AI 中文总结
该研究提出通过具有多消失矩的插值小波（interpolets）压缩核矩阵，推导了线性复杂度的全离散核插值格式，并结合稀疏网格组合技术推广至高维，经数值实验验证了理论结果。
AI 中文摘要
我们研究在单位区间$[0,1]$上，通过具有足够多消失矩的插值小波（interpolets）对核矩阵进行压缩。我们定义了一种压缩规则，该规则会丢弃大部分矩阵系数，同时不损害基础离散化所提供的精度。由于可对插值小波（interpolets）进行缩放，使压缩后的核矩阵具有良好的条件数，因此我们推导了一种全离散格式，可在线性整体复杂度下求解所考虑的核插值问题。最后，我们通过稀疏网格组合技术将该方法推广到单位$n$立方体$[0,1]^n$上。我们开展了数值实验以验证理论结果。
英文摘要
We consider the compression of kernel matrices on the unit interval $[0,1]$ by interpolets that have sufficiently many vanishing moments. We define a compression rule which discards most matrix coefficients without compromising the accuracy offered by the underlying discretization. Since interpolets can be scaled such that the compressed kernel matrices are well conditioned, we derive a fully discrete scheme that solves a kernel interpolation problem under consideration in linear overall complexity. We finally generalize this approach to the unit $n$-cube $[0,1]^n$ by means of the sparse grid combination technique. Numerical experiments are carried out to validate the theoretical findings.
