Sperical splines

Researchers:

• Jorge Stolfi
• Anamaria Gomide

Project description

A spherical spline is a piecewise-polynomial function f defined on the sphere S^d, in terms of a fixed triangulation T of the latter. Such splines have a number of applications, e.g. in geophisics and global weather modeling.

In this project we study the space P_r^g of all spherical splines with a given triangulation T and a given maximum degree g within each triangle, which are also continuous and differentiable up to a given order r. We have determined the dimension of this space, and implemented procedured that construct a locally-supported basis for it.

We are currently improving the basis construction algorithms, and testing the suitability of spline spaces for least-squares approximation and integration of partial differential equations on the sphere, especially for non-uniform triangulations. For this purpose, we have developed the concept of approximation error map [??], a function that shows how the approximation error is distributed over the sphere -- not for a specific target function, but for a whole linear space of them at once.

More details

• Approximation error maps A. Gomide and J. Stolfi. Proceedings of A4A4 - IV International Symposium on Algorithms for Approximation, 446--453. July 2001. (Published in 2002 by the Univ. of Huddersfield, UK). [No PS] [bib]
   
• Bases for non-homogeneous polynomial C_k splines on the sphere. Anamaria Gomide and Jorge Stolfi. Lecture Notes in Computer Science 1380: Proceedings of the 3rd Latin American Theoretical Informatics Conference (LATIN'98), 133--140. Campinas, SP (Brazil); April 1998. [PS,8p,58kB] [bib]
   
• Approximation error maps. A. Gomide and J. Stolfi. Technical report IC-01-01, Institute of Computing, Univ. of Campinas; February 2001. [PS,23p,2.8MB] [bib]
   
• Non-Homogeneous Spline Bases for Approximation on the Sphere, A. Gomide and J. Stolfi. Technical report IC-00-19, Institute of Computing, Univ. of Campinas; December 2000. [No PS] [bib]
   
• Non-homogeneous polynomial C_k splines on the sphere Sⁿ. A. Gomide and J. Stolfi. Technical report IC-00-10, Institute of Computing, Univ. of Campinas; July 2000. [PS,13p,360kB] [bib]
   
• Splines Polinomiais Não Homogêneos na Esfera. Anamaria Gomide. Doctoral thesis, School of Electrical Engineering and Computer Science, Univ. of Campinas; May 1999. [PS,101p,8.2MB] [bib]
   

Additional publications (superseded by the above):

• Ordem de aproximação de splines polinomiais esféricos não homogêneos. A. Gomide and J. Stolfi. Anais do XXIII Congresso Nacional de Matemática Aplicada e Computacional, vol. 1, 246--246 (in Portuguese). September 2000. [No PS] [bib]
   
• Non-homogeneous spline bases for approximation on the sphere. A. Gomide and J. Stolfi Abstracts of the 4th International Conference on Curves and Surfaces, Saint-Malo, France, page 27. July 1999. [No PS] [bib]
   
• Bases para splines polinomiais não homogêneos C_k na esfera. A. Gomide and J. Stolfi. (In Portuguese.) Technical report IC-97-10, Institute of Computing, Univ. of Campinas; August 1997. [PS,9p,39kB] [bib]
   
• Non-Homogeneous Spline Bases for Approximation on the Sphere. A. Gomide and J. Stolfi. (Full version; see the published [abstract].) July 2000. [PS,??p,??KB] [bib]