Fast solvers for nonsmooth optimization problems in phase separation

The phase separation processes are typically modeled by well known Cahn-Hilliard equation with obstacle potential. Solving these equations correspond to a nonsmooth and nonlinear optimization problem. Recently a globally convergent Newton Schur method was proposed for the non-linear Schur complement corresponding to this 2 × 2 non-linear system. The proposed method is similar to an inexact active set method in the sense that the active sets are first identified by solving a quadratic obstacle problem corresponding to the (1, 1) block of the 2 × 2 system, and later solving a reduced linear system by annihilating the rows and columns corresponding to identified active sets. For solving the quadratic obstacle problem, various optimal multi-grid like methods have been proposed. However solving the reduced system remains a major bottleneck. In this paper, we explore an effective preconditioner for the reduced linear system that allows solving large scale optimization problem corresponding to Cahn-Hilliard and to possibly similar models.