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1.
Francis J. Narcowich 《Numerical Algorithms》2005,39(1-3):307-315
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. 相似文献
2.
Francis J. Narcowich Xinping Sun Joseph D. Ward 《Advances in Computational Mathematics》2007,27(1):107-124
Error estimates for scattered data interpolation by “shifts” of a conditionally positive definite function (CPD) for target
functions in its native space, which is its associated reproducing kernel Hilbert space (RKHS), have been known for a long
time. Regardless of the underlying manifold, for example ℝn or S
n, these error estimates are determined by the rate of decay of the Fourier transform (or Fourier series) of the CPD. This
paper deals with the restriction of radial basis functions (RBFs), which are radial CPD functions on ℝn+1, to the unit sphere S
n. In the paper, we first strengthen a result derived by two of us concerning an explicit representation of the Fourier–Legendre
coefficients of the restriction in terms of the Fourier transform of the RBF. In addition, for RBFs that are related to completely
monotonic functions, we derive a new integral representation for these coefficients in terms of the measure generating the
completely monotonic function. These representations are then utilized to show that if an RBF has a native space equivalent
to a Sobolev space H
s(ℝn+1), then the restriction to S
n has a native space equivalent to H
s−1/2(S
n). In addition, they are used to recover the asymptotic behavior of such coefficients for a wide variety of RBFs. Some of
these were known earlier.
Joseph D. Ward: Francis J. Narcowich: Research supported by grant DMS-0204449 from the National Science Foundation. 相似文献
3.
Edward J. Fuselier. 《Mathematics of Computation》2008,77(263):1407-1423
Recently, error estimates have been made available for divergence-free radial basis function (RBF) interpolants. However, these results are only valid for functions within the associated reproducing kernel Hilbert space (RKHS) of the matrix-valued RBF. Functions within the associated RKHS, also known as the ``native space' of the RBF, can be characterized as vector fields having a specific smoothness, making the native space quite small. In this paper we develop Sobolev-type error estimates when the target function is less smooth than functions in the native space.
4.
Francis J. Narcowich Joseph D. Ward Holger Wendland. 《Mathematics of Computation》2005,74(250):743-763
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.
5.
Radial basis functions (RBFs) have found important applications in areas such as signal processing, medical imaging, and neural networks since the early 1980s. Several applications require that certain physical properties are satisfied by the interpolant, for example, being divergence-free in case of incompressible data. In this paper we consider a class of customized (e.g., divergence-free) RBFs that are matrix-valued and have compact support; these are matrix-valued analogues of the well-known Wendland functions. We obtain stability estimates for a wide class of interpolants based on matrix-valued RBFs, also taking into account the size of the compact support of the generating RBF. We conclude with an application based on an incompressible Navier–Stokes equation, namely the driven-cavity problem, where we use divergence-free RBFs to solve the underlying partial differential equation numerically. We discuss the impact of the size of the support of the basis function on the stability of the solution.
AMS subject classification 65D05 相似文献
6.
In this paper, we present sharp estimates for the covering numbers of the embedding of the reproducing kernel Hilbert space (RKHS) associated with the Weierstrass fractal kernel into the space of continuous functions. The method we apply is based on the characterization of the infinite-dimensional RKHS generated by the Weierstrass fractal kernel and it requires estimates for the norm operator of orthogonal projections on the RKHS. 相似文献
7.
In this article we study reproducing kernel Hilbert spaces (RKHS) associated with translation-invariant Mercer kernels. Applying
a special derivative reproducing property, we show that when the kernel is real analytic, every function from the RKHS is
real analytic. This is used to investigate subspaces of the RKHS generated by a set of fundamental functions. The analyticity
of functions from the RKHS enables us to derive some estimates for the covering numbers which form an essential part for the
analysis of some algorithms in learning theory.
The work is supported by City University of Hong Kong (Project No. 7001816), and National Science Fund for Distinguished Young
Scholars of China (Project No. 10529101). 相似文献
8.
We introduce a class of matrix-valued radial basis functions (RBFs) of compact support that can be customized, e.g. chosen to be divergence-free. We then derive and discuss error estimates for interpolants and derivatives based on these matrix-valued RBFs. 相似文献
9.
Edward J. Fuselier 《Advances in Computational Mathematics》2008,29(3):311-313
In this paper we derive several new results involving matrix-valued radial basis functions (RBFs). We begin by introducing
a class of matrix-valued RBFs which can be used to construct interpolants that are curl-free. Next, we offer a characterization
of the native space for divergence-free and curl-free kernels based on the Fourier transform. Finally, we investigate the
stability of the interpolation matrix for both the divergence-free and curl-free cases, and when the kernel has finite smoothness
we obtain sharp estimates.
An erratum to this article can be found at 相似文献
10.
Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions 总被引:1,自引:0,他引:1
Jichun Li 《Advances in Computational Mathematics》2005,23(1-2):21-30
By extending Wendlands meshless Galerkin methods using RBFs, we develop mixed methods for solving fourth-order elliptic and parabolic problems by using RBFs. Similar error estimates as classical mixed finite element methods are proved.
AMS subject classification 35G15, 65N12 相似文献
11.
M. V. Korobkov 《Siberian Mathematical Journal》2002,43(5):827-842
In the framework of Kopylov's -stability concept, we study some stable classes of Lipschitz functions of one real variable. We obtain an exhaustive (nontrivial) classification for these classes and establish the relevant stability estimates in the
-norm. 相似文献
12.
In this paper we investigate the problem of learning an unknown bounded function. We will emphasize special cases where it
is possible to provide very simple (in terms of computation) estimates enjoying, in addition, the property of being universal,
i.e. their construction does not depend on the a priori knowledge of regularity conditions on the unknown object and still
they have almost optimal properties for a whole group of functions spaces. These estimates are constructed using a thresholding
technique, which has proven in the last decade in statistics to have very good properties for recovering signals with inhomogeneous
smoothness but has not been extensively developed in learning theory. We will basically consider two particular situations.
In the first case, we consider the RKHS situation, where we produce a new algorithm and investigate its performances in
. The exponential rates of convergences are proved to be almost optimal, and the regularity assumptions are expressed in
simple terms. The second case considers a more specified situation where the Xi's are one-dimensional and the estimator is a wavelet thresholding estimate. The results are comparable in this setting to
those obtained in the RKHS situation, as concerned the critical value and the exponential rates. The advantage here is that
we are able to state the results in the
-norm and the regularity conditions are expressed in terms of standard Holder spaces. 相似文献
13.
Fornberg Bengt; Flyer Natasha; Hovde Susan; Piret Cecile 《IMA Journal of Numerical Analysis》2008,28(1):121-142
14.
A modified method of approximate particular solutions for solving linear and nonlinear PDEs
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Guangming Yao Ching‐Shyang Chen Hui Zheng 《Numerical Methods for Partial Differential Equations》2017,33(6):1839-1858
The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506–522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matérn and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1839–1858, 2017 相似文献
15.
16.
Recently, Rieger and Zwicknagl (2010) have introduced sampling inequalities for infinitely smooth functions to derive Sobolev-type error estimates. They introduced exponential convergence orders for functions within the native space associated with the given radial basis function (RBF). Our major concern of this paper is to extend the results made in Rieger and Zwicknagl (2010). We derive generalized sampling inequalities for the larger class of infinitely smooth RBFs, including multiquadrics, inverse multiquadrics, shifted surface splines and Gaussians. 相似文献
17.
Gaussian radial basis functions (RBFs) on an infinite interval with uniform grid pacing h are defined by ?(x;α,h)≡exp(-[α2/h2]x2). The only significant numerical parameter is α, the inverse width of the RBF functions relative to h. In the limit α→0, we demonstrate that the coefficients of the interpolant of a typical function f(x) grow proportionally to exp(π2/[4α2]). However, we also show that the approximation to the constant f(x)≡1 is a Jacobian theta function whose coefficients do not blow up as α→0. The subtle interplay between the complex-plane singularities of f(x) (the function being approximated) and the RBF inverse width parameter α are analyzed. For α≈1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(104) and the error saturation is smaller than machine epsilon, so this α is the center of a “safe operating range” for Gaussian RBFs. 相似文献
18.
We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives. 相似文献
19.
A meshless method based on the dual reciprocity method for one‐dimensional stochastic partial differential equations
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Mehdi Dehghan Mohammad Shirzadi 《Numerical Methods for Partial Differential Equations》2016,32(1):292-306
This article describes a new meshless method based on the dual reciprocity method (DRM) for the numerical solution of one‐dimensional stochastic heat and advection–diffusion equations. First, the time derivative is approximated by the time–stepping method to transforming the original stochastic partial differential equations (SPDEs) into elliptic SPDEs. The resulting elliptic SPDEs have been approximated with the new method, which is a combination of radial basis functions (RBFs) method and the DRM method. We have used inverse multiquadrics (IMQ) and generalized IMQ (GIMQ) RBFs, to approximate functions in the presented method. The noise term has been approximated at the source points, at each time step. The developed formulation is verified in two test problems with investigating the convergence and accuracy of numerical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 292–306, 2016 相似文献
20.
Svenja Lowitzsch 《Numerical Algorithms》2005,39(1-3):253-256
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. 相似文献