Approximation of multivariate functions on sparse grids by Kernel-based Quasi-interpolation

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.