The Optimal Radius of the Support of Radial Weights Used in Moving Least Squares Approximation
Y.F. Nie; S.N. Atluri; and C.W. Zuo

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 12, No. 2, pp. 137-148, 2006
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Keywords MLS methods, Radius of support, Scaling, Sobolev norm, Mathematics model, Matrix analysis, Approximate theory.
Abstract Owing to the meshless and local characteristics, moving least squares (MLS) methods have been used extensively to approximate the unknown function of partial differential equation initial boundary value problem. In this paper, based on matrix analysis, a sufficient and necessary condition for the existence of inverse of coefficient matrix used in MLS methods is developed firstly. Then in the light of approximate theory, a practical mathematics model is posed to obtain the optimal radius of support of radial weights used in MLS methods. As an example, while uniform distributed particles and the 4$^{th}$ order spline weight function are adopted in MLS method in two dimension domain and two kinds of norms are used to measure error, optimal results for linear and quadratic basis are gained. Finally, the test data verify that the optimal values are correct. The research idea can be used in 3-dimension problems too.
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