Globalization of nonsmooth Newton methods for optimal control problems

We present a new approach for the globalization of the primal-dual active set or equivalently the nonsmooth Newton method applied to an optimal control problem. The basic result is the equivalence of this method to a nonsmooth Newton method applied to the nonlinear Schur complement of the optimality system. Our approach does not require the construction of an additional merit function or additional descent direction. The nonsmooth Newton directions are naturally appropriate descent directions for a smooth dual energy and guarantee global convergence if standard damping methods are applied

Nonsmooth Schur-Newton methods for nonsmooth saddle point problems.

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

A note on Poincaré- and Friedrichs-type inequalities

We introduce a simple criterion to check coercivity of bilinear forms on subspaces of Hilbert-spaces. The presented criterion allows to derive many standard and non-standard variants of Poincar\'e- and Friedrichs-type inequalities with very little effort

Polyhedral Gauß-Seidel converges

We prove global convergence of an inexact extended polyhedral Gauß-Seidel method for the minimization of strictly convex functionals that are continuously differentiable on each polyhedron of a polyhedral decomposition of their domains of definition. While pure Gauß-Seidel methods are known to be very slow for problems governed by partial differential equations, the presented convergence result also covers multilevel methods that extend the Gauß-Seidel step by coarse level corrections. Our result generalizes the proof of [10] for differentiable functionals on the Gibbs simplex. Example applications are given that require the generality of our approach

Discretization error estimates for penalty formulations of a linearized Canham-Helfrich-type energy

This article is concerned with minimization of a fourth-order linearized Canham–Helfrich energy subject to Dirichlet boundary conditions on curves inside the domain. Such problems arise in the modeling of the mechanical interaction of biomembranes with embedded particles. There, the curve conditions result from the imposed particle–membrane coupling. We prove almost-H. 5. 2 regularity of the solution and then consider two possible penalty formulations. For the combination of these penalty formulations with a Bogner–Fox–Schmit finite element discretization, we prove discretization error estimates that are optimal in view of the solution’s reduced regularity. The error estimates are based on a general estimate for linear penalty problems in Hilbert spaces. Finally, we illustrate the theoretical results by numerical computations. An important feature of the presented discretization is that it does not require the particle boundary to be resolved. This is crucial to avoid re-meshing if the presented problem arises as a subproblem in a model where particles are allowed to move or rotate

The dune-subgrid module and some applications

We present an extension module for the Dune system. This module, called dune-subgrid, allows to mark elements of another Dune hierarchical grid. The set of marked elements can then be accessed as a Dune grid in its own right. dune-subgrid is free software and is available for download (External Dune Modules: www.​dune-project.​org/​downloadext.​html). We describe the functionality and use of dune-subgrid, comment on its implementation, and give two example applications. First, we show how dune-subgrid can be used for micro-FE simulations of trabecular bone. Then we present an algorithm that allows to use exact residuals for the adaptive solution of the spatial problems of time-discretized evolution equations

Multigrid methods for obstacle problems

In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set

On differentiability of the membrane-mediated mechanical interaction energy of discrete-continuum membrane-particle models

We consider a discrete-continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham-Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries resulting in a coupled minimization problem for the membrane shape and particle positions. When considering the energetically optimal membrane shape for each particle position we obtain a reduced energy functional that models the implicitly given interaction potential for the membrane-mediated mechanical particle-particle interactions. We show that this interaction potential is differentiable with respect to the particle positions and orientations. Furthermore we derive a fully practical representation of the derivative only in terms of well defined derivatives of the membrane. This opens the door for the application of minimization algorithms for the computation of minimizers of the coupled system and for further investigation of the interaction potential of membrane-mediated mechanical particle--particle interaction. The results are illustrated with numerical examples comparing the explicit derivative formula with difference quotient approximations. We furthermore demonstrate the application of the derived formula to implement a gradient flow for the approximation of optimal particle configurations

On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints

We consider preconditioned Uzawa iterations for a saddle point problem with inequality constraints as arising from an implicit time discretization of the Cahn-Hilliard equation with an obstacle potential. We present a new class of preconditioners based on linear Schur complements associated with successive approximations of the coincidence set. In numerical experiments, we found superlinear convergence and finite termination

Nonsmooth Newton methods for set-valued saddle point problems

We present a new class of iterative schemes for large scale set-valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be regarded either as nonsmooth Newton-type methods for the nonlinear Schur complement or as Uzawa-type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn–Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach