任意阶的线性成本多谐波样条插值
Linear-cost Polyharmonic Spline Interpolation of Arbitrary Degree
浏览论文内容
中文总结 AI 辅助
该研究结合FMM与Vecchia近似提出线性成本多谐波样条插值方法，实现百万级点的快速高精度薄板样条插值，并提供二维奇数阶PHS插值的高性能软件库。
中文摘要 AI 辅助
我们提出了一种简单且高性能的方法，结合计算静电学中的快速多极子方法（FMM）与高斯过程及稀疏近似逆文献中的Vecchia近似，实现快速且准确的多谐波样条（PHS）插值。利用阿达马乘积和低秩矩阵的基本性质，我们证明采用对数核和距离核这两种核的FMM可实现所有阶PHS的快速插值。此外，我们展示了Matérn协方差模型的稀疏逆近似方法作为预条件器的优异性能。结合对禁止子空间的精细管理，我们描述了一种使用预条件共轭梯度获取预测权重的过程，即便对于包含超过100万个点的问题规模，其收敛迭代次数也少于15次。因此，无需参数调优且精度匹配全稠密O(n³)计算的薄板样条插值（一种特别流行的方法），可在约50至60次FMM的开销下完成。作为本工作的配套成果，我们提供了用于二维奇数阶PHS插值的高性能软件库。
英文摘要
We introduce a simple and performant approach for rapidly and accurately performing polyharmonic spline (PHS) interpolation using a combination of the fast multipole method (FMM) from computational electrostatics and the Vecchia approximation from the Gaussian process and sparse approximate inverse literatures. Using basic properties about Hadamard products and low-rank matrices, we demonstrate that an FMM with two kernels, the logarithmic and distance kernels, results in fast PHS interpolation for all orders. Furthermore, we demonstrate the exceptional performance of sparse inverse approximation methods with the Matérn covariance model for preconditioning. Combined with careful management of disallowed subspaces, we describe a procedure for obtaining prediction weights using preconditioned conjugate gradient that converges in less than $15$ iterations, even for problem sizes with over one million points. As a result, thin-plate spline interpolation---a particularly popular method that does not require parameter tuning---that matches the fully dense $\mathcal{O}(n^3)$ computation in accuracy can be done at the cost of approximately $50-60$ FMMs. A high-performance software library for odd-order PHS interpolation in two dimensions is made available as a companion to this work.
