# Fast Solvers for Nonsmooth Optimization Problems in Phase Separation

## Pawan Kumar

Citation: Proceedings of the 2015 Federated Conference on Computer Science and Information Systems, M. Ganzha, L. Maciaszek, M. Paprzycki (eds). ACSIS, Vol. 5, pages 589594 ()

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Abstract. 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 $2 \times 2$ 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 $2 \times 2$ 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 various effective preconditioners for the reduced linear systems that allow solving large scale optimization problem corresponding Cahn-Hilliard and possibly similar models.