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About 10,000 papers for “Random field”

Random-field random surfaces

Paul Dario, Matan Harel, Ron Peled · arXiv (Cornell University) · 2021

We study how the typical gradient and typical height of a random surface are modified by the addition of quenched disorder in the form of a random independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, in the following cases. It is shown that for real-valued disordered random surfaces of the $\nabla ϕ$ type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions $1\le d\le 2$ and localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions $1\le d\le 4$ and localizes in dimensions $d\ge 5$. It is further shown that for the integer-valued disordered Gaussian free field: (i) The gradient of the surface delocalizes in dimensions $d=1,2$ and localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions $d=1,2$. (iii) The surface localizes in dimensions $d\ge 3$ at weak disorder strength. The behavior in dimensions $d\ge 3$ at strong disorder is left open. The proofs rely on several tools: explicit identities satisfied by the expectation of the random surface, the Efron--Stein concentration inequality, a coupling argument for Langevin dynamics (originally due to Funaki and Spohn) and the Nash--Aronson estimate.

Cited by 1Open access

Random Field, Gaussian

Keith J. Worsley · Encyclopedia of Environmetrics · 2006

Abstract The Gaussian random field Y ( t ), t ∈ T , is one of the most common models used to describe spatial stochastic processes. In many applications, the domain T is a subset of D ‐dimensional Euclidean space (usually D = 2 or D = 3), and the function Y ( t ) is almost surely continuous or smooth in t . The definition is simple: the Gaussian random field must be multivariate Gaussian at all finite sets of points, that is, [ Y ( t 1 ), …, Y ( t n )] must be multivariate Gaussian for all n > 0 and all t j ∈ T . Since the multivariate Gaussian is specified uniquely by its mean vector and variance matrix, then the Gaussian random field is defined uniquely by its mean function μ ( t ) = E[ Y ( t )] and its covariance function C ( s , t ) = cov[ Y ( s ), Y ( t )].

Cited by 1

Random field generation

Laurent, Adissa · Digital Access to Libraries · 2017

This thesis, done in collaboration with Cenaero, contributes to the MACOBIO project. It develops a numerical tool modelling the properties of heterogeneous materials based on random field generation and analysing the mechanical responses of the composite objects. After reviewing the different composite materials and mechanical laws based on homogenization methods, this thesis presents a numerical tool to simulate Gaussian random fields on 3D surfaces. It first defines the covariance function associated with the Gaussian random field. The random field is then discretised on finite elements using Karhunen-Loève expansion. The Karhunen-Loève expansion is computed by solving a Fredholm integral equation of the second kind. The integrals involved by the Fredholm equation are evaluated using Galerking methods. The integrals depend on discrete values of the covariance function, itself a function of the geodesic distances. The geodesic distances are evaluated using the fast marching method or the recursive fast marching method. The recursive fast marching method is applied on 2D and 3D surfaces and the errors of the method are characterized. The random field generation method is applied on a composite chair. Two uncertain properties are modelled: the fibre volume fraction and the angles of the plies. Two mechanical responses (the deflection and the Tsai-Wu criterion) are analysed by means of the Monte Carlo method.

Cited by 0Open access

Random cohesion field

Thibault Duretz · Zenodo (CERN European Organization for Nuclear Research) · 2023

This file contains a 2D random cohesion field used that can be used as an initial condition for the simulation of shear banding.

Cited by 1Open access

ULTRAMETRIC RANDOM FIELD

Andrei Yu. Khrennikov, С. В. Козырев · Infinite Dimensional Analysis Quantum Probability and Related Topics · 2006

Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces. Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the random field under investigation satisfies this property.

Cited by 35

Random Field Optimization

Joshua L. Pulsipher, Benjamin Davidson, Ví­ctor M. Zavala · arXiv (Cornell University) · 2022

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.

Cited by 0Open access

Random cohesion field

Thibault Duretz · Zenodo (CERN European Organization for Nuclear Research) · 2023

This file contains a 2D random cohesion field used that can be used as an initial condition for the simulation of shear banding.

Cited by 0Open access

Measurable Random Field

Yuen-Kwok Chan · Cambridge University Press eBooks · 2021

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 0

Random field of slope

Dong, Xin · Figshare · 2025

This study introduces a methodology for generating constrained random fields (CRFs) by integrating fast Fourier transform (FFT) filtering with co-Kriging interpolation. Through a case study of a representative three-stage bench slope, the distributions of failure modes and statistical characteristics of safety factors are systematically examined for three distinct models: unconditional random field (URF), independent random field (IRF), and constrained random field (CRF). The investigation encompasses varying geological conditions, including rock-soil layer dip angles (15°–75°) and parameter cross-correlation structures (coefficients: -0.2 and 0).

Cited by 0Open access