61 research outputs found
Pseudo-Optimal Reduction of Structured DAEs by Rational Interpolation
In this contribution, we extend the concept of inner product
and pseudo-optimality to dynamical systems modeled by
differential-algebraic equations (DAEs). To this end, we derive projected
Sylvester equations that characterize the inner product in
terms of the matrices of the DAE realization. Using this result, we extend the
pseudo-optimal rational Krylov algorithm for ordinary
differential equations to the DAE case. This algorithm computes the globally
optimal reduced-order model for a given subspace of defined by
poles and input residual directions. Necessary and sufficient conditions for
pseudo-optimality are derived using the new formulation of the
inner product in terms of tangential interpolation conditions.
Based on these conditions, the cumulative reduction procedure combined with the
adaptive rational Krylov algorithm, known as CUREd SPARK, is extended to DAEs.
Important properties of this procedure are that it guarantees stability
preservation and adaptively selects interpolation frequencies and reduced
order. Numerical examples are used to illustrate the theoretical discussion.
Even though the results apply in theory to general DAEs, special structures
will be exploited for numerically efficient implementations
From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation
We propose a new method for the approximate solution of the Lyapunov equation
with rank- right-hand side, which is based on extended rational Krylov
subspace approximation with adaptively computed shifts. The shift selection is
obtained from the connection between the Lyapunov equation, solution of systems
of linear ODEs and alternating least squares method for low-rank approximation.
The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure
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