Applications of the Conjugate Gradient method in optimal surface parameterizations

Abstract

This paper concerns the solutions of very large symmetric semipositive definite (singular) linear systems involved in the problem of optimal surface parameterizations using inverse curvature mapping. Two approaches are presented that transform the singular linear systems into two kinds of symmetric positive definite linear systems, so that the famous Conjugate Gradient (CG) method can be used for solving them. Numerical experiments are run on two practical large problems to illustrate that the CG algorithm works very efficiently.

Keywords:

• symmetric seimidefinite
• positive definite
• CG
• Krylov subspace
• surface parameterization

2000 AMS Subject Classification::

• 65F10
• 65D17
• 68U07

Acknowledgements

The author thanks two referees for their valuable comments and suggestions that helped in improving this paper considerably. Thanks also go to the authors of 7 for providing the problem and data. Supported by the National Natural Science Foundation (No. 10771116) and the Doctoral Program of the Ministry of Education (No. 20060003003).