Nonsmooth Schur Newton Methods and Applications
Tuesday, November 30, 2010 - 9:45am - 10:30am
Keller 3-180
Ralf Kornhuber (Freie Universität Berlin)
The numerical simulation of the coarsening of binary alloys based on the
Cahn-Larch`e equations requires fast, reliable and robust solvers for Cahn-Hilliard equations
with logarithmic potential. After semi-implicit time discretization (cf. Blowey and Elliott 92),
Cahn-Larch`e equations requires fast, reliable and robust solvers for Cahn-Hilliard equations
with logarithmic potential. After semi-implicit time discretization (cf. Blowey and Elliott 92),
the resulting spatial problem can be reformulated as a non-smooth pde-constrained
optimal control problem with cost functional induced by the Laplacian. The associated
Karush-Kuhn-Tucker conditions take the form of a nonsmooth saddle point problem
degenerating to a variational inclusion in the deep quench limit.
Our considerations are based on recent work of Gr¨aser & Kornhuber 09 and the
upcoming dissertation of Gr¨aser 10. The starting point is the elimination of the
primal variable leading to a nonlinear Schur complement which turns out to be the
Fr´ech`et derivative of a convex functional. Now so-called nonsmooth Schur-Newton
methods can be derived as gradient-related descent methods applied to this functional.
In the discrete case we can show global convergence for an exact and
an inexact version independent of any regularization parameters. Local quadratic
convergence or finite termination can be shown for piecewise smooth nonlinearities
or in the deep quench limit respectively. The algorithm can be reinterpreted as a
preconditioned Uzawa method and generalizes the well-known primal-dual active
set strategy by Kunisch, Ito, and Hinterm¨uller 03. A (discrete) Allen-Cahn-type
problem and a linear saddle point problem have to be solved (approximately) in
each iteration step. In numerical computations we observe mesh-independent local
convergence for initial iterates provided by nested iteration. In the deep quench
limit, the numerical complexity of the (approximate) solution of the arising linear
saddle point problem dominates the detection of the actual active set.
Our considerations are based on recent work of Gr¨aser & Kornhuber 09 and the
upcoming dissertation of Gr¨aser 10. The starting point is the elimination of the
primal variable leading to a nonlinear Schur complement which turns out to be the
Fr´ech`et derivative of a convex functional. Now so-called nonsmooth Schur-Newton
methods can be derived as gradient-related descent methods applied to this functional.
In the discrete case we can show global convergence for an exact and
an inexact version independent of any regularization parameters. Local quadratic
convergence or finite termination can be shown for piecewise smooth nonlinearities
or in the deep quench limit respectively. The algorithm can be reinterpreted as a
preconditioned Uzawa method and generalizes the well-known primal-dual active
set strategy by Kunisch, Ito, and Hinterm¨uller 03. A (discrete) Allen-Cahn-type
problem and a linear saddle point problem have to be solved (approximately) in
each iteration step. In numerical computations we observe mesh-independent local
convergence for initial iterates provided by nested iteration. In the deep quench
limit, the numerical complexity of the (approximate) solution of the arising linear
saddle point problem dominates the detection of the actual active set.