209 research outputs found
Efficient Solution of Large-Scale Algebraic Riccati Equations Associated with Index-2 DAEs via the Inexact Low-Rank Newton-ADI Method
This paper extends the algorithm of Benner, Heinkenschloss, Saak, and
Weichelt: An inexact low-rank Newton-ADI method for large-scale algebraic
Riccati equations, Applied Numerical Mathematics Vol.~108 (2016), pp.~125--142,
doi:10.1016/j.apnum.2016.05.006 to Riccati equations associated with Hessenberg
index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise,
e.g., from semi-discretized, linearized (around steady state) Navier-Stokes
equations. The solution of the associated Riccati equation is important, e.g.,
to compute feedback laws that stabilize the Navier-Stokes equations. Challenges
in the numerical solution of the Riccati equation arise from the large-scale of
the underlying systems and the algebraic constraint in the DAE system. These
challenges are met by a careful extension of the inexact low-rank Newton-ADI
method to the case of DAE systems. A main ingredient in the extension to the
DAE case is the projection onto the manifold described by the algebraic
constraints. In the algorithm, the equations are never explicitly projected,
but the projection is only applied as needed. Numerical experience indicates
that the algorithmic choices for the control of inexactness and line-search can
help avoid subproblems with matrices that are only marginally stable. The
performance of the algorithm is illustrated on a large-scale Riccati equation
associated with the stabilization of Navier-Stokes flow around a cylinder.Comment: 21 pages, 2 figures, 4 table
A fast and accurate domain-decomposition nonlinear manifold reduced order model
This paper integrates nonlinear-manifold reduced order models (NM-ROMs) with
domain decomposition (DD). NM-ROMs approximate the FOM state in a
nonlinear-manifold by training a shallow, sparse autoencoder using FOM snapshot
data. These NM-ROMs can be advantageous over linear-subspace ROMs (LS-ROMs) for
problems with slowly decaying Kolmogorov -width. However, the number of
NM-ROM parameters that need to trained scales with the size of the FOM.
Moreover, for "extreme-scale" problems, the storage of high-dimensional FOM
snapshots alone can make ROM training expensive. To alleviate the training
cost, this paper applies DD to the FOM, computes NM-ROMs on each subdomain, and
couples them to obtain a global NM-ROM. This approach has several advantages:
Subdomain NM-ROMs can be trained in parallel, each involve fewer parameters to
be trained than global NM-ROMs, require smaller subdomain FOM dimensional
training data, and training of subdomain NM-ROMs can tailor them to
subdomain-specific features of the FOM. The shallow, sparse architecture of the
autoencoder used in each subdomain NM-ROM allows application of hyper-reduction
(HR), reducing the complexity caused by nonlinearity and yielding computational
speedup of the NM-ROM. This paper provides the first application of NM-ROM
(with HR) to a DD problem. In particular, it details an algebraic DD
formulation of the FOM, trains a NM-ROM with HR for each subdomain, and
develops a sequential quadratic programming (SQP) solver to evaluate the
coupled global NM-ROM. Theoretical convergence results for the SQP method and a
priori and a posteriori error estimates for the DD NM-ROM with HR are provided.
The proposed DD NM-ROM with HR approach is numerically compared to a DD LS-ROM
with HR on 2D steady-state Burgers' equation, showing an order of magnitude
improvement in accuracy of the proposed DD NM-ROM over the DD LS-ROM
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
Maximal parabolic regularity for divergence operators including mixed boundary conditions
We show that elliptic second order operators of divergence type fulfill
maximal parabolic regularity on distribution spaces, even if the underlying
domain is highly non-smooth, the coefficients of are discontinuous and
is complemented with mixed boundary conditions. Applications to quasilinear
parabolic equations with non-smooth data are presented.Comment: 39 pages, 4 postscript figure
- …