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Recent results for moving least squares approximation
 Geometric Modeling and Computing, pages 163 – 176
, 2003
"... Abstract. We describe two experiments recently conducted with the approximate moving least squares (MLS) approximation method. On the one hand, the NFFT library of Kunis, Potts, and Steidl is coupled with the approximate MLS method to obtain a fast and accurate multivariate approximation method. The ..."
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Abstract. We describe two experiments recently conducted with the approximate moving least squares (MLS) approximation method. On the one hand, the NFFT library of Kunis, Potts, and Steidl is coupled with the approximate MLS method to obtain a fast and accurate multivariate approximation method. The second experiment uses approximate MLS approximation in combination with a multilevel approximation algorithm. This method can be used for data compression, or to obtain an approximation with radial functions that employs variable scales and nonuniform center locations. In this paper we address two limitations of approximate moving least squares (MLS) approximation with radial weight functions encountered in our earlier work (see, e.g., [3, 5, 6]). The first problem is that, even though approximate MLS approximation reduces the computational work
Using Meshfree Approximation for MultiAsset American Option Problems
, 2003
"... We study the applicability of meshfree approximation schemes for the solution of multiasset American option problems. In particular, we consider a penalty method which allows us to remove the free and moving boundary by adding a small and continuous penalty term to the BlackScholes equation. A com ..."
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We study the applicability of meshfree approximation schemes for the solution of multiasset American option problems. In particular, we consider a penalty method which allows us to remove the free and moving boundary by adding a small and continuous penalty term to the BlackScholes equation. A comparison with results obtained recently by two of the authors using a linearly implicit finite difference method is included.
Toward Approximate Moving Least Squares Approximation With Irregularly . . .
 COMPUTER METHODS IN APPLIED MECHANICS & ENGINEERING
, 2004
"... By combining the well known moving least squares approximation method and the theory of approximate approximations due to Maz'ya and Schmidt we are able to present an approximate moving least squares method which inherits the simplicity of Shepard's method along with the accuracy of higherorder mov ..."
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By combining the well known moving least squares approximation method and the theory of approximate approximations due to Maz'ya and Schmidt we are able to present an approximate moving least squares method which inherits the simplicity of Shepard's method along with the accuracy of higherorder 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.
Scattered data approximation of noisy data via iterated moving least squares
 Proceedings of Curves and Surfaces Avignon 2006, Nashboro
"... Abstract. In this paper we focus on two methods for multivariate approximation problems with nonuniformly distributed noisy data. The new approach proposed here is an iterated approximate moving leastsquares method. We compare our method to ridge regression which filters out noise by using a smoot ..."
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Abstract. In this paper we focus on two methods for multivariate approximation problems with nonuniformly distributed noisy data. The new approach proposed here is an iterated approximate moving leastsquares method. We compare our method to ridge regression which filters out noise by using a smoothing parameter. Our goal is to find an optimal number of iterations for the iterative method and an optimal smoothing parameter for ridge regression so that the corresponding approximants do not exactly interpolate the given data but are reasonably close. For both approaches we implement variants of leaveoneout crossvalidation in order to find these optimal values. The shape parameter for the basis functions is also optimized in our algorithms. In multivariate data fitting problems we are usually given data (xj, fj), j = 1,..., N with distinct xj ∈ R s and fj ∈ R, and we want to find a
Dual bases and discrete reproducing kernels: A unified framework for RBF and MLS approximation, submitted
"... Moving least squares (MLS) and radial basis function (RBF) methods play a central role in multivariate approximation theory. In this paper we provide a unified framework for both RBF and MLS approximation. This framework turns out to be a linearly constrained quadratic minimization problem. We show ..."
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Moving least squares (MLS) and radial basis function (RBF) methods play a central role in multivariate approximation theory. In this paper we provide a unified framework for both RBF and MLS approximation. This framework turns out to be a linearly constrained quadratic minimization problem. We show that RBF approximation can be considered as a special case of MLS approximation. This sheds new light on both MLS and RBF approximation. Among the new insights are dual bases for the approximation spaces and certain discrete reproducing kernels. 1
Meshfree methods
 Handbook of Theoretical and Computational Nanotechnology. American Scientific Publishers
, 2005
"... Meshfree methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares ..."
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Meshfree methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares (MLS) method. Over the past several years meshfree approximation methods have found their way into many different application areas ranging from artificial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of (partial) differential equations problems. Applications in computational nanotechnology are still somewhat rare, but do exist in the literature. In this chapter we will focus on the mathematical foundation of meshfree methods, and the discussion of various computational techniques presently available for a successful implementation of meshfree methods. At the end of this review we mention some initial applications of meshfree methods to problems in computational nanotechnology, and hope that this introduction will serve as a motivation for others to apply meshfree methods to many other challenging problems in computational nanotechnology.
1.1 History and Outline
"... Originally, the motivation for the basic meshfree approximation methods (radial basis functions and moving least squares methods) came from applications in geodesy, geophysics, mapping, or meteorology. Later, applications were found in many areas such ..."
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Originally, the motivation for the basic meshfree approximation methods (radial basis functions and moving least squares methods) came from applications in geodesy, geophysics, mapping, or meteorology. Later, applications were found in many areas such
Preconditioning of Radial Basis Function Interpolation Systems via Accelerated Iterated Approximate Moving Least Squares Approximation
"... Abstract The standard approach to the solution of the radial basis function interpolation problem has been recognized as an illconditioned problem for many years. This is especially true when infinitely smooth basic functions such as multiquadrics or Gaussians are used with extreme values of their ..."
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Abstract The standard approach to the solution of the radial basis function interpolation problem has been recognized as an illconditioned problem for many years. This is especially true when infinitely smooth basic functions such as multiquadrics or Gaussians are used with extreme values of their associated shape parameters. Various approaches have been described to deal with this phenomenon. These techniques include applying specialized preconditioners to the system matrix, changing the basis of the approximation space or using techniques from complex analysis. In this paper we present a preconditioning technique based on residual iteration of an approximate moving least squares quasiinterpolant that can be interpreted as a change of basis. In the limit our algorithm will produce the perfectly conditioned cardinal basis of the underlying radial basis function approximation space. Although our method is motivated by radial basis function interpolation problems, it can also be adapted for similar problems when the solution of a linear system is involved such as collocation methods for solving differential equations.
A Hybrid MeshlessCollocation/SpectralElement Method for Elliptic Partial Differential Equations
"... We present a hybrid numerical technique that couples spectral element approximation methods with meshless collocation methods for use in solving elliptic boundary value partial differential equations. After briefly reviewing the empirical BackusGilbert meshless collocation method by Blakely in [4], ..."
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We present a hybrid numerical technique that couples spectral element approximation methods with meshless collocation methods for use in solving elliptic boundary value partial differential equations. After briefly reviewing the empirical BackusGilbert meshless collocation method by Blakely in [4], we introduce a domain decomposition procedure which effectively couples nodal spectral element approximations with the meshless collocation method. This domain decomposition approach is an adaptation of the threefield variational formulation by Brezzi et al. [6] which uses two additional functional spaces along the interface between the different approximations. Sufficient and necessary conditions for the hybrid numerical approach to yield stable approximations will be discussed followed by numerical examples using the Helmholtz equation and different choices of discrete threefield spaces.