GP-Recipe：数值方法中线性运算的高斯过程近似
GP-Recipe: Gaussian Process approximation to linear operations in numerical methods
AI总结：
本文提出GP-Recipe方法，利用高斯过程回归对数值方法中的线性运算进行高阶近似，并引入新核处理间断数据，在测试中相比传统有限差分提高了求解精度。
摘要：
我们引入了新的高斯过程（GP）高阶近似，用于各种数值方法中频繁使用的线性运算。我们的方法采用基于核的GP回归建模，这是一种非参数贝叶斯回归方法，作用于拟合观测数据的所有可容许函数的概率分布。在第一部分中，我们首先使用最流行的平方指数核函数，对应用于光滑数据的各种线性算子进行离散数据近似。在第二部分中，我们讨论跨具有陡峭梯度的间断的数据插值，为此我们引入一种新的GP核，能够无振荡地拟合间断数据。当前研究将我们先前在有限差分和有限体积方法中关于无多项式激波捕捉方法的GP工作扩展到光滑数据上的一套线性算子近似。本文引入的公式可以很容易地应用于数值方法的日常实践中，包括有限差分数值近似、求积规则、插值和重构，这些都是现代科学与工程应用中数值建模最常用的方法。在测试问题中，我们证明与传统的有限差分对应方法相比，GP近似解具有更高的解精度。
英文摘要：
We introduce new Gaussian Process (GP) high-order approximations to linear operations that are frequently used in various numerical methods. Our method employs the kernel-based GP regression modeling, a non-parametric Bayesian approach to regression that operates on the probability distribution over all admissible functions that fit observed data. We begin in the first part with discrete data approximations to various linear operators applied to smooth data using the most popular squared exponential kernel function. In the second part, we discuss data interpolation across discontinuities with sharp gradients, for which we introduce a new GP kernel that fits discontinuous data without oscillations. The current study extends our previous GP work on polynomial-free shock-capturing methods in finite difference and finite volume methods to a suite of linear operator approximations on smooth data. The formulations introduced in this paper can be readily adopted in daily practices in numerical methods, including numerical approximations of finite differences, quadrature rules, interpolations, and reconstructions, which are most frequently used in numerical modeling in modern science and engineering applications. In the test problems, we demonstrate that the GP approximated solutions feature improved solution accuracy compared to the conventional finite-difference counterparts.
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