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Approximation of parabolic PDEs on spheres using spherical basis functions

Q. T. Le Gia
2005 Advances in Computational Mathematics  
In this paper we investigate the 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] .  ... 
doi:10.1007/s10444-003-3960-9 fatcat:arohdpwkyjeo7jpjdpwjrnbu5u

Chapter 1. Introduction [chapter]

1997 Spherical Means for PDEs  
In Chapter 8 we give some applications: these are probabilistic numerical algorithms for solving 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.  ... 
doi:10.1515/9783110926026-002 fatcat:4j5gghw7m5c3xbajndcvqqmjda

Isometric embeddings of 2-spheres by embedding flow for applications in numerical relativity

Michael Jasiulek, Mikołaj Korzyński
2012 Classical and quantum gravity  
We employ spectral methods to handle 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  ... 
doi:10.1088/0264-9381/29/15/155010 fatcat:rqccblrqxvfbhanlcyydorq4oq

Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences [chapter]

Natasha Flyer, Grady B. Wright, Bengt Fornberg
2013 Handbook of Geomathematics  
Lecture series: Kernel 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.  ... 
doi:10.1007/978-3-642-27793-1_61-1 fatcat:3ahjcvubkbdxhkugstao3xylh4

Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences [chapter]

Natasha Flyer, Grady B. Wright, Bengt Fornberg
2015 Handbook of Geomathematics  
Lecture series: Kernel 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.  ... 
doi:10.1007/978-3-642-54551-1_61 fatcat:kxa622nqozddlkvcai2e4q36yi

An adaptive meshfree diffusion wavelet method for partial differential equations on the sphere

Kavita Goyal, Mani Mehra
2014 Journal of Computational Physics  
An adaptive meshfree diffusion wavelet method for solving partial differential equations (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.  ... 
doi:10.1016/j.jcp.2014.04.044 fatcat:o6madigoybd2no4sb7mxcgmivu

Asymptotic Methods [chapter]

2011 Partial Differential Equations of Applied Mathematics  
for a Triangle 871 Finite Element 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  ... 
doi:10.1002/9781118033302.ch10 fatcat:dw3utrc34nhmrmvr4kjs6rpafi

A Stable Algorithm for Flat Radial Basis Functions on a Sphere

Bengt Fornberg, Cécile Piret
2008 SIAM Journal on Scientific Computing  
Since interpolation in the flat RBF limit 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.  ... 
doi:10.1137/060671991 fatcat:fbrsjc4pdbc2lpkfb7q3bnmsjq

Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

Quoc Thong Le Gia, William McLean
2013 Advances in Computational Mathematics  
The spatial 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.  ... 
doi:10.1007/s10444-013-9311-6 fatcat:pltngcw5ljdazcp4w4elkeqw3e

The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces

Cécile Piret
2012 Journal of Computational Physics  
Much work has been done 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.  ... 
doi:10.1016/j.jcp.2012.03.007 fatcat:6mzytwzx5re5xp3ogopf3k6cki

Scale Space Analysis and Active Contours for Omnidirectional Images

I. Bogdanova, X. Bresson, J.-P. Thiran, P. Vandergheynst
2007 IEEE Transactions on Image Processing  
We derive new energy 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 .  ... 
doi:10.1109/tip.2007.899008 pmid:17605386 fatcat:xqflyykt2jcppjnl7tbm3nuoie

A finite element method of the self-consistent field theory on general curved surfaces

Huayi Wei, Ming Xu, Wei Si, Kai Jiang
2019 Journal of Computational Physics  
Numerical results illustrate the efficiency 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.  ... 
doi:10.1016/j.jcp.2019.02.047 fatcat:5wzbsdlvnrhl5fbtb5vijb2234

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.  ... 

Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

Elena Franchini, Serena Morigi, Fiorella Sgallari
2010 Numerical Mathematics: Theory, Methods and Applications  
The distance 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".  ... 
doi:10.4208/nmtma.2010.m9009 fatcat:7ezwfg3s5rbxfmf6zgmdvjnx2e

Transport schemes on a sphere using radial basis functions

Natasha Flyer, Grady B. Wright
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  ... 
doi:10.1016/j.jcp.2007.05.009 fatcat:plmzslcrl5g3fevkeaxyasuxhe
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