3,378 research outputs found
Order reduction approaches for the algebraic Riccati equation and the LQR problem
We explore order reduction techniques for solving the algebraic Riccati
equation (ARE), and investigating the numerical solution of the
linear-quadratic regulator problem (LQR). A classical approach is to build a
surrogate low dimensional model of the dynamical system, for instance by means
of balanced truncation, and then solve the corresponding ARE. Alternatively,
iterative methods can be used to directly solve the ARE and use its approximate
solution to estimate quantities associated with the LQR. We propose a class of
Petrov-Galerkin strategies that simultaneously reduce the dynamical system
while approximately solving the ARE by projection. This methodology
significantly generalizes a recently developed Galerkin method by using a pair
of projection spaces, as it is often done in model order reduction of dynamical
systems. Numerical experiments illustrate the advantages of the new class of
methods over classical approaches when dealing with large matrices
Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
This paper introduces tensorial calculus techniques in the framework of
Proper Orthogonal Decomposition (POD) to reduce the computational complexity of
the reduced nonlinear terms. The resulting method, named tensorial POD, can be
applied to polynomial nonlinearities of any degree . Such nonlinear terms
have an on-line complexity of , where is the
dimension of POD basis, and therefore is independent of full space dimension.
However it is efficient only for quadratic nonlinear terms since for higher
nonlinearities standard POD proves to be less time consuming once the POD basis
dimension is increased. Numerical experiments are carried out with a two
dimensional shallow water equation (SWE) test problem to compare the
performance of tensorial POD, standard POD, and POD/Discrete Empirical
Interpolation Method (DEIM). Numerical results show that tensorial POD
decreases by times the computational cost of the on-line stage of
standard POD for configurations using more than model variables. The
tensorial POD SWE model was only slower than the POD/DEIM SWE model
but the implementation effort is considerably increased. Tensorial calculus was
again employed to construct a new algorithm allowing POD/DEIM shallow water
equation model to compute its off-line stage faster than the standard and
tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table
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
On the ADI method for the Sylvester Equation and the optimal- points
The ADI iteration is closely related to the rational Krylov projection
methods for constructing low rank approximations to the solution of Sylvester
equation. In this paper we show that the ADI and rational Krylov approximations
are in fact equivalent when a special choice of shifts are employed in both
methods. We will call these shifts pseudo H2-optimal shifts. These shifts are
also optimal in the sense that for the Lyapunov equation, they yield a residual
which is orthogonal to the rational Krylov projection subspace. Via several
examples, we show that the pseudo H2-optimal shifts consistently yield nearly
optimal low rank approximations to the solutions of the Lyapunov equations
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