SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A953-A979, January 2021.
In this study, we present a new class of quasi-interpolation schemes for the approximation of multivariate functions on sparse grids. Each scheme in this class is based on shifts of kernels constructed from one-dimensional radial basis functions such as multiquadrics. The kernels are modified near the boundaries to prevent deterioration of the fidelity of the approximation. We implement our scheme using the standard single-level method as well as the multilevel technique designed to improve rates of approximation. Advantages of the proposed quasi-interpolation schemes are twofold. First, our sparse approximation attains almost the same level convergence order as the optimal approximation on the full grid related to the Strang--Fix condition, reducing the amount of data required significantly compared to full grid methods. Second, the single-level approximation performs nearly as well as the multilevel approximation, with much less computation time. We provide a rigorous proof for the approximation orders of our quasi-interpolations. In particular, compared to another quasi-interpolation scheme in the literature based on the Gaussian kernel using the multilevel technique, we show that our methods provide significantly better rates of approximation. Finally, some numerical results are presented to demonstrate the performance of the proposed schemes.

中文翻译：

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基于核的拟插值逼近稀疏网格上的多元函数

SIAM科学计算杂志，第43卷，第2期，第A953-A979页，2021年1月。
在这项研究中，我们为稀疏网格上的多元函数逼近提供了一类新的拟插值方案。此类中的每种方案都基于由一维径向基函数（例如多二次方）构造的内核移位。在边界附近对内核进行修改，以防止近似值的保真度变差。我们使用标准的单级方法以及旨在提高近似率的多级技术来实施我们的方案。拟内插方案的优点是双重的。首先，我们的稀疏近似与与Strang-Fix条件有关的全网格上的最佳近似具有几乎相同的层级收敛阶，与全网格方法相比，显着减少了所需的数据量。第二，单级逼近的性能几乎与多级逼近的好，而计算时间却少得多。我们为准插值的逼近阶数提供了严格的证明。特别是，与文献中使用多级技术基于高斯核的另一种拟插值方案相比，我们证明了我们的方法提供了明显更高的逼近率。最后，一些数值结果被提出来证明所提出的方案的性能。我们证明了我们的方法可以提供更好的近似率。最后，一些数值结果被提出来证明所提出的方案的性能。我们证明了我们的方法可以提供更好的近似率。最后，一些数值结果被提出来证明所提出的方案的性能。