Detection of discontinuities in scattered data approximation

A Detection Algorithm for the localisation of unknown fault lines of a surface from scattered data is given. The method is based on a local approximation scheme using thin plate splines, and we show that this yields approximation of second order accuracy instead of first order as in the global case. Furthermore, the Detection Algorithm works with triangulation methods, and we show their utility for the approximation of the fault lines. The output of our method provides polygonal curves which can be used for the purpose of constrained surface approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multilevel scattered data approximation by adaptive domain decomposition

A new multilevel approximation scheme for scattered data is proposed. The scheme relies on an adaptive domain decomposition strategy using quadtree techniques (and their higher-dimensional generalizations). It is shown in the numerical examples that the new method achieves an improvement on the approximation quality of previous well-established multilevel interpolation schemes. AMS subject classification 65D15, 65D05, 65D07, 65D17     

Error estimates for interpolation of rough data using the scattered shifts of a radial basis function

The error between appropriately smooth functions and their radial basis function interpolants, as the interpolation points fill out a bounded domain in R^d, is a well studied artifact. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function – the native space. The native space contains functions possessing a certain amount of smoothness. This paper establishes error estimates when the function being interpolated is conspicuously rough. AMS subject classification 41A05, 41A25, 41A30, 41A63R.A. Brownlee: Supported by a studentship from the Engineering and Physical Sciences Research Council.     

Recent developments in error estimates for scattered-data interpolation via radial basis functions

Error estimates for scattered data interpolation by shifts of a positive definite function for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. However, apart from special cases where data is gridded, these interpolation estimates do not apply when the target functions generating the data are outside of the associated RKHS, and in fact until very recently no estimates were known in such situations. In this paper, we review these estimates in cases where the underlying space is Rⁿ and the positive definite functions are radial basis functions (RBFs). AMS subject classification 41A25, 41A05, 41A63, 42B35Research supported by grant DMS-0204449 from the National Science Foundation.     

Recursive Kernels

This paper is an extension of earlier papers [8, 9] on the “native” Hilbert spaces of functions on some domain Ω ⊂ R ^d in which conditionally positive definite kernels are reproducing kernels. Here, the focus is on subspaces of native spaces which are induced via subsets of Ω, and we shall derive a recursive subspace structure of these, leading to recursively defined reproducing kernels. As an application, we get a recursive Neville-Aitken-type interpolation process and a recursively defined orthogonal basis for interpolation by translates of kernels.     

RECURSIVE KERNELS

This paper is an extension of earlier papers [8, 9] on the ＂native＂ Hilbert spaces of functions on some domain Ωbelong toR^d Rd in which conditionally positive definite kernels are reproducing kernels. Here, the focus is on subspaces of native spaces which are induced via subsets of Ω, and we shall derive a recursive subspace structure of these, leading to recur- sively defined reproducing kernels. As an application, we get a recursive Neville-Aitken- type interpolation process and a recursively defined orthogonal basis for interpolation by translates of kernels.     

函数的径向基表示

本文是关于函数的径向表示的综述性文章，基于我们对这方面的研究工作的了解，介绍了国际上近年来这方面的主要的有关研究结果以及在一些领域中应用的情况，文后附有主要的参考文献以便感兴趣的读者查阅。     

SOME SHAPE-PRESERVING QUASI-INTERPOLANTS TO NON-UNIFORMLY DISTRIBUTED DATABY MQ-B-SPLINES

Based on the definition of MQ-B-Splines, this article constructs five types of univariate quasi-interpolants to non-uniformly distributed data. The error estimates and the shape-preserving properties are shown in details. And examples are shown to demonstrate the capacity of the quasi-interpolants for curve representation.     

Approximation by radial bases and neural networks

In this paper, we study approximation by radial basis functions including Gaussian, multiquadric, and thin plate spline functions, and derive order of approximation under certain conditions. Moreover, neural networks are also constructed by wavelet recovery formula and wavelet frames.     

Multiscale support vector regression method in Sobolev spaces on bounded domains

In this paper, we investigate the multiscale support vector regression (SVR) method for approximation of functions in Sobolev spaces on bounded domains. The Vapnik ?-intensive loss function, which has been developed well in learning theory, is introduced to replace the standard l² loss function in multiscale least squares methods. Convergence analysis is presented to verify the validity of the multiscale SVR method with scaled versions of compactly supported radial basis functions. Error estimates on noisy observation data are also derived to show the robustness of our proposed algorithm. Numerical simulations support the theoretical predictions.     

Radial basis interpolation on homogeneous manifolds: convergence rates

Pointwise error estimates for approximation on compact homogeneous manifolds using radial kernels are presented. For a positive definite kernel κ the pointwise error at x for interpolation by translates of κ goes to 0 like ρ ^r , where ρ is the density of the interpolating set on a fixed neighbourhood of x. Tangent space techniques are used to lift the problem from the manifold to Euclidean space, where methods for proving such error estimates are well established. Partially supported by NSF Grant DMS-9972004.     

径向基插值BFGP算法的一种异步并行格式

本文研究了多变量散乱数据插值问题,利用径向基函数方法,得到了并行迭代格式及其收敛性,改进了BFGP算法.     

Constructive Approximation by Superposition of Sigmoidal Functions

In this paper, a constructive theory is developed for approximating functions of one or more variables by superposition of sigmoidal functions. This is done in the uniform norm as well as in the $L^p$ norm. Results for the simultaneous approximation, with the same order of accuracy, of a function and its derivatives (whenever these exist), are obtained. The relation with neural networks and radial basis functions approximations is discussed. Numerical examples are given for the purpose of illustration.     

Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does not belong to the native space of the RBF.

     

Nonlinear approximation using Gaussian kernels

It is well known that nonlinear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron (in press) [2] for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore-Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.     

Surface Reconstruction of 3D Scattered Data with Radial Basis Functions

We use Radial Basis Functions （RBFs） to reconstruct smooth surfaces from 3D scattered data. An object＇s surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improvements on the methods of surface reconstruction with radial basis functions. A sparse approximation set of scattered data is constructed by reducing the number of interpolating points on the surface. We present an adaptive method for finding the off-surface normal points. The order of the equation decreases greatly as the number of the off-surface constraints reduces gradually. Experimental results are provided to illustrate that the proposed method is robust and may draw beautiful graphics.     

Local polynomial reproduction and moving least squares approximation

Local polynomial reproduction is a key ingredient in providingerror estimates for several approximation methods. To boundthe Lebesgue constants is a hard task especially in a multivariatesetting. We provide a result which allows us to bound the Lebesgueconstants uniformly and independently of the space dimensionby oversampling. We get explicit and small bounds for the Lebesgueconstants. Moreover, we use these results to establish errorestimates for the moving least squares approximation scheme,also with special emphasis on the involved constants. We discussthe numerical treatment of the method and analyse its effort.Finally, we give large scale examples.     

Unconditional basis recursively generated in L p on compact sets

We construct an unconditional basis in the Banach space L ^p(Ω) for p 1 by using the refinement equation and the basic operation of translation and scale, where Ω is a compact subset in ℝⁿ. We also give an algorithm of how to construct an unconditional basis in L ^p(Ω p). At the end of this paper, we give the characterization of the functions in L ^p(Ω p) by using the wavelet coefficients.     

A density theorem for matrix-valued radial basis functions

Recently a new class of customized radial basis functions (RBFs) was introduced. We revisit this class of RBFs and derive a density result guaranteeing that any sufficiently smooth divergence-free function can be approximated arbitrarily closely by a linear combination of members of this class. This result has potential applications to numerically solving differential equations, such as fluid flows, whose solution is divergence free. AMS subject classification 41Axx, 41A30, 41A35, 41A60Svenja Lowitzsch: The results are part of the authorss dissertation written at Texas A&M University, College Station, TX 77843, USA.     

Unconditional basis recursively generated in L~p on compact sets

We construct an unconditional basis in the Banach space Lp(Ω, ρ) for p > 1 by using the refinement equation and the basic operation of translation and scale, where Ω is a compact subset in Rn. We also give an algorithm of how to construct an unconditional basis in Lp(Ω,ρ). At the end of this paper, we give the characterization of the functions in Lp (Ω,ρ) by using the wavelet coefficients.