41 research outputs found
Multiobjective Optimization of Non-Smooth PDE-Constrained Problems
Multiobjective optimization plays an increasingly important role in modern
applications, where several criteria are often of equal importance. The task in
multiobjective optimization and multiobjective optimal control is therefore to
compute the set of optimal compromises (the Pareto set) between the conflicting
objectives. The advances in algorithms and the increasing interest in
Pareto-optimal solutions have led to a wide range of new applications related
to optimal and feedback control - potentially with non-smoothness both on the
level of the objectives or in the system dynamics. This results in new
challenges such as dealing with expensive models (e.g., governed by partial
differential equations (PDEs)) and developing dedicated algorithms handling the
non-smoothness. Since in contrast to single-objective optimization, the Pareto
set generally consists of an infinite number of solutions, the computational
effort can quickly become challenging, which is particularly problematic when
the objectives are costly to evaluate or when a solution has to be presented
very quickly. This article gives an overview of recent developments in the
field of multiobjective optimization of non-smooth PDE-constrained problems. In
particular we report on the advances achieved within Project 2 "Multiobjective
Optimization of Non-Smooth PDE-Constrained Problems - Switches, State
Constraints and Model Order Reduction" of the DFG Priority Programm 1962
"Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation
and Hierarchical Optimization"
Model Order Reduction by Proper Orthogonal Decomposition
We provide an introduction to POD-MOR with focus on (nonlinear) parametric
PDEs and (nonlinear) time-dependent PDEs, and PDE constrained optimization with
POD surrogate models as application. We cover the relation of POD and SVD, POD
from the infinite-dimensional perspective, reduction of nonlinearities,
certification with a priori and a posteriori error estimates, spatial and
temporal adaptivity, input dependency of the POD surrogate model, POD basis
update strategies in optimal control with surrogate models, and sketch related
algorithmic frameworks. The perspective of the method is demonstrated with
several numerical examples.Comment: arXiv admin note: substantial text overlap with arXiv:1701.0505
Space-time POD-Galerkin approach for parametric flow control
In this contribution we propose reduced order methods to fast and reliably
solve parametrized optimal control problems governed by time dependent
nonlinear partial differential equations. Our goal is to provide a tool to deal
with the time evolution of several nonlinear optimality systems in many-query
context, where a system must be analysed for various physical and geometrical
features. Optimal control can be used in order to fill the gap between
collected data and mathematical model and it is usually related to very time
consuming activities: inverse problems, statistics, etc. Standard
discretization techniques may lead to unbearable simulations for real
applications. We aim at showing how reduced order modelling can solve this
issue. We rely on a space-time POD-Galerkin reduction in order to solve the
optimal control problem in a low dimensional reduced space in a fast way for
several parametric instances. The proposed algorithm is validated with a
numerical test based on environmental sciences: a reduced optimal control
problem governed by viscous Shallow Waters Equations parametrized not only in
the physics features, but also in the geometrical ones. We will show how the
reduced model can be useful in order to recover desired velocity and height
profiles more rapidly with respect to the standard simulation, not losing
accuracy
Optimality conditions in terms of Bouligand generalized differentials for a nonsmooth semilinear elliptic optimal control problem with distributed and boundary control pointwise constraints
We prove a novel optimality condition in terms of Bouligand generalized
differentials for a local minimizer of optimal control problems governed by a
nonsmooth semilinear elliptic partial differential equation with both
distributed and boundary unilateral pointwise control constraints, in which the
nonlinear coefficient in the state equation is not differentiable at one point.
Therefore, the Bouligand subdifferential of this nonsmooth coefficient in
every point consists of one or two elements that will be used to construct the
two associated Bouligand generalized derivatives of the control-to-state
operator in any admissible control.
We also establish the optimality conditions in the form of multiplier
existence. There, in addition to the existence of the adjoint state and of the
nonnegative multipliers associated with the pointwise constraints as usual,
other nonnegative multipliers exist and correspond to the nondifferentiability
of the control-to-state mapping.
The latter type of optimality conditions shall be applied to the optimal
control problems without distributed and boundary pointwise constraints to
derive the so-called \emph{strong} stationarity conditions, where the sign of
the associated adjoint state does not vary on the level set of the
corresponding optimal state at the value of nondifferentiability.Comment: 33 page
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
Air Force Institute of Technology Research Report 2000
This report summarizes the research activities of the Air Force Institute of Technology’s Graduate School of Engineering and Management. It describes research interests and faculty expertise; lists student theses/dissertations; identifies research sponsors and contributions; and outlines the procedures for contacting the school. Included in the report are: faculty publications, conference presentations, consultations, and funded research projects. Research was conducted in the areas of Aeronautical and Astronautical Engineering, Electrical Engineering and Electro-Optics, Computer Engineering and Computer Science, Systems and Engineering Management, Operational Sciences, and Engineering Physics