49M29 Methods involving duality
We introduce and analyze nonsmooth Schur-Newton methods for a class of nonsmooth saddle point problems. The method is able to solve problems where the primal energy decomposes into a convex smooth part and a convex separable but nonsmooth part. The method is based on nonsmooth Newton techniques for an equivalent unconstrained dual problem. Using this we show that it is globally convergent even for inexact evaluation of the linear subproblems.
In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and Newton type methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy compression
maintaining a certain error bound turns out to be sufficient.
Motivated by an obstacle problem for a membrane
subject to cohesion forces, constrained minimization problems involving
a non-convex and non-differentiable objective functional
representing the total potential energy are considered. The associated
first order optimality system leads to a hemi-variational inequality,
which can also be interpreted as a special complementarity
problem in function space. Besides an analytical investigation
of first-order optimality, a primal-dual active set solver is introduced.
It is associated to a limit case of a semi-smooth Newton
method for a regularized version of the underlying problem class.
For the numerical algorithms studied in this paper, global as well as
local convergence properties are derived and verified numerically.