90 research outputs found
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
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
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