An optimal ansatz space for moving least squares approximation on spheres An optimal ansatz space for moving least squares approximation

• Published in: Advances in computational mathematics Vol. 50; no. 6
• Main Authors: Hielscher, Ralf, Pöschl, Tim
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2024
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Visualization; Sphere; Lebesgue constant; Moving least squares; 65D10; Local approximation; 65D15; Tangent space; 65D07
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10201-z

We revisit the moving least squares (MLS) approximation scheme on the sphere S d - 1 ⊂ R d , where d > 1 . It is well known that using the spherical harmonics up to degree L ∈ N as ansatz space yields for functions in C L + 1 ( S d - 1 ) the approximation order O h L + 1 , where h denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to L , while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as h → 0 . Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space.