On estimating the entropy of random fields
Random fields entropy estimation technique taking into account higher than immediately adjacent spatial dependencies
Kaplan, A. J.↗
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Random field optimization
Herein we present a new modeling paradigm for optimization that we call random field optimization. Random fields are a powerful modeling abstraction that aims to capture the behavior of random variables that live on infinite-dimensional spaces (e.g., space and time) such as stochastic processes (e.g., time series, Gaussian processes, and Markov processes), random matrices, and random spatial fields. This paradigm involves sophisticated mathematical objects (e.g., stochastic differential equations and space-time kernel functions) and has been widely used in neuroscience, geoscience, physics, civil engineering, and computer graphics. Despite of this, however, random fields have seen limited use in optimization; specifically, existing optimization paradigms that involve uncertainty (e.g., stochastic programming and robust optimization) mostly focus on the use of finite random variables. This trend is rapidly changing with the advent of statistical optimization (e.g., Bayesian optimization) and multi-scale optimization (e.g., integration of molecular sciences and process engineering). Our work extends a recently-proposed abstraction for infinite-dimensional optimization problems by capturing more general uncertainty representations. Moreover, we discuss solution paradigms for this new class of problems based on finite transformations and sampling, and identify open questions and challenges.
97 MATHEMATICS AND COMPUTING↗
Fast estimation of the look-elsewhere effect using Gaussian random fields
Abstract We discuss the use of Gaussian random fields to estimate the look-elsewhere effect correction. We show that Gaussian random fields can be used to model the null-hypothesis significance maps from a large set of statistical problems commonly encountered in physics, such as template matching and likelihood ratio tests. Some specific examples are searches for dark matter using pixel arrays, searches for astronomical transients, and searches for fast-radio bursts. Gaussian random fields can be sampled efficiently in the frequency domain, and the excursion probability can be fitted with these samples to extend any estimation of the look-elsewhere effect to lower p values. In addition, in cases where the Gaussian random field is stationary and the parameter space is Euclidean, the look-elsewhere effect correction can be computed analytically. We demonstrate these methods using two example template matching problems. Finally, we apply these methods to estimate the trial factor of a $$4^3$$ 4 3 accelerometer array for the detection of dark matter tracks in the Windchime project. When a global significance of $$3\sigma $$ 3 σ is required, the estimated trial factor for such an accelerometer array is $$10^{14}$$ 10 14 for a one-second search, and $$10^{22}$$ 10 22 for a 1-year search.
Qin, Juehang (ORCID:0000000182288949)↗
Spatio-temporal contextual classification based on Markov random field model
A contextural classifier based on a Markov random field model, which can utilize both spatial and temporal contexts, is investigated. Spatial and temporal neighbors are defined, and the class assignment of each pixel is assumed to be dependent only on the measurement vectors of itself and those of its spatial and temporal neighbors according to the Markov random field property. Only interpixel class dependency context is used in the classification. The joint prior probability of the classes of each pixel and its spatial and temporal neighbors are modeled by a Gibbs random field. The classification is performed in a recursive manner. Experiments with multi-temporal Thematic Mapper data show promising results.
Jeon, Byeungwoo↗
COBRA:COMPUTED-TOMOGRAPHY BASED RANDOM-FIELD APPROXIMATION
SF-25-115 COBRA (COmputed-tomography Based Random-field Approximation) is a Python application for generating statistically equivalent random fields from CT-scan imagery. It leverages Karhunen–Loève expansions to model microstructural variability, enabling users to: Preprocess CT scans (filtering and Gaussian transformation); Fit covariance kernels fromempirical data; Solve eigenproblems to obtain KL modes; Sample random fields onsistent with fitted statistics; Postprocess samples back into the physical domain.
Hu, Tianchen↗
Distributed memory parallel Markov random fields using graph partitioning
Markov random fields (MRF) based algorithms have attracted a large amount of interest in image analysis due to their ability to exploit contextual information about data. Image data generated by experimental facilities, though, continues to grow larger and more complex, making it more difficult to analyze in a reasonable amount of time. Applying image processing algorithms to large datasets requires alternative approaches to circumvent performance problems. Aiming to provide scientists with a new tool to recover valuable information from such datasets, we developed a general purpose distributed memory parallel MRF-based image analysis framework (MPI-PMRF). MPI-PMRF overcomes performance and memory limitations by distributing data and computations across processors. The proposed approach was successfully tested with synthetic and experimental datasets. Additionally, the performance of the MPI-PMRF framework is analyzed through a detailed scalability study. We show that a performance increase is obtained while maintaining an accuracy of the segmentation results higher than 98%. The contributions of this paper are: (a) development of a distributed memory MRF framework; (b) measurement of the performance increase of the proposed approach; (c) verification of segmentation accuracy in both synthetic and experimental, real-world datasets
97 MATHEMATICS AND COMPUTING↗
Review of Recent Developments in the Random-Field Ising Model
A lot of progress has been made recently in our understanding of the random-field Ising model thanks to large-scale numerical simulations. In particular, it has been shown that, contrary to previous statements: the critical exponents for different probability distributions of the random fields and for diluted antiferromagnets in a field are the same. Therefore, critical universality, which is a perturbative renormalization-group prediction, holds beyond the validity regime of perturbation theory. Most notably, dimensional reduction is restored at five dimensions, i.e., the exponents of the random-field Ising model at five dimensions and those of the pure Ising ferromagnet at three dimensions are the same.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Ordering and phase transitions in random-field Ising systems
An exact analysis of the Ising model with infinite-range interactions in a random field and a local mean-field theory in three dimensions is carried out leading to a phase diagram with several coexistence surfaces and lines of critical points. The results show that the phase diagram depends crucially on whether the distribution of random fields is symmetric or not. Thus, Ising-like phase transitions in a porous medium (the asymmetric case) are in a different universality class from the conventional random-field model (symmetric case).
Maritan, Amos↗
Augmented Gaussian random field: Theory and computation
We propose the novel augmented Gaussian random field (AGRF), which is a universal framework incorporating the data of observable and derivatives of any order. Rigorous theory is established. We prove that under certain conditions, the observable and its derivatives of any order are governed by a single Gaussian random field, which is the aforementioned AGRF. As a corollary, the statement "the derivative of a Gaussian process remains a Gaussian process" is validated, since the derivative is represented by a part of the AGRF. Moreover, a computational method corresponding to the universal AGRF framework is constructed. Both noiseless and noisy scenarios are considered. Formulas of the posterior distributions are deduced in a nice closed form. A significant advantage of our computational method is that the universal AGRF framework provides a natural way to incorporate arbitrary order derivatives and deal with missing data. We use four numerical examples to demonstrate the effectiveness of the computational method. The numerical examples are composite function, damped harmonic oscillator, Korteweg-De Vries equation, and Burgers' equation.
Mathematics↗
A Note on Exponential Decay in the Random Field Ising Model
For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (i) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (ii) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kertész line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes
This work describes a domain embedding technique between two nonmatching meshes used for generating realizations of spatially correlated random fields with applications to large-scale sampling-based uncertainty quantification. The goal is to apply the multilevel Monte Carlo (MLMC) method for the quantification of output uncertainties of PDEs with random input coefficients on general and unstructured computational domains. We propose a highly scalable, hierarchical sampling method to generate realizations of a Gaussian random field on a given unstructured mesh by solving a reaction–diffusion PDE with a stochastic right-hand side. The stochastic PDE is discretized using the mixed finite element method on an embedded domain with a structured mesh, and then, the solution is projected onto the unstructured mesh. This work describes implementation details on how to efficiently transfer data from the structured and unstructured meshes at coarse levels, assuming that this can be done efficiently on the finest level. We investigate the efficiency and parallel scalability of the technique for the scalable generation of Gaussian random fields in three dimensions. An application of the MLMC method is presented for quantifying uncertainties of subsurface flow problems. Here, we demonstrate the scalability of the sampling method with nonmatching mesh embedding, coupled with a parallel forward model problem solver, for large-scale 3D MLMC simulations with up to 1.9·109 unknowns.
97 MATHEMATICS AND COMPUTING↗
Absence of Replica Symmetry Breaking in the Transverse and Longitudinal Random Field Ising Model
It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin–Kasteleyn–Ginibre property in this model and the Ghirlanda–Guerra identities in artificial models in a path integral representation based on the Lie–Trotter–Suzuki formula enable us to extend Chatterjee’s proof for the random field Ising model to the quantum model.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Characterisation of the spatial variability of material properties of Gilsocarbon and NBG-18 using random fields
Graphite is a candidate material for Generation IV concepts and is used as a moderator in Advanced Gas-cooled Reactors (AGR) in the UK. Spatial material variability is present within billets causing different material property values between different components. Variations in material properties and irradiation effects can produce stress concentrations and diverse mechanical responses in a nuclear reactor graphite core. In order to characterise the material variability, geostatistical techniques called variography and random field theory were adapted for studying the density and Young's modulus of a billet of Gilsocarbon and NBG-18 graphite grades. Variography is a technique for estimating the distance over which material property values have significant spatial correlation, known as the scale of fluctuation or spatial correlation length. The paper uses random field theory to create models that mimic the original spatial and statistical distributions of the original data set. This study found different values of correlation length for density and Young's modulus around the edges of a Gilsocarbon billet, while in the case of NBG-18, similar correlation lengths where found across the billet. Furthermore examples of several random fields are given to reproduce the spatial patterns and values found in the original data.
36 MATERIALS SCIENCE↗
Experimental demonstration and analysis of random field effects in ferromagnet/antiferromagnet bilayers
More than 30 years ago, Malozemoff [Phys. Rev. B 35, 3679 (1987)] hypothesized that exchange interaction at the interface between a ferromagnet (F) and an antiferromagnet (AF) can act as an effective random field, which can profoundly affect the magnetic properties of the system. However, until now this hypothesis has not been directly experimentally tested. We utilize magnetoelectronic measurements to analyze the effective exchange fields at permalloy/CoO interface. Our results cannot be explained in terms of quasiuniform effective exchange fields but are in agreement with the random-field hypothesis of Malozemoff. Finally, the presented approach opens a new route for the quantitative analysis of effective exchange fields and anisotropies in magnetic heterostructures for memory, sensing and computing applications.
36 MATERIALS SCIENCE↗
A Markov Random Field Model for Texture-based Segmentation of Small Cracks in Thin InconelTubes
Markov random fields have been used for image segmentation since their introduction in the1980s. This work applies a method from Principal Component Thermography to enhance thecontrast in damage regions in 3D images derived from X-ray computed tomography (CT)inspections. The developed method is applied to sizing of small cracks in thin Inconel tubes designed as probability of detection (POD) samples for radiographic inspection.Misclassification errors arising from artifacts due to beam-hardening are reduced by fitting the boundary of the segmented damage region with the arc of an ellipse. Results are comparedagainst those obtained through manual inspection.
Image segmentation↗
Using porous random fields to predict the elastic modulus of unoxidized and oxidized superfine graphite
Nuclear graphite is a candidate material for Generation IV nuclear power plants. Porous materials such as graphite can contain complex networks of pores that influence the material's mechanical and irradiation response. A methodology known as the random finite element method (RFEM) was adapted to create synthetic microstructures and predict the influence of porosity on the elastic properties of graphite during oxidation. RFEM combines random field theory and the finite element method in a Monte Carlo framework to estimate the mechanical response of a given grade of graphite. In this research, the random fields were verified through experimental characterization to predict the elastic response of three nuclear graphite grades, ETU-10, IG-110, and 2114. Finite element models (FEM) were generated using segmentations of x-ray computed tomography (XCT) data known as image-based models (IBMs) to validate and compare with the RFEM results and better understand the effects of uniform oxidation in these graphite grades. The RFEM predictions appear to correlate well with the experimental values of the measured Young’s modulus of the three graphite grades and display the same trends as IBMs.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Reduced Wiener Chaos representation of random fields via basis adaptation and projection
A new characterization of random fields appearing in physical models is presented that is based on their well-known Homogeneous Chaos expansions. We take advantage of the adaptation capabilities of these expansions where the core idea is to rotate the basis of the underlying Gaussian Hilbert space, in order to achieve reduced functional representations that concentrate the induced probability measure in a lower dimensional subspace. For a smooth family of rotations along the domain of interest, the uncorrelated Gaussian inputs are transformed into a Gaussian process, thus introducing a mesoscale that captures intermediate characteristics of the quantity of interest.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Random field estimation approach to robot dynamics
The difference equations of Kalman filtering and smoothing recursively factor and invert the covariance of the output of a linear state-space system driven by a white-noise process. Here it is shown that similar recursive techniques factor and invert the inertia matrix of a multibody robot system. The random field models are based on the assumption that all of the inertial (D'Alembert) forces in the system are represented by a spatially distributed white-noise model. They are easier to describe than the models based on classical mechanics, which typically require extensive derivation and manipulation of equations of motion for complex mechanical systems. With the spatially random models, more primitive locally specified computations result in a global collective system behavior equivalent to that obtained with deterministic models. The primary goal of applying random field estimation is to provide a concise analytical foundation for solving robot control and motion planning problems.
Rodriguez, Guillermo↗