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Copyright c 2001 Tech Science Press CMES, vol.2, no.4, pp.447-462, 2001
 

Summary: Copyright c

2001 Tech Science Press CMES, vol.2, no.4, pp.447-462, 2001
On the Equivalence Between Least-Squares and Kernel Approximations in
Meshless Methods
Xiaozhong Jin1, Gang Li2 and N. R. Aluru3
Abstract: Meshless methods using least-squares ap-
proximations and kernel approximations are based on
non-shifted and shifted polynomial basis, respectively.
We show that, mathematically, the shifted and non-
shifted polynomial basis give rise to identical interpola-
tion functions when the nodal volumes are set to unity in
kernel approximations. This result indicates that math-
ematically the least-squares and kernel approximations
are equivalent. However, for large point distributions or
for higher-order polynomial basis the numerical errors
with a non-shifted approach grow quickly compared to
a shifted approach, resulting in violation of consistency
conditions. Hence, a shifted polynomial basis is better
suited from a numerical implementation point of view.

  

Source: Aluru, Narayana R. - Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign

 

Collections: Engineering; Materials Science