论文标题
基于平滑颗粒流体动力学
Boltzmann-Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics
论文作者
论文摘要
Boltzmann-Gibbs随机场是根据指数表达EXP（-h）定义的，其中H是磁场状态x（s）的适当定义的能量功能。本文提出了一种新的Boltzmann-Gibbs模型，该模型具有能量功能中的局部相互作用。这些相互作用体现在空间耦合函数中，该空间耦合函数使用了从平滑粒子流体动力学理论中启发的空间衍生物的平滑核函数近似值。研究了基于拉普拉斯操作员二级多项式的相互作用的特定模型。对于平方指数（高斯）平滑核的情况，得出了空间耦合函数（精度函数）的显式，无网格表达。该耦合函数允许模型从离散的数据向量无缝扩展到连续性字段。建立了与高斯马尔可夫随机字段和带有$ν= 1 $的Matérn场的连接。
Boltzmann-Gibbs random fields are defined in terms of the exponential expression exp(-H), where H is a suitably defined energy functional of the field states x(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 $ν=1$ are established.
