The Numerical Solution of a Control Problem Governed by a
Phase Field Model
M. Heinkenschloss
Department of Computational and Applied Mathematics
Rice University
Optimization Methods and Software, Vol. 7, 1997, pp. 211-263.
Abstract
The application of a multilevel Newton method for
the numerical solution of an optimal
control problem governed by a phase field model is investigated.
The phase field model is used to describe the phase change
(e.g. liquid to solid) of materials and is
given as a system of two semilinear parabolic equations.
For the solution of the optimal control problem the
necessary optimality conditions are reformulated as a compact fixed
point problem.
In this formulation, controls, states, and adjoints are viewed
as independent variables, the state equation is treated as a constraint.
A multilevel Newton method due to Atkinson and Brakhage
is applied for its solution.
The multilevel Newton method operates on a sequence of grids and
uses inexpensive coarse grid information to obtain good approximations
for the Newton step. If the coarse grid is sufficiently fine,
this yields a fast linearly convergent method which considerably reduces
the amount of work per iteration and also requires fewer data to handle.
Other advantages of this approach are that the reformulation as a
compact fixed point problem yields a decoupling of the phase field
model and generates functions that only involve the solution of
linear parabolic equations.
Numerical examples indicate that the work per iteration of the resulting
algorithm is linear in the number of variables, if fast methods for the
solution of the linear PDEs are used.
Keywords
Optimal control problems, phase field model,
multilevel Newton methods, inexact Newton methods,
sequential quadratic programming methods.
AMS subject classifications
49M05, 65H10