Approximation of parabolic PDEs on spheres using spherical basis functions

Gia, Quôc Thông Lê

doi:10.1007/s10444-003-3960-9

Cited by 32 publications

(19 citation statements)

References 21 publications

(17 reference statements)

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“…The commuting property is essential to doing so for it allows the filtered equations to resemble the original PDEs, therefore, making its analysis tractable. The results here can also be a useful contribution to approximation theory, where there are promising and exciting efforts to use radial basis functions for solving PDEs on the sphere [60,61,34,35,33,37,36].…”

Section: Discussionmentioning

confidence: 83%

Convolutions on the sphere: commutation with differential operators

Aluie

2019

Int J Geomath

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We generalize the definition of convolution of vectors and tensors on the 2-sphere, and prove that it commutes with differential operators. Moreover, vectors and tensors that are normal/tangent to the spherical surface remain so after the convolution. These properties make the new filtering operation particularly useful to analyzing and modeling nonlinear dynamics in spherical systems, such as in geophysics, astrophysics, and in inertial confinement fusion applications. An essential tool we use is the theory of scalar, vector, and tensor spherical harmonics. We then show that our generalized filtering operation is equivalent to the (traditional) convolution of scalar fields of the Helmholtz decomposition of vectors and tensors.

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Section: Discussionmentioning

confidence: 83%

Convolutions on the sphere: commutation with differential operators

Aluie

2019

Int J Geomath

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“…To overcome this drawback, [22] treated the approximation of functions from fixed large Sobolev-type spaces via the same assumption on the kernels, that is, the same asymptotic behavior on the Fourier-Jacobi coefficients of K . Others applications of this assumption may be found in [17], [19], [21], [26], [37]. Also, it is important to add that (2.11) is a reasonable assumption for the present setting due to the behaviour of the sequence {λ k } ensured by Lemma 2.2.…”

Section: Theorem 21 (The Funk-hecke Formula) Let S Be a Spherical Hamentioning

confidence: 93%

Covering numbers of isotropic reproducing kernels on compact two‐point homogeneous spaces

Azevedo

Barbosa²

2017

Mathematische Nachrichten

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In this paper we present upper and lower estimates for the covering numbers of the unit ball of a reproducing kernel Hilbert space associated to a continuous isotropic kernel on a compact two‐point homogeneous space (CTPHS). These estimates are obtained from estimates on the decay of the Fourier–Jacobi coefficients of the kernel via applications of the Funk–Hecke formula and the Schoenberg series representation of an isotropic kernel on CTPHS and also by the use of cubature formulas on these spaces.

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“…The coefficients of A must be independent of t, and later we restrict our attention to the case when −A is the LaplaceBeltrami operator. In that case, equation (1) describes the diffusion of heat on the surface of the sphere with a given density f of sources [2,4].…”

Section: Introductionmentioning

confidence: 99%

“…Instead of using time stepping [2], our approach here is based on an approximation by quadrature of the representation (5), with an appropriately deformed contour of integration. This idea was introduced by Sheen, Sloan and Thomée [7] for a parabolic problem on a bounded domain Ω in R n , and used in conjunction with a spatial discretisation by finite elements.…”

Section: Introductionmentioning

confidence: 99%