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SKETCHING WITH spherical DESIGNS FOR NOISY DATA FITTING ON SPHERES

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2024年第1期46卷 A313-A337页

作者： Lin, Shao-bo Wang, Di Zhou, Ding-xuan Xi An Jiao Tong Univ Ctr Intelligent Decis Making & Machine Learning Sch Management Xian 710049 Peoples R China Univ Sydney Sch Math & Stat Sydney NSW 2006 Australia

This paper proposes a sketching strategy with spherical designs to equip the classical spherical basis function (SBF) approach for massive spherical data fitting. We conduct theoretical analysis and numerical verifications to demonstrate the feasibility of the proposed sketching strategy. From the theoretical side, we prove that sketching based on spherical designs can reduce the computational burden of the SBF approach without sacrificing its approximation capability. In particular, we provide upper and lower bounds for the proposed sketching strategy to fit noisy data on spheres. From the experimental side, we numerically illustrate the feasibility of the sketching strategy by showing its comparable fitting performance with the SBF approach. These interesting findings show that the proposed sketching strategy is effective in fitting massive and noisy data on spheres.

关键词： spherical data fitting spherical basis functions sketching spherical designs

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Generalised Wendland functions for the sphere

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2023年第1期49卷 1-21页

作者： Hubbert, Simon Jaeger, Janin Birkbeck Univ London Dept Econ Math & Stat Malet St London WC1E 7HX England Justus Liebig Univ Lehrstuhl Numer Math D-35392 Giessen Germany

In this paper, we compute the spherical Fourier expansion coefficients for the restriction of the generalised Wendland functions from d-dimensional Euclidean space to the (d - 1)-dimensional unit sphere. We use results from the theory of special functions to show that they can be expressed in a closed form as a multiple of a certain F-3(2) hypergeometric function. We present tight asymptotic bounds on the decay rate of the spherical Fourier coefficients and, in the case where d is odd, we are able to provide the precise asymptotic rate of decay. Numerical evidence suggests that this precise asymptotic rate also holds when d is even and we pose this as an open problem. Finally, we observe a close connection between the asymptotic decay rate of the spherical Fourier coefficients and that of the corresponding Euclidean Fourier transform.

关键词： Positive definite kernels spherical basis functions Compact support

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Polya-type criteria for conditional strict positive definiteness of functions on spheres

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JOURNAL OF APPROXIMATION THEORY 2020年 257卷

作者： Buhmann, Martin Jaeger, Janin Justus Liebig Univ Lehrstuhl Numer Math D-35392 Giessen Germany

Identifying (conditionally) strictly positive definite functions is of great importance as they allow the unique solution of certain interpolation problems. We introduce new sufficient (and some necessary) conditions for functions to be conditionally strictly positive definite on all spheres Sd-1, d > 2, only employing monotonicity properties. For strictly positive definite and conditionally negative definite functions on all spheres we give a characterisation in terms of monotonicity properties. Further, we prove that multiply monotone functions are positive definite on spheres up to a certain dimension. (C) 2020 Elsevier Inc. All rights reserved.

关键词： spherical basis functions Isotropic kernels Dot product kernels Completely monotone functions Interpolation of scattered data Conditionally positive definite kernels

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A Petrov-Galerkin RBF method for diffusion equation on the unit sphere

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NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 2020年第6期36卷 1682-1698页

作者： Darani, Mohammadreza Ahmadi Mirzaei, Davoud Shahrekord Univ Fac Math Sci Dept Appl Math Shahrekord Iran Univ Isfahan Fac Math & Stat Dept Appl Math & Comp Sci Esfahan *** Iran Inst Res Fundamental Sci IPM Sch Math Tehran Iran

This paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov-Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation and the use of local weak forms instead of a global variational formulation. The local Petrov-Galerkin formulation allows to compute the integrals on small independent spherical caps without any dependence on a connected background mesh. Experimental results show the accuracy and the efficiency of the new method.

关键词： diffusion on the sphere meshless methods Petrov-Galerkin method radial basis functions spherical basis functions

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A PETROV-GALERKIN KERNEL APPROXIMATION ON THE SPHERE

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2018年第1期56卷 274-295页

作者： Mirzaei, Davoud Univ Isfahan Dept Math Esfahan *** Iran

In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov-Galerkin test discretization. The solvability of the scheme is proved, and the error bounds are obtained for functions in appropriate Sobolev spaces. The condition number of the final system is estimated in terms of discretization parameters. The method is meshless because in the trial side the numerical solution parameterizes entirely in terms of scattered points and in the test side everything breaks down to simple numerical integrations over independent spherical caps. This means that no connected background mesh is required for either approximation or integration.

关键词： radial basis functions spherical basis functions Petrov-Galerkin method convolution data error analysis condition numbers

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Using spherical basis functions on a Polar Grid for Iterative Image Reconstruction in Small Animal PET

Using Spherical Basis Functions on a Polar Grid for Iterativ...

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Conference on Medical Imaging 2011 - Physics of Medical Imaging

作者： Cabello, Jorge Oliver, Josep F. Rafecas, Magdalena Univ Valencia CSIC Inst Fis Corpuscular IFIC Valencia Spain

ISBN: (纸本)9780819485038

Statistical iterative methods have been extensively demonstrated to outperform analytical methods in terms of image quality in nuclear imaging. In the method, the mathematically unknown biodistribution is usually represented by cubic basis functions in 3D. Alternatively, spherical basis functions have demonstrated lower noise produced in the resulting reconstructed images. Additionally, the system response matrix (SRM), a key element required by iterative methods, is usually too large to be stored in random access memory of a regular computer. The SRM can be calculated prior to reconstruction and stored on-disk, and thus be directly accessed during the reconstruction process. But this approach usually makes the process too time consuming. To reduce the number of elements to be computed and stored, a common approach uses the scanner symmetries. In this work we use polar voxels, reducing the number of non-zero elements to be computed by a factor 72, allowing us to speed up the Monte-Carlo simulations in which the computation of the SRM is based. In this work we combine the use blobs, a type of spherical basis function, as basis functions and a polar representation in the SRM. The latter is especially important for blobs, given that due to the overlapping of blobs, the size of the SRM significantly increases. This work will show a quantitative comparison of reconstructed images using Cartesian voxels, polar voxels and blobs. We will show that blobs reduce image noise compared to voxels, producing a lower spatial resolution degradation, compared to polar voxels after Gaussian filtering.

关键词： PET iterative statistical reconstruction Monte Carlo system response matrix spherical basis functions blobs polar voxels

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L^p BERNSTEIN ESTIMATES AND APPROXIMATION BY spherical basis functions

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MATHEMATICS OF COMPUTATION 2010年第271期79卷 1647-1679页

作者： Mhaskar, H. N. Narcowich, F. J. Prestin, J. Ward, J. D. Calif State Univ Los Angeles Dept Math Los Angeles CA 90032 USA Texas A&M Univ Dept Math College Stn TX 77843 USA Med Univ Lubeck Inst Math D-23560 Lubeck Germany

The purpose of this paper is to establish L-p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L-p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L-p norm of the function itself. An important step in its proof involves measuring the L-p stability of functions in the approximating space in terms of the l(p) norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L-P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.

关键词： Sphere Bernstein estimates approximation spherical basis functions

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A combination of downward continuation and local approximation for harmonic potentials

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INVERSE PROBLEMS 2014年第8期30卷

作者： Gerhards, C. Univ Kaiserslautern Geomath Grp D-67663 Kaiserslautern Germany

This paper presents a method for the approximation of harmonic potentials that combines downward continuation of globally available data on a sphere Omega(R) of radius R (e.g., a satellite's orbit) with locally available data in a subregion Gamma(r) of the sphere Omega(r) of radius r < R (e.g., the spherical Earth's surface). The approximation is based on a two-step algorithm motivated by spherical multiscale expansions: first, a convolution with a scaling kernel Phi(N) deals with the downward continuation from Omega(R) to Omega(r), while in a second step, the result is locally refined by a convolution on Omega(r) with a wavelet kernel <(Psi)over tilde>(N). The kernels Phi(N) and (Psi) over tilde (N) are optimized in such a way that the former behaves well for the downward continuation while the latter shows a good localization in Gamma(r) The concept is indicated for scalar as well as vector potentials.

关键词： downward continuation harmonic potentials multiscale methods spatial localization spherical basis functions

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Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions

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FOUNDATIONS OF COMPUTATIONAL MATHEMATICS 2007年第3期7卷 369-390页

作者： Narcowich, Francis J. Sun, Xingping Ward, Joseph D. Wendland, Holger Texas A&M Univ Dept Math College Stn TX 77843 USA SW Missouri State Univ Dept Math Springfield MO 65804 USA Univ Gottingen D-37083 Gottingen Germany

The purpose of this paper is to get error estimates for spherical basis function (SBF) interpolation and approximation for target functions in Sobolev spaces less smooth than the SBFs, and to show that the rates achieved are, in a sense, best possible. In addition, we establish a Bernstein-type theorem, where the smallest separation between data sites plays the role of a Nyquist frequency. We then use these Berstein-type estimates to derive inverse estimates for interpolation via SBFs.

关键词： scattered data spherical basis functions radial basis functions error estimates inverse theorem Bernstein theorem

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Approximation power of RBFs and their associated SBFs: a connection

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2007年第1期27卷 107-124页

作者： Narcowich, Francis J. Sun, Xinping Ward, Joseph D. Texas A&M Univ Dept Math College Stn TX 77843 USA SW Missouri State Univ Dept Math Springfield MO 65804 USA

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 R-n 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 Hs-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.

关键词： interpolation radial basis functions spherical basis functions native space Fourier-Legendre coefficients

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