demonstrate that spatial variability of the hydrothermal properties considerably affects the estimation of closure time. This study reveals the importance of random analysis in coping with the variation in hydrothermal properties and provides references for the design and construction of AGF.

Fast and realistic large-scale structure from machine-learning-augmented random field simulations

ABSTRACT Producing thousands of simulations of the dark matter distribution in the Universe with increasing precision is a challenging but critical task to facilitate the exploitation of current and forthcoming cosmological surveys. Many inexpensive substitutes to full N-body simulations have been proposed, even though they often fail to reproduce the statistics of the smaller non-linear scales. Among these alternatives, a common approximation is represented by the lognormal distribution, which comes with its own limitations as well, while being extremely fast to compute even for high-resolution density fields. In this work, we train a generative deep learning model, mainly made of convolutional layers, to transform projected lognormal dark matter density fields to more realistic dark matter maps, as obtained from full N-body simulations. We detail the procedure that we follow to generate highly correlated pairs of lognormal and simulated maps, which we use as our training data, exploiting the information of the Fourier phases. We demonstrate the performance of our model comparing various statistical tests with different field resolutions, redshifts, and cosmological parameters, proving its robustness and explaining its current limitations. When evaluated on 100 test maps, the augmented lognormal random fields reproduce the power spectrum up to wavenumbers of $1 \, h \, \rm {Mpc}^{-1}$, and the bispectrum within 10 per cent, and always within the error bars, of the fiducial target simulations. Finally, we describe how we plan to integrate our proposed model with existing tools to yield more accurate spherical random fields for weak lensing analysis.

EFFICIENT APPROXIMATION OF HIGH-DIMENSIONAL EXPONENTIALS BY TENSOR NETWORKS

In this work a general approach to compute a compressed representation of the exponential $\exp(h)$ of a high-dimensional function $h$ is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of log-normal random fields or the evaluation of Bayesian posterior measures. Usually, these high-dimensional objects are intractable numerically and can only be accessed pointwise in sampling methods. In contrast, the proposed method constructs a functional representation of the exponential by exploiting its nature as a solution of an ordinary differential equation. The application of a Petrov--Galerkin scheme to this equation provides a tensor train representation of the solution for which we derive an efficient and reliable a posteriori error estimator. Numerical experiments with a log-normal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent low-rank representations for the respective applications. Although the present work considers only a specific differential equation, the presented method can be applied in a more general setting. We show that the composition of a generic holonomic function and a high-dimensional function corresponds to a differential equation that can be used in our method. Moreover, the differential equation can be modified to adapt the norm in the a posteriori error estimates to the problem at hand.

Map-based cosmology inference with lognormal cosmic shear maps

ABSTRACT Most cosmic shear analyses to date have relied on summary statistics (e.g. ξ+ and ξ−). These types of analyses are necessarily suboptimal, as the use of summary statistics is lossy. In this paper, we forward-model the convergence field of the Universe as a lognormal random field conditioned on the observed shear data. This new map-based inference framework enables us to recover the joint posterior of the cosmological parameters and the convergence field of the Universe. Our analysis properly accounts for the covariance in the mass maps across tomographic bins, which significantly improves the fidelity of the maps relative to single-bin reconstructions. We verify that applying our inference pipeline to Gaussian random fields recovers posteriors that are in excellent agreement with their analytical counterparts. At the resolution of our maps – and to the extent that the convergence field can be described by the lognormal model – our map posteriors allow us to reconstruct all summary statistics (including non-Gaussian statistics). We forecast that a map-based inference analysis of LSST-Y10 data can improve cosmological constraints in the σ8–Ωm plane by $\approx\!{30}{{\ \rm per\ cent}}$ relative to the currently standard cosmic shear analysis. This improvement happens almost entirely along the $S_8=\sigma _8\Omega _{\rm m}^{1/2}$ directions, meaning map-based inference fails to significantly improve constraints on S8.

Bayesian inference for random field parameters with a goal-oriented quality control of the PGD forward model’s accuracy

Numerical models built as virtual-twins of a real structure (digital-twins) are considered the future of monitoring systems. Their setup requires the estimation of unknown parameters, which are not directly measurable. Stochastic model identification is then essential, which can be computationally costly and even unfeasible when it comes to real applications. Efficient surrogate models, such as reduced-order method, can be used to overcome this limitation and provide real time model identification. Since their numerical accuracy influences the identification process, the optimal surrogate not only has to be computationally efficient, but also accurate with respect to the identified parameters. This work aims at automatically controlling the Proper Generalized Decomposition (PGD) surrogate’s numerical accuracy for parameter identification. For this purpose, a sequence of Bayesian model identification problems, in which the surrogate’s accuracy is iteratively increased, is solved with a variational Bayesian inference procedure. The effect of the numerical accuracy on the resulting posteriors probability density functions is analyzed through two metrics, the Bayes Factor (BF) and a criterion based on the Kullback-Leibler (KL) divergence. The approach is demonstrated by a simple test example and by two structural problems. The latter aims to identify spatially distributed damage, modeled with a PGD surrogate extended for log-normal random fields, in two different structures: a truss with synthetic data and a small, reinforced bridge with real measurement data. For all examples, the evolution of the KL-based and BF criteria for increased accuracy is shown and their convergence indicates when model refinement no longer affects the identification results.

Direct Simulation Methods for a Class of Normal and Lognormal Random Fields with Applications in Modeling Material Properties

Direct Simulation Methods for a Class of Normal and Lognormal Random Fields with Applications in Modeling Material Properties

Reliability based design optimization of bridges considering bridge-vehicle interaction by Kriging surrogate model

This paper presents a reliability-based design optimization method for bridge structures, considering the uncertainties of the material parameters and the effect of bridge-vehicle interaction. The uncertainties of system parameters, such as Young’s modulus and mass density, are considered, which are simulated as Gaussian and/or lognormal random fields. The formulas used to represent the random fields of material properties and system matrices are derived. The Gaussian random inputs are approximated with Karhunen-Loeve (KL) expansion, while the lognormal random inputs are approximated with a combination of KL expansion and Polynomial Chaos (PC) expansion. Reliability-based design optimization analysis is conducted to determine the minimum required cross-section area of bridge structures under probability constraints, in which the failure probability is estimated from the Kriging surrogate model and Monte Carlo Simulation (MCS) method. Numerical studies on a simply-supported beam under a moving force and a three-dimensional box-section bridge under a moving vehicle are conducted to investigate the efficiency and accuracy of the proposed approach. The reliability analysis results are compared with those obtained from MCS method, validating the accuracy of the proposed approach. With the proposed method, the minimum value of cross-section area of the bridge under the preset probability constraint can be obtained.

Modeling Seepage Flow and Spatial Variability of Soil Thermal Conductivity during Artificial Ground Freezing for Tunnel Excavation

Artificial ground freezing (AGF) technology has been commonly applied in tunnel construction. Its primary goal is to create a frozen wall around the tunnel profile as a hydraulic barrier and temporary support, but it is inevitably affected by two natural factors. Firstly, seepage flows provide large and continuous heat energy to prevent the soil from freezing. Secondly, as a key soil parameter in heat transfer, the soil thermal conductivity shows inherent spatial variability, binging uncertainties in freezing effects and efficiency. However, few studies have explored the influence of spatially varied soil thermal conductivity on AGF. In this study, a coupled hydro-thermal numerical model was developed to examine the effects of seepage on the formation of frozen wall. The soil thermal conductivity is simulated as a lognormal random field and analyzed by groups of Monte-Carlo simulations. The results confirmed the adverse effect of groundwater flow on the formation of frozen wall, including the uneven development of frozen body towards the downstream side and the higher risk of water leakage on the upstream face of the tunnel. Based on random finite element analysis, this study quantitively tabulated the required additional freezing time above the deterministic scenario. Two levels of the additional freezing time are provided, namely the average level and conservative level, which aim to facilitate practitioners in making a rule-of-thumb estimation in the design of comparable situations. The findings can offer practitioners a rule of thumb for estimating the additional freezing times needed in artificial ground freezing, accounting for the seepage flow and spatial variation in soil thermal conductivity.

Lognormal Random Field models to identify temporal Land cover changes using full polarimetric L-Band SAR imagery

Multiplicative Autoregressive Random Field (MAR) based texture models have been identified as one of the most appropriate models for SAR intensity images to capture the stochastic spatial interaction among neighboring pixels. But very few studies have tested their viability particularly in disaster applications. In this paper, we analyse the MAR texture models for their advantageous in land cover change detection compared to the changes resulting from logarithm of SAR image intensity and speckle filtered SAR imagery. The paper shows that lognormal random fields with multiplicative spatial interactions in the form of MAR models can be an effective alternative to suppress speckle noise and model SAR image intensity in time series data analysis. The pre and post disaster observational data of the Tohoku earthquake, in the east coast of Japan, acquired by the Advanced Land Observation Satellite (ALOS)/phased array type L-band synthetic aperture radar (PALSAR) were synthesized using MAR model based texture measures. Two of the main texture descriptors of the MAR model were considered primarily in this study. Those are the neighborhood weighting and the noise variance parameters. A 2nd order neighborhood configuration was used to estimate them. We present a variogram based analysis, structural similarity index measure (SSIM), and the mean ratio detector (MRD) as three different approaches to analyse the changes in land cover using radar texture. The change detection results of the MRD were further tested using area error proportion (AEP), root mean square error (RMSE) and correlation coefficient (CC), keeping normalized ratio, principle component analysis (PCA) and adaptive Lee filtered polarimetric intensity based change as the references.

Non-intrusive reliability analysis of unsaturated embankment slopes accounting for spatial variabilities of soil hydraulic and shear strength parameters

Existing reliability analyses of embankment slopes usually considered the spatial variabilities of shear strength parameters and saturated hydraulic conductivity separately. Additionally, the variability in the soil–water characteristic curve (SWCC) was ignored. A non-intrusive approach is proposed for reliability analyses of unsaturated embankment slopes under four different cases on a steady-state seepage analysis. The spatial variabilities of hydraulic parameters (including fitting parameters, a and n, of SWCC and saturated hydraulic conductivity) and shear strength parameters are accounted for simultaneously. To illustrate the proposed approach, a hypothetical embankment under unsaturated seepage is investigated. The soil parameters of the embankment and foundation are modeled as lognormal random fields and lognormal random variables, respectively, to take into account the inherent uncertainties. Parametric sensitivity studies are then performed to account explicitly for the influence of the spatial variation of hydraulic parameters on the reliability of embankment slope stability. The proposed approach can act as a practical and rigorous tool for estimating the reliability of embankment slopes with small levels of probability of failure (i.e., 10−4–10−3). An interesting finding that the probability of failure of the unsaturated embankment slope decreases with the variability of the fitting parameter n of SWCC is observed and explained.

Effect of Spatial Variability on Disc Cutters Failure During TBM Tunneling in Hard Rock

This study examines the effect of variations of the rock property and TBM parameters on the cutter failure. A new disc cutter wear model is developed by coupling Archard’s model and TBM cutter kinematics. The rock property and the TBM parameters are modelled as anisotropic cross-correlated log-normal random fields. The proposed model with a discretized algorithm allows a quantitative evaluation of the possible cutter failure during the TBM tunneling. The failure of the cutter head system is characterized by a random failure pattern and its failure can be evaluated using a probability method. With this technique, it can be a useful tool in providing an indication for the cutters replacement. Both the cutter and its mechanical system life are determined based on their allowed working distance of the TBM at the targeted safety level. The obtained results indicate that the life of the cutters and its cutter head system decreases with the presence of rock spatial heterogeneity. Finally, the obtained attenuation factors incorporating different randomness levels is practically useful for the evaluation of the cutter life.

Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: Random lognormal permeability

SummaryEfficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen‐Loève expansion (KLE). The stochastic variability of permeability is modeled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss‐Seidel used as a preconditioner for CG is shown to be efficient and robust for stochastic finite element method.

Stochastic turbulence modeling in RANS simulations via multilevel Monte Carlo

A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method. The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al. (2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level.