2,056 research outputs found
Model Reduction of Descriptor Systems by Interpolatory Projection Methods
In this paper, we investigate interpolatory projection framework for model
reduction of descriptor systems. With a simple numerical example, we first
illustrate that employing subspace conditions from the standard state space
settings to descriptor systems generically leads to unbounded H2 or H-infinity
errors due to the mismatch of the polynomial parts of the full and
reduced-order transfer functions. We then develop modified interpolatory
subspace conditions based on the deflating subspaces that guarantee a bounded
error. For the special cases of index-1 and index-2 descriptor systems, we also
show how to avoid computing these deflating subspaces explicitly while still
enforcing interpolation. The question of how to choose interpolation points
optimally naturally arises as in the standard state space setting. We answer
this question in the framework of the H2-norm by extending the Iterative
Rational Krylov Algorithm (IRKA) to descriptor systems. Several numerical
examples are used to illustrate the theoretical discussion.Comment: 22 page
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
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
Buildings-to-Grid Integration Framework
This paper puts forth a mathematical framework for Buildings-to-Grid (BtG)
integration in smart cities. The framework explicitly couples power grid and
building's control actions and operational decisions, and can be utilized by
buildings and power grids operators to simultaneously optimize their
performance. Simplified dynamics of building clusters and building-integrated
power networks with algebraic equations are presented---both operating at
different time-scales. A model predictive control (MPC)-based algorithm that
formulates the BtG integration and accounts for the time-scale discrepancy is
developed. The formulation captures dynamic and algebraic power flow
constraints of power networks and is shown to be numerically advantageous. The
paper analytically establishes that the BtG integration yields a reduced total
system cost in comparison with decoupled designs where grid and building
operators determine their controls separately. The developed framework is
tested on standard power networks that include thousands of buildings modeled
using industrial data. Case studies demonstrate building energy savings and
significant frequency regulation, while these findings carry over in network
simulations with nonlinear power flows and mismatch in building model
parameters. Finally, simulations indicate that the performance does not
significantly worsen when there is uncertainty in the forecasted weather and
base load conditions.Comment: In Press, IEEE Transactions on Smart Gri
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
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