Combining Deflation and Nested Iteration for Computing Multiple Solutions of Nonlinear Variational Problems

Abstract

Many physical systems support multiple equilibrium states that enable their use in modern science and engineering applications. Having the ability to reliably compute such states facilitates more accurate physical analysis and understanding of experimental behavior. This paper adapts and extends a deflation technique for the computation of multiple distinct solutions in the context of nonlinear systems and applies the method to the modeling of equilibrium configurations of nematic and cholesteric liquid crystals. In particular, the deflation approach is interwoven with nested iteration, creating an efficient and effective method that further enables the discovery of distinct solutions. The combined methodology is applied as part of an overall free-energy variational approach within the framework of optimization of a functional with constraints imposed via Lagrange multipliers. A key feature in the combined algorithm is the reuse of effective preconditioners designed for the undeflated systems within the Newton iteration for the deflated systems. Four numerical experiments are performed, demonstrating the efficacy and accuracy of the algorithm in detecting important physical phenomena, including bifurcation and disclination behaviors.

Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

“…The area of each bounding ellipse is held fixed at 3π 2 . Our multilevel finite-element code used to compute stationary points of the free energy ( 1) is thoroughly described elsewhere [21,[26][27][28]. Briefly, the code uses the Cartesian representation of the director n = (n x , n y , n z ) and directly finds equilibrium points of the Frank energy (1) by applying Newton linearisation to the firstorder optimality conditions in variational form, resulting from the constrained minimisation.…”

“…Numerical experiments have found the effectiveness of the deflation operator to be relatively insensitive to the choice of deflation parameters. However, for certain problems, performance improvements may be obtained by varying p and α [20,21]. Typical values, used everywhere in this paper, are α = 1 and p = 2.…”

“…In this paper, we adapt a new technique known as the deflation method [20,21] to a model problem in this class, the configuration of a cholesteric in an elliptical channel. The method is generalisable, robust and computationally efficient for large-scale applications.…”

Bifurcation analysis of stationary solutions of two-dimensional coupled Gross–Pitaevskii equations using deflated continuation

“…Second, branch switching algorithms (as implemented in, e.g., AUTO [39]) could be applied to branches identified via deflation, combining their advantages. Third, deflation can be combined with nested iteration to greatly enhance its robustness [40]. Fourth, the use of more robust nonlinear solvers (improved line searches or trust region techniques) would make the solution of deflated problems more successful on average.…”