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- optimal control (9)
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The paper presents a new affine invariant theory on asymptotic mesh
independence of Newton’s method for discretized nonlinear operator equations.
Compared to earlier attempts, the new approach is both much simpler
and more intuitive from the algorithmic point of view. The theory
is exemplified at collocation methods for ODE boundary value problems
and at finite element methods for elliptic PDE problems.

In this paper a nonlocal phase-field model for non-isothermal phase transitions
with a non-conserved order parameter is studied. The paper complements
recent investigations by S. Zheng and the second author and treats
the case when the part of the free energy density forcing the order parameter
to attain values within the physically meaningful range [0; 1] is not given
by a logarithmic expression but by the indicator function of [0; 1] . The resulting
field equations form a system of integro-partial differential inclusions
that are highly nonlinearly coupled. For this system, results concerning global
existence, uniqueness and large-time asymptotic behaviour are derived. The
main results are proved by first transforming the system of inclusions into an
equivalent system of equations in which hysteresis operators occur, and then
employing techniques similar to those recently developed by the authors for
phase-field systems involving hysteresis operators.

We study the optimization of three dimensional curved rods and of shells
under minimal regularity assumptions for the geometry. The results that we
establish concern the existence of optimal shapes and the sensitivity analysis.
We also compute several numerical examples for the curved rods. The models
that we use have been investigated in our previous work [11], [16] and a
complete study of the Kirchhoff-Love arches and their optimization has been
performed in [10].

We prove new properties for the linear isotropic elasticity system and for
thickness minimization problems. We also present very recent results concerning
shape optimization problems for three-dimensional curved rods and
for shells. The questions discussed in this paper are related to the control
variational method and to control into coefficients problems.

We consider stochastic programs with risk measures in the objective and study
stability properties as well as decomposition structures. Thereby we place emphasis on dynamic
models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we
define the class of polyhedral risk measures such that stochastic programs with risk measures taken
from this class have favorable properties. Polyhedral risk measures are defined as optimal values of
certain linear stochastic programs where the arguments of the risk measure appear on the right-hand
side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and
used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we
propose multiperiod extensions of the Conditional-Value-at-Risk.

We discuss a parallel library of efficient algorithms for model reduction of largescale
systems with state-space dimension up to O(104). We survey the numerical
algorithms underlying the implementation of the chosen model reduction methods.
The approach considered here is based on state-space truncation of the system
matrices and includes absolute and relative error methods for both stable and unstable
systems. In contrast to serial implementations of these methods, we employ
Newton-type iterative algorithms for the solution of the major computational tasks.
Experimental results report the numerical accuracy and the parallel performance of
our approach on a cluster of Intel Pentium II processors.

An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise constant functions. The state and
the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous
optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these
approximations have convergence order h2.

Dynamics of a surface-tension-gradient-driven liquid film rising from a reservoir onto a substrate
(2006)

On a tilted heated substrate, surface tension gradients can draw liquid up
out of a reservoir. The resulting film thickness profile is controlled by the tilt
of the substrate, the imposed temperature gradient, and the thickness of a
postulated thin precursor layer. The evolution of this film in time is studied
using a lubrication model. A number of distinct behaviours are possible as
the substrate tilt angle and other parameters are varied. Recent results for
the multiple stationary profiles possible near the meniscus are used, and the
interaction of these profiles with the advancing front is examined. It is shown
how to systematically determine the evolution of the entire film profile from
the meniscus to the apparent contact line. This allows a categorisation of the
range of behaviours for a transversely uniform profile, in a twodimensional
parameter space. In addition to capillary fronts, and double shock structures,
a new combination of a Type I meniscus with a rarefaction fan, and either an
undercompressive or a classical wave at the advancing front, that arises for
certain ranges of large substrate tilt and precursor thickness is described.