AI 中文总结
本文提出一种基于各向异性核的多层插值方法，在稀疏网格上逼近高维函数，推导显式表示并证明误差估计，给出三种权重策略，数值实验验证了理论。
AI 中文摘要
张量积多层方法（TPML）通过将Smolyak稀疏网格构造与基于核的多层插值相结合，在一般有界域上从散乱数据逼近高维函数，并在每一层将核尺度与填充距离耦合，使得条件数保持一致有界。其原始表述仅隐式依赖于目标函数，这阻碍了数值评估。利用低维多层算子的一种近期节点表示，我们推导出高维插值函数的显式、可计算表示。在此基础上，我们在混合正则Sobolev范数和连续范数下证明了误差估计，扩展了现有的$L_2$理论，并刻画了能够达到相对于自由度数量的最优收敛速率的所有各向异性权重范围。随后提出了三种显式权重策略，分别均衡精度、自由度或成本效益比。在多达十维的数值实验验证了该理论。
英文摘要
The tensor product multilevel method (TPML) approximates high-dimensional functions from scattered data on general bounded domains by combining Smolyak's sparse grid construction with the kernel-based multilevel interpolation, coupling the kernel scale to the fill distance at each level so that condition numbers stay uniformly bounded. Its original formulation depended only implicitly on the target function, which precluded numerical evaluation. Using a recent nodal representation of the low-dimensional multilevel operators, we derive an explicit, computable representation of the high-dimensional interpolant. On this basis we prove error estimates in mixed-regularity Sobolev and continuity norms, extending the existing $ L_2 $-theory, and we characterise the full range of anisotropy weights that attain the optimal rate against the number of degrees of freedom. Three explicit weight strategies follow, equilibrating the accuracy, the degrees of freedom, or the cost-benefit ratio. Numerical experiments in up to ten dimensions illustrate the theory.
