论文标题
使用匹配的内核和设计快速自动贝叶斯立方体
Fast Automatic Bayesian Cubature Using Matching Kernels and Designs
论文作者
论文摘要
自动库将积分近似于用户指定的误差公差。对于高维问题，很难自适应地更改采样模式以专注于峰值，因为峰可以更容易隐藏在高维空间中。但是，给定合理的固定采样模式，可以自动确定样本量$ n $。这种方法是在Jagadeeswaran和Hickernell，Stat。\ Comput。，29：1214-1229，2019中采用的，其中使用贝叶斯的观点来构建积分不可或缺的可靠间隔，并且当间隔的半宽度不大于所需的误差时，计算被终止。我们较早的工作采用了集成晶格采样，并且快速傅立叶变换加快了计算，因为选择了集成媒体上的高斯过程的协方差内核，以选择转换。在本章中，我们通过\ emph {Digitally} Shift-Invariant协方差内核和快速WALSH变换将快速自动贝叶斯立方体扩展到数字净采样。 我们的算法在MATLAB保证自动集成库（GAIL）和QMCPY PYTHON库中实现。
Automatic cubatures approximate integrals to user-specified error tolerances. For high dimensional problems, it is difficult to adaptively change the sampling pattern to focus on peaks because peaks can hide more easily in high dimensional space. But, one can automatically determine the sample size, $n$, given a reasonable, fixed sampling pattern. This approach is pursued in Jagadeeswaran and Hickernell, Stat.\ Comput., 29:1214-1229, 2019, where a Bayesian perspective is used to construct a credible interval for the integral, and the computation is terminated when the half-width of the interval is no greater than the required error tolerance. Our earlier work employs integration lattice sampling, and the computations are expedited by the fast Fourier transform because the covariance kernels for the Gaussian process prior on the integrand are chosen to be shift-invariant. In this chapter, we extend our fast automatic Bayesian cubature to digital net sampling via \emph{digitally} shift-invariant covariance kernels and fast Walsh transforms. Our algorithm is implemented in the MATLAB Guaranteed Automatic Integration Library (GAIL) and the QMCPy Python library.
