Title: Reduced-Order Modeling of Complex Systems

Abstract:

The computational approximation of solutions of complex systems such as the Navier-Stokes 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 large-scale finite element or finite-volume or spectral codes. For this reason, there has been much interest in the development of low-dimensional models that can accurately be used to simulate and control complex systems. We review some of the existing reduced-order modeling approaches, including reduced-basis 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 reduced-order modeling in general. We also discuss the use of design of experiment strategies for helping to construct effective reduced bases.

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 co-state, 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 quasi-orthogonality 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.

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 semi-smooth Newton methods in functions spaces. Here, in particular, a new path-following algorithm for proper increase of a penalty parameter will be introduced.

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 multi-dimensional networks. Transport processes in networks are of importance in the control-management 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.

Title: Certified Reduced-Basis Methods for Reliable Rapid Solution of Parametrized Partial Differential Equations: Application to Real-Time 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 input-parametrized (mu-parametrized) partial differential equation. System behavior is thus described by an input-output 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 real-time. 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 real-time 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 (linear-functional) 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 reduced-basis 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 many-query and real-time 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(100-1000) are often observed: two-dimensional Navier-Stokes natural convection at moderate Grash of number - from prescribed input to output and error bound in 0.01 seconds on a Pentium 1.6GHz; three-dimensional 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. Real-time applications may thus be considered: we can quickly arrive at conclusions which will be feasible, optimal, and robust. Of particular interest are "assess-act" scenarios: real-time parameter "assess" followed by fail-safe optimal "act". We draw our examples from heat transfer, elasticity, incompressible fluid flow (Navier-Stokes), 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.

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.

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.