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Approximation of parabolic PDEs on spheres using spherical basis functions

2005
*
Advances in Computational Mathematics
*

In this paper we investigate the

doi:10.1007/s10444-003-3960-9
fatcat:arohdpwkyjeo7jpjdpwjrnbu5u
*approximation**of*a class*of**parabolic*partial differential equations*on*the unit*spheres*S n ⊂ R n+1*using**spherical**basis**functions*. ... Le Gia /*Approximation**of**parabolic**PDEs**on**spheres*Based*on*[4, theorem 4.7], we have the following theorem. Theorem 2.1. ... The*spherical**basis**functions**used*to construct the*approximate*solution are derived from a class*of*locally supported radial*basis**function*proposed by Wendland [21] . ...##
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Chapter 1. Introduction
[chapter]

1997
*
Spherical Means for PDEs
*

In Chapter 8 we give some applications: these are probabilistic numerical algorithms for solving

doi:10.1515/9783110926026-002
fatcat:4j5gghw7m5c3xbajndcvqqmjda
*PDEs*(the so-called Random Walk*on**Spheres*algorithms) which we construct*on*the*basis**of*the integral formulations ... are resolved*on*the*basis**of*the well developed Fredholm theory. ...##
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Isometric embeddings of 2-spheres by embedding flow for applications in numerical relativity

2012
*
Classical and quantum gravity
*

We employ spectral methods to handle

doi:10.1088/0264-9381/29/15/155010
fatcat:rqccblrqxvfbhanlcyydorq4oq
*functions**on*the surface and to solve various (non)-linear elliptic*PDEs*. ... The method is based*on*a construction introduced by Weingarten and was*used*in Nirenberg's proof*of*Weyl's conjecture. ... This leads*us*to the following numerical problem: Given the gradient*of*a*function**on*S 2 expanded in terms*of**spherical*harmonics, what is the*spherical*harmonics decomposition*of*the original*function*...##
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Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences
[chapter]

2013
*
Handbook of Geomathematics
*

Lecture series: Kernel

doi:10.1007/978-3-642-27793-1_61-1
fatcat:3ahjcvubkbdxhkugstao3xylh4
*approximation**on*the*sphere*with applications to computational geosciences. 2. ... SCHOLARLY ACTIVITY RESEARCH AREAS Applied and computational mathematics: radial*basis**function*, pseudospectral, and high resolution finite volume methods for numerically solving partial differential equations ... Customized*Approximation*with Radial*Basis**Functions*. 18.*PDEs**on*the*Sphere*. Santa Fe, NM, April 27-30, 2009. Invited. ...##
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Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences
[chapter]

2015
*
Handbook of Geomathematics
*

Lecture series: Kernel

doi:10.1007/978-3-642-54551-1_61
fatcat:kxa622nqozddlkvcai2e4q36yi
*approximation**on*the*sphere*with applications to computational geosciences. 2. ... SCHOLARLY ACTIVITY RESEARCH AREAS Applied and computational mathematics: radial*basis**function*, pseudospectral, and high resolution finite volume methods for numerically solving partial differential equations ... Customized*Approximation*with Radial*Basis**Functions*. 18.*PDEs**on*the*Sphere*. Santa Fe, NM, April 27-30, 2009. Invited. ...##
###
An adaptive meshfree diffusion wavelet method for partial differential equations on the sphere

2014
*
Journal of Computational Physics
*

An adaptive meshfree diffusion wavelet method for solving partial differential equations (

doi:10.1016/j.jcp.2014.04.044
fatcat:o6madigoybd2no4sb7mxcgmivu
*PDEs*)*on*the*sphere*is developed. ...*Approximation*formulae for Laplacian-Beltrami (∇ 2 ) and gradient ( ∇) operators are derived*using*radial*basis**functions*(RBFs), and the convergence*of*these*approximations*to ∇ 2 and ∇ is verified for ... Acknowledgements The first author would like to thank Council*of*Scientific and Industrial Research for providing Ph.D. scholarship. We would also like to acknowledge prof. ...##
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Asymptotic Methods
[chapter]

2011
*
Partial Differential Equations of Applied Mathematics
*

for a Triangle 871 Finite Element

doi:10.1002/9781118033302.ch10
fatcat:dw3utrc34nhmrmvr4kjs6rpafi
*Basis**Functions*872 Plots*of**Basis**Functions*874 Full Set*of*Finite Element*Basis**Functions*874 Finite Element Representations in Terms*of**Basis**Functions*and Their Plots ... in*One*Dimension 418 Green's*Functions*for Nonself-Adjoint Elliptic Equations Finite Difference Methods for*PDEs*with Variable Coefficients 821 Method*of*Lines for Linear and Semilinear*Parabolic*Equations ...##
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A Stable Algorithm for Flat Radial Basis Functions on a Sphere

2008
*
SIAM Journal on Scientific Computing
*

Since interpolation in the flat RBF limit

doi:10.1137/060671991
fatcat:fbrsjc4pdbc2lpkfb7q3bnmsjq
*on*a*sphere*is found to coincide with*spherical*harmonics interpolation, new insights are gained as to why the RBF approach (with non-flat*basis**functions*) often ... Because the most obvious method to calculate an RBF interpolant becomes a numerically unstable algorithm for a stable problem in case*of*near-flat*basis**functions*, there will typically also be a separate ... It follows from the RBF-QR algorithm that ε → 0 leads to the same results as when*using*SPH*basis**functions*.*One*might therefore ask why not just*use*SPH as a computational*basis**on*the*sphere*. ...##
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Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

2013
*
Advances in Computational Mathematics
*

The spatial

doi:10.1007/s10444-013-9311-6
fatcat:pltngcw5ljdazcp4w4elkeqw3e
*approximation**of*the solution employs radial*basis**functions*restricted to the*sphere*. The method allows*us*to construct high accuracy numerical solutions in parallel. ... We propose a method to construct numerical solutions*of**parabolic*equations*on*the unit*sphere*. The time discretization*uses*Laplace transforms and quadrature. ...*Approximation**of**parabolic**PDEs**on**spheres**using*spher- ical*basis**functions*. Adv. Comput. Math., 22:377397, 2005. [5] W. McLean and V. Thomée. ...##
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The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces

2012
*
Journal of Computational Physics
*

Much work has been done

doi:10.1016/j.jcp.2012.03.007
fatcat:6mzytwzx5re5xp3ogopf3k6cki
*on*reconstructing arbitrary surfaces*using*the radial*basis**function*(RBF) method, but*one*can hardly find any work done*on*the*use**of*RBFs to solve partial differential equations ... (*PDEs*)*on*arbitrary surfaces. ... Acknowledgements The work*of*this author was supported by a FSR post-doctoral grant from the catholic University*of*Louvain. ...##
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Scale Space Analysis and Active Contours for Omnidirectional Images

2007
*
IEEE Transactions on Image Processing
*

We derive new energy

doi:10.1109/tip.2007.899008
pmid:17605386
fatcat:xqflyykt2jcppjnl7tbm3nuoie
*functionals*and*PDEs*for segmenting images obtained from catadioptric cameras and show that they can be implemented robustly*using*classical finite difference schemes. ... Various experimental results illustrate the potential*of*these new methods*on*both synthetic and natural images. ... contour*on*a*parabolic*manifold (44)*using*. ...##
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A finite element method of the self-consistent field theory on general curved surfaces

2019
*
Journal of Computational Physics
*

Numerical results illustrate the efficiency

doi:10.1016/j.jcp.2019.02.047
fatcat:5wzbsdlvnrhl5fbtb5vijb2234
*of*the proposed method. The obtained ordered structures are consistent with the previous results*on*standard surfaces, such as*sphere*and torus. ... However, most*of*these focus*on*the bulk systems, and little work*on*the confined systems, especially*on*general curved surfaces. ... For a special surface, such as a*sphere*, the global*basis**of**spherical*harmonics can be*used*to expand the spatial*functions*. ...##
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Page 2633 of Mathematical Reviews Vol. , Issue 88e
[page]

1988
*
Mathematical Reviews
*

An orthonormal

*basis*{9 ;,} is*used**on*A; and a finite-element ba- sis {yj}*on*B;. A Petrov-Galerkin method is*used*, in which the*basis**on*Q; is a finite set*of*the*functions*{gj x yj}. ... A bivariate tau*approximation*, applied to linear*PDEs*by Ortiz and H. ...##
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Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

2010
*
Numerical Mathematics: Theory, Methods and Applications
*

The distance

doi:10.4208/nmtma.2010.m9009
fatcat:7ezwfg3s5rbxfmf6zgmdvjnx2e
*function*is computed by implicit local interpolants defined in terms*of*radial*basis**functions*. ... Space discretization*of*the*PDE*model is obtained by finite co-volume schemes and semi-implicit approach is*used*in time/scale. ... Acknowledgements This work has been supported by PRIN-MIUR-Cofin 2006, project, by "Progetti Strategici EF2006" University*of*Bologna, and by University*of*Bologna "Funds for selected research topics". ...##
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Transport schemes on a sphere using radial basis functions

2007
*
Journal of Computational Physics
*

*used*spectral methods

*on*a

*sphere*such as

*spherical*harmonics, double Fourier series, and spectral element methods. ...

*on*a

*sphere*. ... Summary The main goal

*of*this paper is to illustrate the effectiveness and performance

*of*the RBF methodology for solving purely hyperbolic

*PDEs*

*on*a

*sphere*,

*using*test cases in the numerical climate modeling ...

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