Quasi-interpolation for high-dimensional function approximation

• Published in: Numerische Mathematik Vol. 156; no. 5; pp. 1855 - 1885
• Main Authors: Gao, Wenwu, Wang, Jiecheng, Sun, Zhengjie, Fasshauer, Gregory E.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2024; Springer Nature B.V
• Subjects: Approximation; Convolution; Dimensional analysis; Discretization; Error analysis; Interpolation; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Operators (mathematics); Quadratures; Regularization; Simulation; Theoretical; 41A25; 65D40; 41A30; 41A63; 65D15; 41A05; 42B05
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-024-01435-6

The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rule at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and is capable of mitigating the curse of dimensionality for approximating high-dimensional functions.