M. D. Gunzburger (Florida State University)

Title:
ReducedOrder Modeling of Complex Systems
Abstract:
The computational approximation of solutions of complex systems such as the
NavierStokes equations is often a formidable task. For example, in feedback control
settings where one often needs solutions of the complex systems in real time,
it would be impossible to use largescale finite element or finitevolume or
spectral codes.
For this reason, there has been much interest in the development of
lowdimensional models that can accurately be used to simulate and control
complex systems. We review some of the existing reducedorder modeling approaches, including reducedbasis methods and especially methods based on proper orthogonal decompositions techniques. We also discuss a new approach based on centroidal Voronoi tessellations. We discuss the relative merits and deficiencies of the different approaches and also the inherent limitations of reducedorder
modeling in
general. We also discuss the use of design of experiment strategies for helping
to construct effective reduced bases.

R. H. W. Hoppe (University of Augsburg)

Title:
Adaptive Finite Element Methods for Optimally Controlled Elliptic Problems with Control
Abstract:
We are concerned with the development and analysis of
adaptive finite element methods for optimally controlled elliptic
PDEs with constraints on the control. In the unconstrained case,
such methods have been considered previously (cf., e.g., [Bangerth
and Rannacher 2003] and the references therein), whereas in the
presence of control constraints some results have been obtained in
[Li, Liu, Ma, and Tang 2002] and [Liu and Yan 2003]. The methods
presented in this contribution provide an error reduction and thus
guarantee convergence of the adaptive loop which consists of the
essential steps SOLVE, ESTIMATE, MARK, and REFINE. Here, SOLVE
stands for the efficient solution of the finite element
discretized problems which is taken care of by appropriate active
set strategies (cf., e.g., [Hintermüller, Ito, and Kunisch
2003]). The following step ESTIMATE is devoted to the a posteriori
error estimation of the global discretization errors in the state,
the costate, and the control by easily computable local
quantities. A greedy algorithm is the core of the step MARK to
indicate selected elements for refinement, whereas the final step
REFINE deals with the technical realization of the refinement
process itself. The analysis is carried out for the lowest order
P1 conforming finite elements. Important tools in the convergence
proof are the reliability of the estimator, a strong discrete
local efficiency, and quasiorthogonality properties (see, e.g.,
[Carstensen and Hoppe 2005] and [Morin, Nochetto, and Siebert
2000] for various finite element approximations of elliptic PDEs).
The proof does not require regularity of the solution nor does it
make use of duality arguments.

K. Kunisch (University of Graz)

Title:
Optimal Control with State Constraints: Theory and Numerical Methods
Abstract:
First a brief summary on the existence and structure of Lagrange multipliers for
state constrained optimal control problems is given. Then two rather recent
numerical solution techniques are discussed. One approach involves the use
of level sets describing the interfaces between active and inactive sets of the
constrained state variables. The other one is based on semismooth Newton
methods in functions spaces. Here, in particular, a new pathfollowing algorithm for proper increase of a penalty parameter will be introduced.

G. Leugering (Univ. ErlangenNürnb.)

Title:
Modelling, Control and Optimization of Distributed Parameter Systems on
Networked Domains
Abstract:
We consider modelling, analysis, optimization and control of transport
processes and problems of wave propagation on multidimensional networks.
Transport processes in networks are of importance in the controlmanagement of
water, gas and electrical power systems, as well as in the simulation and the
control of blood and traffic flow, while waves propagating within networked
domains occur in flexible structures, mechanical microstructures,
macromolecules and seismic problems. The mathematical modelling along with its
analysis requires model hierarchies, multiple joint conditions, homogenization,
domain decomposition or substructuring and hybrid systems approaches. The
optimization and control problems are formulated for the entire network problem
followed by a reduction to tractable subproblems via homogenization and domain
decomposition. As for domain decomposition methods, a posteriori estimates will
also be discussed, and numerical examples will be provided.

A. T. Patera (MIT)

Title:
Certified ReducedBasis Methods for Reliable Rapid Solution of
Parametrized Partial Differential Equations: Application to RealTime
Parameter Estimation
Abstract:
Engineering analysis requires the prediction of selected
"outputs" relevant to ultimate component and system
performance. These outputs are functions of "inputs" mu that
serve to identify a particular configuration of the component or
system. In many cases, the output is best articulated as a (say)
linear functional l of a field variable u(mu) that is the
solution to an inputparametrized (muparametrized) partial
differential equation. System behavior is thus described by an
inputoutput relation s(mu) = l(u(mu)) the evaluation of
which requires solution of the underlying partial differential
equation. Characterization, design, optimization, and control
typically require thousands of evaluations mu
> s(mu)  and often in effectively realtime. Most classical numerical approaches consider
effectively "dense" approximation subspaces: the computational
requirement for a particular value of mu is thus
typically measured in minutes, hours, or even days. This has two
distinct but related implications. First, we can not adequately
explore the parameter domain: thus we can not determine which parts
of the parameter domain are indeed "representative"; and hence we
can not rationally decide which expensive "next experiment"
should be accorded highest priority. Second, we can not address
the many "design" and "operations" applications that require
either realtime response to
 or simply very many  queries mu > s(mu): thus we can
not perform optimization of
components and processes, robust parameter estimation of
properties and state, or adaptive design and control of assets and
missions. The goal of our work is to address this deficiency: we
present a technique for the rapid and reliable prediction
of (linearfunctional) outputs of partial differential equations
with approximately affine parameter dependence. The method is
applicable to a wide variety of coercive and noncoercive, linear
and nonlinear, and elliptic and parabolic equations. The essential
components of our approach are threefold: (i) rapidly and
uniformly convergent reducedbasis approximations  Galerkin
projection onto a space W_N spanned by solutions of the
governing partial differential equation at N optimally selected
points in parameter space; (ii) a posteriori error
estimation
 relaxations of the residual equation that provide inexpensive yet sharp and
rigorous bounds for the error in
the outputs of interest; and (iii) offline/onlinecomputational procedures  strategies that
accept increased initial preprocessing (offline) expense in
exchange for greatly reduced subsequent (online) marginal
cost. The method is ideally suited to the manyquery and
realtime contexts (in which marginal cost is clearly the critical
computational metric): the operation count for the online, or
"deployed", stage depends only on N
 typically very small  and the parametric complexity of the problem. We can
thus provide predictions that
are certifiably as good as classical "truth"
approximations but literally several orders of magnitude less
expensive. Online factors of improvement of O(1001000) are often observed: twodimensional
NavierStokes natural convection at moderate Grash of number
 from prescribed input to output and error bound in 0.01 seconds
on a Pentium 1.6GHz; threedimensional exterior Dirichlet acoustic
Helmholtz (albeit in relatively simple geometry)  from input to
output and error bound in 0.04 seconds. The advantages of our
approach can be further leveraged in service of parameter
estimation, optimization, and adaptive design. In particular, not
only can we rigorously accommodate numerical uncertainty, but 
through the extensive exploration of parameter space afforded by
our rapid evaluation technology  we can also accommodate (some)
model uncertainty. Realtime applications may thus be considered:
we can quickly arrive at conclusions which will be feasible,
optimal, and robust. Of particular interest are "assessact"
scenarios: realtime parameter "assess" followed by failsafe
optimal "act". We draw our examples from heat transfer,
elasticity, incompressible fluid flow (NavierStokes), and quantum
mechanics, with application to robust parameter estimation 
nondestructive crack assessment, acoustic inverses cattering for
mine detection, material properties from quantum models; adaptive
design  maximal safe load prediction, vehicle stealth, thermal
management; and optimal control  contaminant remediation, heat
treatment, reactor design.

R. Rannacher (University of Heidelberg)

Title:
On the Discretization of Optimization Problems by Adaptive
Finite Element Methods
Abstract:
In this talk we summarize recent developments in a posteriori error
estimation and mesh adaptation for the discretization of optimization problems.
In the context of the finite element Galerkin method we describe
the Dual Weighted Residual (DWR) method exploiting 'Galerkin orthogonality',
'residual evaluation' and 'dual stability'. The applications discussed range
from simple boundary
control over parameter identification and model callibration to first steps
towards experimental design.

E. W. Sachs (University of Trier)

Title:
Preconditioning Techniques for Optimal Control Problems
Abstract:
In this talk we consider the preconditioning
for linear systems arising in problems of optimal control
that are governed by partial differential equations.
We address first the infinite dimensional problem and
apply some classical results to these cases.
In the sequel we refine the approach by looking at discretized
problems and derive results including the mesh size parameters.
This is applied to some examples of optimal control problems.

