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Volume 32, Issue 1
A Natural Gradient Descent Algorithm for the Solution of Lyapunov Equations Based on the Geodesic Distance

Xiaomin Duan, Huafei Sun & Zhenning Zhang

DOI: 10.4208/jcm.1310-m4225

J. Comp. Math., 32 (2014), pp. 93-106.

Published online: 2014-02

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  • Abstract

A new framework based on the curved Riemannian manifold is proposed to calculate the numerical solution of the Lyapunov matrix equation by using a natural gradient descent algorithm and taking the geodesic distance as the objective function. Moreover, a gradient descent algorithm based on the classical Euclidean distance is provided to compare with this natural gradient descent algorithm. Furthermore, the behaviors of two proposed algorithms and the conventional modified conjugate gradient algorithm are compared and demonstrated by two simulation examples. By comparison, it is shown that the convergence speed of the natural gradient descent algorithm is faster than both of the gradient descent algorithm and the conventional modified conjugate gradient algorithm in solving the Lyapunov equation.

  • Keywords

Lyapunov equation, Geodesic distance, Natural gradient descent algorithm.

  • AMS Subject Headings

65F10, 53B21, 90C26, 93C05.

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-32-93, author = {Xiaomin Duan, Huafei Sun and Zhenning Zhang}, title = {A Natural Gradient Descent Algorithm for the Solution of Lyapunov Equations Based on the Geodesic Distance}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {1}, pages = {93--106}, abstract = {

A new framework based on the curved Riemannian manifold is proposed to calculate the numerical solution of the Lyapunov matrix equation by using a natural gradient descent algorithm and taking the geodesic distance as the objective function. Moreover, a gradient descent algorithm based on the classical Euclidean distance is provided to compare with this natural gradient descent algorithm. Furthermore, the behaviors of two proposed algorithms and the conventional modified conjugate gradient algorithm are compared and demonstrated by two simulation examples. By comparison, it is shown that the convergence speed of the natural gradient descent algorithm is faster than both of the gradient descent algorithm and the conventional modified conjugate gradient algorithm in solving the Lyapunov equation.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1310-m4225}, url = {http://global-sci.org/intro/article_detail/jcm/9871.html} }
TY - JOUR T1 - A Natural Gradient Descent Algorithm for the Solution of Lyapunov Equations Based on the Geodesic Distance AU - Xiaomin Duan, Huafei Sun & Zhenning Zhang JO - Journal of Computational Mathematics VL - 1 SP - 93 EP - 106 PY - 2014 DA - 2014/02 SN - 32 DO - http://doi.org/10.4208/jcm.1310-m4225 UR - https://global-sci.org/intro/article_detail/jcm/9871.html KW - Lyapunov equation, Geodesic distance, Natural gradient descent algorithm. AB -

A new framework based on the curved Riemannian manifold is proposed to calculate the numerical solution of the Lyapunov matrix equation by using a natural gradient descent algorithm and taking the geodesic distance as the objective function. Moreover, a gradient descent algorithm based on the classical Euclidean distance is provided to compare with this natural gradient descent algorithm. Furthermore, the behaviors of two proposed algorithms and the conventional modified conjugate gradient algorithm are compared and demonstrated by two simulation examples. By comparison, it is shown that the convergence speed of the natural gradient descent algorithm is faster than both of the gradient descent algorithm and the conventional modified conjugate gradient algorithm in solving the Lyapunov equation.

Xiaomin Duan, Huafei Sun and Zhenning Zhang. (2014). A Natural Gradient Descent Algorithm for the Solution of Lyapunov Equations Based on the Geodesic Distance. Journal of Computational Mathematics. 32 (1). 93-106. doi:10.4208/jcm.1310-m4225
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