91 research outputs found
Distributed optimal control of a nonstandard system of phase field equations
We investigate a distributed optimal control problem for a phase field model
of Cahn-Hilliard type. The model describes two-species phase segregation on an
atomic lattice under the presence of diffusion; it has been recently introduced
by the same authors in arXiv:1103.4585v1 [math.AP] and consists of a system of
two highly nonlinearly coupled PDEs. For this reason, standard arguments of
optimal control theory do not apply directly, although the control constraints
and the cost functional are of standard type. We show that the problem admits a
solution, and we derive the first-order necessary conditions of optimality.Comment: Key words: distributed optimal control, nonlinear phase field
systems, first-order necessary optimality condition
Local Error Analysis of Discontinuous Galerkin Methods for Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems
This paper analyzes the local properties of the symmetric interior penalty upwind
discontinuous Galerkin (SIPG) method for the numerical solution of optimal control problems governed
by linear reaction-advection-diffusion equations with distributed controls. The theoretical and
numerical results presented in this paper show that for advection-dominated problems the convergence
properties of the SIPG discretization can be superior to the convergence properties of stabilized
finite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. For
example, we show that for a small diffusion parameter the SIPG method is optimal in the interior
of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the
existence of boundary layers can pollute the numerical solution of optimal control problems everywhere
even into domains where the solution is smooth and, as a consequence, in general reduces
the convergence rates to only first order. In order to prove the nice convergence properties of the
SIPG discretization for optimal control problems, we first improve local error estimates of the SIPG
discretization for single advection-dominated equations by showing that the size of the numerical
boundary layer is controlled not by the mesh size but rather by the size of the diffusion parameter.
As a result, for small diffusion, the boundary layers are too “weak” to pollute the SIPG solution into
domains of smoothness in optimal control problems. This favorable property of the SIPG method is
due to the weak treatment of boundary conditions, which is natural for discontinuous Galerkin methods,
while for SUPG methods strong imposition of boundary conditions is more conventional. The
importance of the weak treatment of boundary conditions for the solution of advection dominated
optimal control problems with distributed controls is also supported by our numerical results
An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations
We discuss model reduction for a particular class of quadratic-bilinear (QB)
descriptor systems. The main goal of this article is to extend the recently
studied interpolation-based optimal model reduction framework for QBODEs
[Benner et al. '16] to a class of descriptor systems in an efficient and
reliable way. Recently, it has been shown in the case of linear or bilinear
systems that a direct extension of interpolation-based model reduction
techniques to descriptor systems, without any modifications, may lead to poor
reduced-order systems. Therefore, for the analysis, we aim at transforming the
considered QB descriptor system into an equivalent QBODE system by means of
projectors for which standard model reduction techniques for QBODEs can be
employed, including aforementioned interpolation scheme. Subsequently, we
discuss related computational issues, thus resulting in a modified algorithm
that allows us to construct \emph{near}--optimal reduced-order systems without
explicitly computing the projectors used in the analysis. The efficiency of the
proposed algorithm is illustrated by means of a numerical example, obtained via
semi-discretization of the Navier-Stokes equations
Model Order Reduction for Rotating Electrical Machines
The simulation of electric rotating machines is both computationally
expensive and memory intensive. To overcome these costs, model order reduction
techniques can be applied. The focus of this contribution is especially on
machines that contain non-symmetric components. These are usually introduced
during the mass production process and are modeled by small perturbations in
the geometry (e.g., eccentricity) or the material parameters. While model order
reduction for symmetric machines is clear and does not need special treatment,
the non-symmetric setting adds additional challenges. An adaptive strategy
based on proper orthogonal decomposition is developed to overcome these
difficulties. Equipped with an a posteriori error estimator the obtained
solution is certified. Numerical examples are presented to demonstrate the
effectiveness of the proposed method
Constrained dogleg methods for nonlinear systems with simple bounds
We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problem
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Optimization and geophysical inverse problems
A fundamental part of geophysics is to make inferences about the interior of the earth on the basis of data collected at or near the surface of the earth. In almost all cases these measured data are only indirectly related to the properties of the earth that are of interest, so an inverse problem must be solved in order to obtain estimates of the physical properties within the earth. In February of 1999 the U.S. Department of Energy sponsored a workshop that was intended to examine the methods currently being used to solve geophysical inverse problems and to consider what new approaches should be explored in the future. The interdisciplinary area between inverse problems in geophysics and optimization methods in mathematics was specifically targeted as one where an interchange of ideas was likely to be fruitful. Thus about half of the participants were actively involved in solving geophysical inverse problems and about half were actively involved in research on general optimization methods. This report presents some of the topics that were explored at the workshop and the conclusions that were reached. In general, the objective of a geophysical inverse problem is to find an earth model, described by a set of physical parameters, that is consistent with the observational data. It is usually assumed that the forward problem, that of calculating simulated data for an earth model, is well enough understood so that reasonably accurate synthetic data can be generated for an arbitrary model. The inverse problem is then posed as an optimization problem, where the function to be optimized is variously called the objective function, misfit function, or fitness function. The objective function is typically some measure of the difference between observational data and synthetic data calculated for a trial model. However, because of incomplete and inaccurate data, the objective function often incorporates some additional form of regularization, such as a measure of smoothness or distance from a prior model. Various other constraints may also be imposed upon the process. Inverse problems are not restricted to geophysics, but can be found in a wide variety of disciplines where inferences must be made on the basis of indirect measurements. For instance, most imaging problems, whether in the field of medicine or non-destructive evaluation, require the solution of an inverse problem. In this report, however, the examples used for illustration are taken exclusively from the field of geophysics. The generalization of these examples to other disciplines should be straightforward, as all are based on standard second-order partial differential equations of physics. In fact, sometimes the non-geophysical inverse problems are significantly easier to treat (as in medical imaging) because the limitations on data collection, and in particular on multiple views, are not so severe as they generally are in geophysics. This report begins with an introduction to geophysical inverse problems by briefly describing four canonical problems that are typical of those commonly encountered in geophysics. Next the connection with optimization methods is made by presenting a general formulation of geophysical inverse problems. This leads into the main subject of this report, a discussion of methods for solving such problems with an emphasis upon newer approaches that have not yet become prominent in geophysics. A separate section is devoted to a subject that is not encountered in all optimization problems but is particularly important in geophysics, the need for a careful appraisal of the results in terms of their resolution and uncertainty. The impact on geophysical inverse problems of continuously improving computational resources is then discussed. The main results are then brought together in a final summary and conclusions section
A Spatial Domain Decomposition Method for Parabolic Optimal Control Problems
We present a non-overlapping spatial domain decomposition method for the solution of linear-quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space-time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space-time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space-time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann-Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou
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