Quasi-interpolation for high-dimensional function approximation

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...

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Vydáno v:Numerische Mathematik Ročník 156; číslo 5; s. 1855 - 1885
Hlavní autoři: Gao, Wenwu, Wang, Jiecheng, Sun, Zhengjie, Fasshauer, Gregory E.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2024
Springer Nature B.V
Témata:
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
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Popis
Shrnutí: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.
Bibliografie:ObjectType-Article-1
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-024-01435-6