共查询到20条相似文献,搜索用时 312 毫秒
1.
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
2.
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
3.
Adaptive frame methods for elliptic operator equations: the steepest descent approach 总被引:2,自引:0,他引:2
Dahlke Stephan; Raasch Thorsten; Werner Manuel; Fornasier Massimo; Stevenson Rob 《IMA Journal of Numerical Analysis》2007,27(4):717-740
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Au L.; Byrnes G. B.; Bain C. A.; Fackrell M.; Brand C.; Campbell D. A.; Taylor P. G. 《IMA Journal of Management Mathematics》2009,20(1):39-49
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Barrett John W.; Garcke Harald; Nurnberg Robert 《IMA Journal of Numerical Analysis》2008,28(2):292-330
8.
Azaiez M.; Ben Belgacem F.; Bernardi C.; El Rhabi M. 《IMA Journal of Numerical Analysis》2008,28(1):106-120
9.
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
10.
Bai Xiaoming; Yang Xiao-Song; Li Huimin 《IMA Journal of Mathematical Control and Information》2007,24(4):483-491
11.
In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by Lévy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points. 相似文献
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Numerical treatment of retarded boundary integral equations by sparse panel clustering 总被引:1,自引:0,他引:1
14.
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
15.
Gavrilyuk I. P.; Klimenko A. V.; Makarov V. L.; Rossokhata N. O. 《IMA Journal of Numerical Analysis》2007,27(4):818-838
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Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated
reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply
to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes
of functions still were not "large" enough to be applicable to a number of practical situations. In this paper we obtain Sobolev-type
error estimates on compact regions of Rn when the RBFs have Fourier transforms that decay algebraically. In addition, we derive a Bernstein inequality for spaces
of finite shifts of an RBF in terms of the minimal separation parameter. 相似文献
18.
Shale E. A.; Boylan John E.; Johnston F. R. 《IMA Journal of Management Mathematics》2008,19(2):137-143
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