Tuned Local Regression Estimators for the Numerical Solution of Differential Equations
The first meshfree method was SPH. Central to its formulation is the statistical “kernel density estimator” which generalizes the histogram and was introduced by Rosenblatt in 1956. Stone introduced the improved “local regression estimator” in 1977, which
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Computer and Computation Sciences Division, Los Alamos National Laboratory, Los Alamos NM, USA. School of Computational Sciences, George Mason University, Fairfax VA, USA.
Abstract. The first meshfree method was SPH. Central to its formulation is the statistical "kernel density estimator" which generalizes the histogram and was introduced by Rosenblatt in 1956. Stone introduced the improved "local regression estimator" in 1977, which generalizes the least-squares linear fit. Given a set of data, local regression finds a locally-weighted least-squares fit for a Taylor series expansion from a given evaluation point to nearby data points. The coefficients in the Taylor series fit are estimates for all derivatives of the data, computed simultaneously. The moving-least-squares estimator used in EFG, RKPM, and MLSPH is algebraically identical to the zeroth-derivative estimate of local regression. As with finite element and smoothing kernel interpolations, local regression estimates can be expressed as an expansion in shape functions. In the finite-element and meshfree methods, derivatives of the data are estimated by taking explicit derivatives of the shape function expansion. With local regression, the estimate of the derivative is equal to the derivative of the estimate only in the limit of small smoothing length. In familiar finite element and meshfree methods for solving differential equations, a Galerkin variational approach leads to a finite-dimensional system involving derivatives of the shape functions. Since local regression estimates all derivatives of the data simultaneously, constraining those estimates to satisfy a differential equation results in an estimate of a solution near the data. A constrained least-squares approach is the result. The paper proposes a formal setting for these ideas and supplies some very simple examples. Indications are made as to how the technique might be expanded.
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Some Meshfree Concerns
The original meshfree method for PDE's, smoothed-particle hydrodynamics (SPH) [8, 11] has been successfully applied to diverse physical phenomena. Numerous difficulties with the technique have been pointed out in various places, including lack of completeness, tension instability, lack of boundary conditions, etc. Many fixes have been proposed: re-normalized, corrected or moving least squares derivatives, stress points, etc. Despite these attempts there still remain fundamental difficulties with enhanced SPH-type methods. Correcting these seems to require the imposition of conflicting demands on the technique the simultaneous resolution of which is not known at this time. It was pointed out in [3] that SPH is not complete in the sense that estimates of linear data are not strictly linear, as is the case in finite eleM. Griebel et al. (eds.), Meshfree Methods for Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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G. A. Dilts, A. Haque, J. Wallin
ments. This leads of course to a loss of accuracy and a sensitivity to particle configuration but it was shown in [5] that
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