Combining deflation and nested iteration for computing multiple solutions of nonlinear variational problems

Journal article

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.

Authors: Adler, J; Emerson, D; Farrell, P; MacLachlan, S

• Publication status: In press
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Scientific Computing
• Volume: 39
• Issue: 1
• Pages: B29–B52
• Publication date: 2017-01-24
• Acceptance date: 2016-10-14
• DOI: 10.1137/16M1058728
• EISSN: 1095-7197
• ISSN: 1064-8275
• Keywords: energy optimization; distinct solutions; deflation methods; liquid crystal simulation; nested iteration