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 multigrid 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.(original abstract)
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