Abstract
A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.
Highlights
Input-output systems map an input function to an output function via a dynamical system
For this cross-Gramian-based dominant subspace method, an a-priori error indicator is developed, and the numerical issues arising in the wake of large-scale systems are addressed, by utilizing the hierarchical approximate proper orthogonal decomposition (HAPOD) [18]
Two incremental HAPODs are performed for the Gramian partitions respectively, and subsequently, a distributed HAPOD of the resulting singular vectors from both sub-trees yield the dominant subspace projection
Summary
Input-output systems map an input function to an output function via a dynamical system. Himpe applications from natural sciences and engineering, the dimensionality of the dynamical system may render the numerical computation of outputs from inputs excessively expensive or at least demanding Model reduction addresses this computational challenge by algorithms that provide surrogate systems, which approximate the input-output mapping of the original system with a low(er) dimensional intermediate dynamical system. The approach proposed in this work combines the method from [32] with the cross Gramian (matrix) [13], which encodes controllability and observability information of an underlying input-output system For this cross-Gramian-based dominant subspace method, an a-priori error indicator is developed, and the numerical issues arising in the wake of large-scale systems are addressed, by utilizing the hierarchical approximate proper orthogonal decomposition (HAPOD) [18].
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