Volume 14, Issue 1
Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model

Edison E. Chukwuemeka & Shawn W. Walker

Adv. Appl. Math. Mech., 14 (2022), pp. 1-32.

Published online: 2021-11

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  • Abstract

This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals. Three (alternating direction) minimization schemes are applied to a structure preserving finite element discretization of the uniaxial model: a standard gradient descent method, the "heavy-ball" method, and Nesterov's method. The performance of the schemes is measured in terms of the number of iterations required to obtain the equilibrium state, as well as the total computational time (wall time).

The numerical experiments clearly show that the accelerated gradient descent schemes reduce the number of iterations and computational time significantly, despite the hard uniaxial constraint that is not "smooth'' when defects are present. Moreover, our results show that accelerated schemes are not hindered when combined with an alternating direction minimization algorithm and are easy to implement.

  • Keywords

Liquid crystals, Landau-de Gennes, uniaxial, heavy-ball method, Nesterov's method.

  • AMS Subject Headings

65N30, 49M25, 35J70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-14-1, author = {Chukwuemeka , Edison E. and Walker , Shawn W.}, title = {Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {14}, number = {1}, pages = {1--32}, abstract = {

This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals. Three (alternating direction) minimization schemes are applied to a structure preserving finite element discretization of the uniaxial model: a standard gradient descent method, the "heavy-ball" method, and Nesterov's method. The performance of the schemes is measured in terms of the number of iterations required to obtain the equilibrium state, as well as the total computational time (wall time).

The numerical experiments clearly show that the accelerated gradient descent schemes reduce the number of iterations and computational time significantly, despite the hard uniaxial constraint that is not "smooth'' when defects are present. Moreover, our results show that accelerated schemes are not hindered when combined with an alternating direction minimization algorithm and are easy to implement.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0075}, url = {http://global-sci.org/intro/article_detail/aamm/19974.html} }
TY - JOUR T1 - Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model AU - Chukwuemeka , Edison E. AU - Walker , Shawn W. JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 1 EP - 32 PY - 2021 DA - 2021/11 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0075 UR - https://global-sci.org/intro/article_detail/aamm/19974.html KW - Liquid crystals, Landau-de Gennes, uniaxial, heavy-ball method, Nesterov's method. AB -

This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals. Three (alternating direction) minimization schemes are applied to a structure preserving finite element discretization of the uniaxial model: a standard gradient descent method, the "heavy-ball" method, and Nesterov's method. The performance of the schemes is measured in terms of the number of iterations required to obtain the equilibrium state, as well as the total computational time (wall time).

The numerical experiments clearly show that the accelerated gradient descent schemes reduce the number of iterations and computational time significantly, despite the hard uniaxial constraint that is not "smooth'' when defects are present. Moreover, our results show that accelerated schemes are not hindered when combined with an alternating direction minimization algorithm and are easy to implement.

Edison E. Chukwuemeka & Shawn W. Walker. (1970). Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model. Advances in Applied Mathematics and Mechanics. 14 (1). 1-32. doi:10.4208/aamm.OA-2021-0075
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