Stochastic Random Field Research Articles

Uncertainty quantification for Fisher-Kolmogorov equation on graphs with application to patient-specific Alzheimer's disease

The Fisher-Kolmogorov equation is a diffusion-reaction PDE that is used to model the accumulation of prionic proteins, which are responsible for many different neurological disorders. Likely, the most important and studied misfolded protein in literature is the Amyloid-$\beta$, responsible for the onset of Alzheimer disease. Starting from medical images we construct a reduced-order model based on a graph brain connectome. The reaction coefficient of the proteins is modelled as a stochastic random field, taking into account all the many different underlying physical processes, which can hardly be measured. Its probability distribution is inferred by means of the Monte Carlo Markov Chain method applied to clinical data. The resulting model is patient-specific and can be employed for predicting the disease's future development. Forward uncertainty quantification techniques (Monte Carlo and sparse grid stochastic collocation) are applied with the aim of quantifying the impact of the variability of the reaction coefficient on the progression of protein accumulation within the next 20 years.

Computational generation of multiphase asphalt nanostructures using random fields

AbstractThis study presents a novel methodology to generate computational replicates of nanostructures of multiphase materials, such as asphalt binders, by integrating image analysis techniques with stochastic random field (RF) modeling. Image analysis techniques are used to identify and segment nanostructure images obtained by atomic force microscopy, while RF is used to model the spatial distribution of their material properties. The results of this process are images showing probable arrangements of nanostructures with stochastic material properties that replicate the experimentally obtained images. The computationally generated nanostructures are then used as inputs in a finite element model to evaluate the effect of heterogeneity on their mechanical response. The efficacy of the developed approach is demonstrated through simulations of asphalt binders’ nanostructures, which reveal novel insights regarding their nanoscale mechanical behavior and response. The FE simulations provided the link between the distribution of nanoscale properties of asphalt binders and variations in their mechanical response. The application of this methodology expands the body of knowledge beyond the deterministic analysis of asphalt binders toward probabilistic analysis and uncertainty quantification that considers their heterogeneous, multiphase structures. Consequently, the methodology can be used to design multiphase materials, such as asphaltic blends, with tailored properties and enhanced performance.

Stochastic reconstruction of 3D microstructures from 2D cross-sectional images using machine learning-based characterization

The availability of high-quality three-dimensional (3D) microstructures is an essential prerequisite to understanding the macroscopic behaviours of heterogeneous media. Considering the high cost of 3D microscopy imaging, it is an economical way to derive large numbers of virtual 3D microstructures from limited morphological information of 2D cross-sectional images. This paper presents an efficient method that incorporates machine learning-based characterization of 2D images to statistically reconstruct 3D microstructures. The latent stochasticity of 2D images is mastered by fitting supervised machine learning models, which essentially characterize the local morphological statistics. A morphology integration scheme is developed to project the 2D morphological statistics into the 3D space, and new equivalent 3D microstructures can then be synthesized by sequentially generating voxel values from probability sampling. The new method is tested on a series of heterogeneous media with distinct morphologies, and it is also compared with two classical methods (i.e. stochastic optimization-based reconstruction and Gaussian random field transformation) in terms of reconstruction accuracy and efficiency. Besides, various microstructural descriptors are used to quantify the discrepancies between the reconstructed and target microstructures. The results confirm that the proposed method provides a highly cost-effective and widely applicable way to reproduce 3D realizations that precisely preserve the statistical characteristics, geometrical irregularities, long-distance connectivity and anisotropy that exist in 2D cross-sectional images.

Stochastic field simulation of slope stability problems: Improvement and reduction of computational effort

In recent years, stochastic simulation has obtained increasing attention in geomechanics as it allows to consider uncertainties in geomaterials’ properties. Spatial uncertainties may influence simulation results and lead to either unsafe or uneconomic designs. By implementing stochastic random field algorithms in Finite-Element simulations, the impact of such uncertainty can be integrated in the simulation results. However, the application of such algorithms implicates new requirements in understanding the relationships between correlation length, mesh coarseness, and required number of Finite-Element simulations. The present study intends to provide a deeper understanding of such aspects. In the presented approach, stochastic random fields analysis is applied in a Finite-Element code by remote scripting. Considering a slope stability analysis and using Monte-Carlo Simulations, the impact of varying mesh coarseness and the correlation length on the obtained safety factor is investigated. As for large numbers of mesh elements, computational costs increase rapidly, the study expands its investigations by employing two further advanced methods, namely mesh adaptivity and subset simulation. These state-of-the-art methods allow to reduce calculation efforts, once by iterative remeshing, once by selective sampling according to failure probability. The results show promising improvements in the concept of computational efforts.

Interval K-L expansion of interval process model for dynamic uncertainty analysis

The interval process model describes a time-variant or dynamic uncertain parameter by the upper and lower bounds rather than the precise probability distribution at each time point, providing an effective structural dynamic uncertainty quantification model with insufficient sample information. By reference to the Karhunen-Loève (K-L) expansion for stochastic process and random field models, a novel expansion method for the interval process model, namely, the interval K-L expansion is proposed in this work. The interval K-L expansion describes the continuous uncertainty of the interval process in time domain by superposition of infinite deterministic time-related functions with uncorrelated interval coefficients, which makes the interval process model much more convenient to use. By introducing a splicing technique, the interval K-L expansion method for multiple correlated interval processes is also presented. Based on the interval K-L expansion, the vibration of structures subject to interval process excitations is analyzed, where the analytic formulation of dynamic response bounds for linear elastic systems and continuum structures are derived. Finally, several numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Deep Markov models of multidimensional random fields

Abstract The paper discusses the use of a doubly stochastic random field model based on autoregressions with multiple roots of characteristic equations for solving the problem of image generation and segmentation. It was proposed to form an image taking into account hidden Markov parameter fields. In particular, the correlation coefficients of the model, as well as the average values and brightness variances, are realizations of random fields, the assessment of which in explicit form is not possible. In addition, a modification of a variational autoencoder for a doubly stochastic model is considered. Finally, based on the use of a modified variational auto encoder, classification and segmentation of satellite images is performed. The combined use of the developed segmentation algorithm with the Unet neural network provides an increase in segmentation accuracy of 3%.

Development of hybrid low salinity water polymer flooding numerical reservoir simulator and smart proxy model for chemical enhanced oil recovery (CEOR)

A MATLAB based finite volume CEOR simulator was developed for modeling the recovery of oil by low salinity water polymer flooding (LSWPF) in a proposed geostatistical method of generating porous media. Firstly, this paper coupled ion exchange, wettability and fluid flow transport models to the proposed heterogeneous porous media to investigate LSWPF oil recovery performance. Secondly, for the first time, a new hybrid smart proxy model of particle swarm optimization artificial neural network (PSO-ANN) recovery factor (RF) correlation was derived for LSWPF oil recovery performance using the extracted weights and biases. The proposed porous media as well as stochastic multi-fractal random field and circulant embedding method (CEM) permeability field realizations did not exhibit artificial banding unlike turning bands method (TBM) realizations when compared. A fast convergence pre-conditioner solution technique of algebraic multigrid (AMG) linear solver combined with an acceleration method of biconjugate gradients stabilized method (BI-CGSTAB) was used to solve the fully implicit system of coupled linearized equations. The simulator was validated with PHREEQC geochemical package and a secondary LSWPF coreflood test with outstanding history match. Numerical 2D examples study in this paper revealed the synergetic effects of LSWPF and high salinity water polymer flooding (HSWPF) with higher oil RFs of 39% and 31%, respectively than the standalone LSWF of 23% and normal waterflood of 11% for 50 days of production. The optimal architecture of the PSO-ANN RF model is 7 inputs, 3 hidden neurons and 1 output. Also, the PSO-ANN RF model predictions were compared to the LSWPF numerical simulation results with the former remarkably having reduced computational time in seconds with training and blind testing errors <1% average absolute percentage error (AAPE) for various permeability field realizations investigated. Finally, the PSO-ANN RF model will serve as a fast-alternative reservoir management and planning tool to supplement the high-fidelity LSWPF CEOR simulator developed to minimize computational overheads, risks and uncertainties in oil recovery performance modeling.

Stochastic reconstruction of spatio-temporal rainfall patterns by inverse hydrologic modelling

Abstract. Knowledge of spatio-temporal rainfall patterns is required as input for distributed hydrologic models used for tasks such as flood runoff estimation and modelling. Normally, these patterns are generated from point observations on the ground using spatial interpolation methods. However, such methods fail in reproducing the true spatio-temporal rainfall pattern, especially in data-scarce regions with poorly gauged catchments, or for highly dynamic, small-scale rainstorms which are not well recorded by existing monitoring networks. Consequently, uncertainties arise in distributed rainfall–runoff modelling if poorly identified spatio-temporal rainfall patterns are used, since the amount of rainfall received by a catchment as well as the dynamics of the runoff generation of flood waves is underestimated. To address this problem we propose an inverse hydrologic modelling approach for stochastic reconstruction of spatio-temporal rainfall patterns. The methodology combines the stochastic random field simulator Random Mixing and a distributed rainfall–runoff model in a Monte Carlo framework. The simulated spatio-temporal rainfall patterns are conditioned on point rainfall data from ground-based monitoring networks and the observed hydrograph at the catchment outlet and aim to explain measured data at best. Since we infer a three-dimensional input variable from an integral catchment response, several candidates for spatio-temporal rainfall patterns are feasible and allow for an analysis of their uncertainty. The methodology is tested on a synthetic rainfall–runoff event on sub-daily time steps and spatial resolution of 1 km2 for a catchment partly covered by rainfall. A set of plausible spatio-temporal rainfall patterns can be obtained by applying this inverse approach. Furthermore, results of a real-world study for a flash flood event in a mountainous arid region are presented. They underline that knowledge about the spatio-temporal rainfall pattern is crucial for flash flood modelling even in small catchments and arid and semiarid environments.

Uncertainty quantification for linear hyperbolic equations with stochastic process or random field coefficients

In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media. Two types of models are presented: The first has a time-dependent coefficient modeled by the Ornstein–Uhlenbeck process. The second has a random field coefficient with a given covariance in space. For the former a formula for the exact solution in terms of moments is derived. In both cases stable numerical schemes are introduced to solve these random partial differential equations. Simulation results including convergence studies conclude the theoretical findings.

A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators

The aim of this work is to study the thermo-elastic quality factor (Q) of micro-resonators with a stochastic multi-scale approach. In the design of high-Q micro-resonators, thermo-elastic damping is one of the major dissipation mechanisms, which may have detrimental effects on the quality factor, and has to be predicted accurately. Since material uncertainties are inherent to and unavoidable in micro-electromechanical systems (MEMS), the effects of those variations have to be considered in the modeling in order to ensure the required MEMS performance. To this end, a coupled thermo-mechanical stochastic multi-scale approach is developed in this paper. Thermo-mechanical micro-models of polycrystalline materials are used to represent micro-structure realizations. A computational homogenization procedure is then applied on these statistical volume elements to obtain the stochastic characterizations of the elasticity, thermal expansion, and conductivity tensors at the meso-scale. Spatially correlated meso-scale random fields can thus be generated to represent the stochastic behavior of the homogenized material properties. Finally, the distribution of the thermo-elastic quality factor of MEMS resonators is studied through a stochastic finite element method using as input the generated stochastic random field.

Negative-Saturation Approach for Compositional Flow Simulations of Mixed CO2/Water Injection Into Geothermal Reservoirs Including Phase Appearance and Disappearance

Summary Cold mixed CO2/water injection into hot-water reservoirs can be used for simultaneous geothermal-energy (heat) production and subsurface CO2 storage. This paper studies this process in a 2D geothermal homogeneous reservoir, a layered reservoir, and a heterogeneous reservoir represented by a stochastic-random field. We give a set of simulations for a variety of CO2/water-injection ratios. In this process, often regions of two-phase flow are connected to regions of single-phase flow. Different systems of equations apply for single-phase and two-phase regions. We develop a solution approach, called the nonisothermal-negative-saturation (NegSat) solution approach, to solve efficiently nonisothermal compositional flow problems (e.g., CO2/water injection into geothermal reservoirs) that involve phase appearance, phase disappearance, and phase transitions. The advantage of this solution approach is that it circumvents using different equations for single-phase and two-phase regions and the ensuing unstable switching procedure. In the NegSat approach, a single-phase multicomponent fluid is replaced by an equivalent fictitious two-phase fluid with specific properties. The equivalent properties are such that the extended saturation of a fictitious gas is negative in the single-phase aqueous region. We discuss the salient features of the simulations in detail. When two phases are present at the injection side, heterogeneity and layering lead to more CO2 storage compared with the homogeneous case because of capillary trapping. In addition, layering avoids movement of the CO2 to the upper part of the reservoir and thus reduces the risk of leakage. Our results also show that heterogeneity and layering change the character of the solution in terms of useful-energy production and CO2 storage. The simulations can be used to construct a plot of the recovered useful energy vs. maximally stored CO2. Increasing the amount of CO2 in the injection mixture leads to bifurcation points at which the character of the solution in terms of energy production and CO2 storage changes. For overall injected-CO2 mole fractions less than 0.04, the result with gravity is the same as the result without gravity. For larger overall injected-CO2 mole fractions, however, the plot without gravity differs from the plot with gravity because of early breakthrough of a supercritical-CO2 tongue near the caprock. The plot of the useful energy (exergy) vs. the CO2-storage capacity in the presence of gravity shows a Z-shape. The top horizontal part represents a branch of high exergy recovery and a relatively lower storage capacity, whereas the bottom part represents a branch of lower exergy recovery and a higher storage capacity.

ℓ1-Penalized Likelihood Smoothing and Segmentation of Volatility Processes Allowing for Abrupt Changes

We consider the problem of estimating the volatility of a financial asset from a time series record. We believe the underlying volatility process is smooth, possibly stationary, and with potential small to large jumps due to market news. By drawing parallels between time series and regression models, in particular between stochastic volatility models and Markov random field smoothers, we propose a semiparametric estimator of volatility. For the selection of the smoothing parameter, we derive a universal rule borrowed from wavelet smoothing. Our Bayesian posterior mode estimate is the solution to an ℓ1-penalized likelihood optimization that we solve with an interior point algorithm that is efficient since its complexity is bounded by O(T 3/2), where T is the length of the time series. We apply our volatility estimator to real financial data, diagnose the model, and perform back-testing to investigate forecasting power by comparison to standard methods. Supplemental materials (Appendixes, Matlab code, and data) are available online.

Stochastic identification of composite material properties from limited experimental databases, Part II: Uncertainty modelling

The objective of this work is to characterise stochastic macroscopic material properties of heterogeneous composite fabrics from limited-size macro-scale experimental measurements. The work is presented in a sequence of two papers. In the first paper (Part I), a database consisting of observations of heterogeneous Young's modulus fields is obtained by a set of deterministic inverse problems. In this paper (Part II), a data assimilation framework is considered to identify a stochastic random field model of the Young's modulus. Such a model is set up to account for both aleatory uncertainties, related to sample inter-variabilities, as well as epistemic uncertainties due to insufficiency of the available data. This uncertainty characterisation is achieved by discretising the random field using a spectral decomposition procedure known as the Karhunen–Loève expansion. Random variables of this representation are expanded in a Hermite Polynomial Chaos (PC) basis whose coefficients themselves are considered as random variables. While the Gaussian variables of the PC basis model the aleatory uncertainty, the PC coefficients represent the epistemic uncertainty. A Bayesian inference scheme with Markov Chain Monte Carlo sampler is implemented to characterise the PC coefficients according to the Maximum A posteriori Probability (MAP) estimator.

The EXOR gate under uncertainty: A case study

Probabilistic AND/EXOR networks have been defined, in the past, as a class of Reed-Muller circuits, which operate on random signals. In contemporary logic network design, it is classified as behavioral notation of probabilistic logic gates and networks. In this paper, we introduce additional notations of probabilistic AND/EXOR networks: belief propagation, stochastic, decision diagram, neuromorphic models, and Markov random field model. Probabilistic logic networks, and, in particular, probabilistic AND/EXOR networks, known as turbo-decoders (used in cell phones and iPhone) are in demand in the coding theory. Another example is intelligent decision support in banking and security applications. We argue that there are two types of probabilistic networks: traditional logic networks assuming random signals, and belief propagation networks. We propose the taxonomy for this design, and provide the results of experimental study. In addition, we show that in forthcoming technologies, in particular, molecular electronics, probabilistic computing is the platform for developing the devices and systems for low-power low-precise data processing.

Model selection for a class of stochastic processes or random fields with bounded range

Methods are developed for finding an optimal model for a non-Gaussian stationary stochastic process or homogeneous random field under limited information. The available information consists of: (i) one or more finite length samples of the process or field; and (ii) knowledge that the process or field takes values in a bounded interval of the real line whose ends may or may not be known. The methods are developed and applied to the special case of non-Gaussian processes or fields belonging to the class of beta translation processes. Beta translation processes provide a flexible model for representing physical phenomena taking values in a bounded range, and are therefore useful for many applications. Numerical examples are presented to illustrate the utility of beta translation processes and the proposed methods for model selection.

Modeling multiscale diffusion processes in random heterogeneous media

A simple stochastic up-scaling strategy is presented for incorporating the effects of micro-scale variations of thermal conductivity in the thermal analysis of macro-scale systems. Incomplete characterization of the fine-scale variation necessitates posing the thermal conductivity as a stochastic multiscale random field. A data-driven approach coupled with a non-linear dimension reduction algorithm is used to construct a low-dimensional stochastic descriptor of the fine-scale variation of the stochastic conductivity. A representative volume element (RVE) argument is utilized to construct a simple model for up-scaling the stochastic thermal conductivity. In this model, the effective stochastic thermal conductivity is estimated by virtually interrogating the stochastic RV. A realistic example of the proposed stochastic multiscale diffusion strategy is illustrated utilizing limited experimental data.

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.

Root-Water Uptake Model at Heterogeneous Soil Fields

A new stochastic model for one-dimensional root-water uptake is developed with main emphasis on its probabilistic structure from random variations in saturated hydraulic conductivity. The resulting model has the form of a two-dimensional Fokker-Planck equation, and its applicability as a model for the probabilistic evolution of the nonlinear stochastic root-water uptake process is explored with the saturated hydraulic conductivity taken as stochastic random field. In order to perform this exploration, the generalization of a one-dimensional numerical scheme for the numerical solution of a two-dimensional Fokker-Planck equation is attempted. The proposed model has the advantage of providing a probabilistic solution to soil-water flow under root-water uptake, from which one can obtain the ensemble average behavior of the soil-water system at the scale of a heterogeneous field soil.

A Krylov Subspace Method for Covariance Approximation and Simulation of Random Processes and Fields

This paper proposes a new iterative algorithm for simultaneously computing an approximation to the covariance matrix of a random vector and drawing a sample from that approximation. The algorithm is especially suited to cases for which the elements of the random vector are samples of a stochastic process or random field. The proposed algorithm has close connections to the conjugate gradient method for solving linear systems of equations. A comparison is made between our algorithm's structure and complexity and other methods for simulation and covariance matrix approximation, including those based on FFTs and Lanczos methods. The convergence of our iterative algorithm is analyzed both analytically and empirically, and a preconditioning technique for accelerating convergence is explored. The numerical examples include a fractional Brownian motion and a random field with the spherical covariance used in geostatistics.

Scanning and prediction in multidimensional data arrays

The problem of sequentially scanning and predicting data arranged in a multidimensional array is considered. We introduce the notion of a scandictor, which is any scheme for the sequential scanning and prediction of such multidimensional data. The scandictability of any finite (probabilistic) data array is defined as the best achievable expected performance on that array. The scandictability of any (spatially) stationary random field on /spl Zopf//sup m/ is defined as the limit of its scandictability on finite boxes (subsets of /spl Zopf//sup m/), as their edges become large. The limit is shown to exist for any stationary field, and essentially be independent of the ratios between the box dimensions. Fundamental limitations on scandiction performance in both the probabilistic and the deterministic settings are characterized for the family of difference loss functions. We find that any stochastic process or random field that can be generated autoregressively with a maximum-entropy innovation process is optimally scandicted the way it was generated. These results are specialized for cases of particular interest. The scandictability of any stationary Gaussian field under the squared-error loss function is given a single-letter expression in terms of its spectral measure and is shown to be attained by the raster scan. For a family of binary Markov random fields (MRFs), the scandictability under the Hamming distortion measure is fully characterized.