A low complexity Lie group method on the Stiefel manifold
نویسنده
چکیده
A low complexity Lie group method for numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is presented. Based on the quotient space representation of the Stiefel manifold we provide a representation of the tangent space suitable for Lie group methods. According to this representation a special type of generalized polar coordinates (GPC) is defined and used as a coordinate map. The GPC maps, recently proposed by Munthe-Kaas and Zanna, prove to adapt well to the Stiefel manifold. For the n×k matrix representation of the Stiefel manifold the arithmetic complexity of the method presented is of order nk, and for n ≫ k this leads to huge savings in computation time compared to ordinary Lie group methods. Numerical experiments compare the method to a standard Lie group method using the matrix exponential, and conclude that on the examples presented, the methods perform equally on both accuracy and maintaining orthogonality.
منابع مشابه
On the Implementation of Lie Group Methods on the Stiefel Manifold on the Implementation of Lie Group Methods on the Stiefel Manifold *
There are several applications in which one needs to integrate a system of ODEs whose solution is an n × p matrix with orthonormal columns. In recent papers algorithms of arithmetic complexity order n×p have been proposed. The class of Lie group integrators may seem like a worth while alternative for this class of problems, but it has not been clear how to implement such methods with O(np) comp...
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