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**Toward approximate moving least squares approximation with irregularly spaced centers.**
*(English)*
Zbl 1060.65041

Summary: By combining the well-known moving least squares approximation method and the theory of approximate approximations due to V. Maz’ya and G. Schmidt [J. Approximation Theory 110, No. 2, 125–145 (2001; Zbl 0976.41004)] we are able to present an approximate moving least squares method which inherits the simplicity of D. Shepard’s method [A two-dimensional interpolation function for irregularly spaced data, in: Proc. 23rd Nat. Conf., ACM. New York, 517–523 (1968)] along with the accuracy of higher-order moving least squares approximations.

In this paper we focus our interest on practical implementations for irregularly spaced data sites. The two schemes described here along with some first numerical experiments are to be viewed as exploratory work only. These schemes apply to centers that are obtained from gridded centers via a smooth parametrization. Further work to find a robust numerical scheme applicable to arbitrary scattered data is needed.

In this paper we focus our interest on practical implementations for irregularly spaced data sites. The two schemes described here along with some first numerical experiments are to be viewed as exploratory work only. These schemes apply to centers that are obtained from gridded centers via a smooth parametrization. Further work to find a robust numerical scheme applicable to arbitrary scattered data is needed.

### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65D15 | Algorithms for approximation of functions |

### Keywords:

Moving least squares; Approximate approximation; Irregularly spaced centers; Numerical experiments; scattered data### Citations:

Zbl 0976.41004### Software:

ANN
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\textit{G. E. Fasshauer}, Comput. Methods Appl. Mech. Eng. 193, No. 12--14, 1231--1243 (2004; Zbl 1060.65041)

### References:

[1] | Fasshauer, G.E., Matrix-free multilevel moving least-squares methods, (), 271-281 · Zbl 1028.65011 |

[2] | Fasshauer, G.E., Approximate moving least-squares approximation with compactly supported weights, (), 105-116 · Zbl 1014.65014 |

[3] | G.E. Fasshauer, Approximate moving least-squares approximation for time-dependent PDEs, in: H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner (Eds.), Fifth World Congress on Computational Mechanics (WCCM V), Vienna University of Technology, 2002, available from <http://wccm.tuwien.ac.at> · Zbl 1014.65014 |

[4] | Fasshauer, G.E., Approximate moving least-squares approximation: a fast and accurate multivariate approximation method, (), 139-148 · Zbl 1037.65009 |

[5] | Lancaster, P.; Šalkauskas, K., Surfaces generated by moving least squares methods, Math. comp., 37, 141-158, (1981) · Zbl 0469.41005 |

[6] | Levin, D., The approximation power of moving least-squares, Math. comp., 67, 1517-1531, (1998) · Zbl 0911.41016 |

[7] | Li, S.; Liu, W.K., Meshfree and particle methods and their applications, Appl. mech. rev., 55, 1-34, (2002) |

[8] | Maz’ya, V.; Schmidt, G., On quasi-interpolation with non-uniformly distributed centers on domains and manifolds, J. approx. theory, 110, 125-145, (2001) · Zbl 0976.41004 |

[9] | D.M. Mount, S. Arya, ANN: library for Approximate Nearest Neighbor Searching, available from <http://www.cs.umd.edu/ mount/ANN/> · Zbl 1204.68103 |

[10] | Niederreiter, H., Random number generation and quasi-Monte Carlo methods, CBMS-NSF regional conference series in applied mathematics, vol. 63, (1992), SIAM Philadelphia, PA · Zbl 0761.65002 |

[11] | Shepard, D., A two dimensional interpolation function for irregularly spaced data, (), 517-523 |

[12] | Wendland, H., Local polynomial reproduction and moving least squares approximation, IMA J. numer. anal., 21, 285-300, (2001) · Zbl 0976.65013 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.