Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics

IF 0.4 Q4 STATISTICS & PROBABILITY
D. Hristopulos
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引用次数: 0

Abstract

Boltzmann–Gibbs random fields are defined in terms of the exponential expression exp ⁡ ( − H ) \exp \left (-\mathcal {H}\right ) , where H \mathcal {H} is a suitably defined energy functional of the field states x ( s ) x(\mathbf {s}) . This paper presents a new Boltzmann–Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free expression of the spatial coupling function (precision function) is derived for the case of the squared exponential (Gaussian) smoothing kernel. This coupling function allows the model to seamlessly extend from discrete data vectors to continuum fields. Connections with Gaussian Markov random fields and the Matérn field with ν = 1 \nu =1 are established.
基于光滑粒子流体力学的无网格精度算子Boltzmann-Gibbs随机场
玻尔兹曼-吉布斯随机场用指数表达式exp (- H) \exp\left (- \mathcal H{}\right)来定义,其中H \mathcal H{是场态x(s) x(}\mathbf s{)的适当定义的能量函数。本文提出了一种新的具有能量泛函局部相互作用的玻尔兹曼-吉布斯模型。相互作用体现在一个空间耦合函数中,该函数使用了受光滑粒子流体力学理论启发的空间导数的光滑核函数近似。研究了一种基于拉普拉斯算子二阶多项式的特定模型。对于平方指数(高斯)平滑核,导出了空间耦合函数(精度函数)的显式无网格表达式。这种耦合功能允许模型从离散数据向量无缝扩展到连续域。建立了高斯马尔可夫随机场与ν =1 }\nu =1的mat rn场的连接。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
22
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