Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model

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.