Abstract
The Loewner framework is one of the most successful data-driven model order reduction techniques. If N is the cardinality of a given data set, the so-called Loewner and shifted Loewner matrices {mathbb {L}}in {mathbb {C}}^{Ntimes N} and {mathbb {S}}in {mathbb {C}}^{Ntimes N} can be defined by solely relying on information encoded in the considered data set and they play a crucial role in the computation of the sought rational model approximation.In particular, the singular value decomposition of a linear combination of {mathbb {S}} and {mathbb {L}} provides the tools needed to construct accurate models which fulfill important approximation properties with respect to the original data set. However, for highly-sampled data sets, the dense nature of {mathbb {L}} and {mathbb {S}} leads to numerical difficulties, namely the failure to allocate these matrices in certain memory-limited environments or excessive computational costs. Even though they do not possess any sparsity pattern, the Loewner and shifted Loewner matrices are extremely structured and, in this paper, we show how to fully exploit their Cauchy-like structure to reduce the cost of computing accurate rational models while avoiding the explicit allocation of {mathbb {L}} and {mathbb {S}}. In particular, the use of the hierarchically semiseparable format allows us to remarkably lower both the computational cost and the memory requirements of the Loewner framework obtaining a novel scheme whose costs scale with N log N.
Highlights
The Loewner framework, originally proposed in [30] for solving the generalized realization problem coupled with tangential interpolation, was successfully employed for data-driven model order reduction from frequency domain data [26]
Greedy-type approaches were proposed in [26], reducing memory requirements, from O(N 2) for storing the dense Loewner and shifted Loewner matrices to O(N + n2), and the computational cost, from O(N 3) for computing the singular value decomposition (SVD) to O(N n3) and O(N n4), where N is the size of the data set and n is the order of the model
The proposed strategy is supported by a thorough analysis of the computational cost, showing that, for very large data sets for which carrying out the full SVD is intractable, our strategy leads to remarkable reductions in both the computational efforts and the storage demand for building minimal realizations in the Loewner framework
Summary
The Loewner framework, originally proposed in [30] for solving the generalized realization problem coupled with tangential interpolation, was successfully employed for data-driven model order reduction from frequency domain data [26]. The main drawbacks, are the large storage requirements paired with the significant CPU cost inherent to the full SVD computation for data sets with a large number of measurements (values in the range 105 are common in industrial applications) To bypass these inconveniences, greedy-type approaches were proposed in [26], reducing memory requirements, from O(N 2) for storing the dense Loewner and shifted Loewner matrices to O(N + n2), and the computational cost, from O(N 3) for computing the SVD to O(N n3) and O(N n4), where N is the size of the data set and n is the order of the model. The factored ADI-Galerkin method for computing these matrices as solutions to certain Sylvester equations with a factored right-hand side was investigated in [18] Such a scheme computes low-rank approximations to the dense Loewner matrix to speed-up the SVD computation. 3 presents results showcasing the special structure of the Loewner and shifted Loewner matrices as Cauchy-like matrices and their approximation as hierarchically semiseparable matrices allowing for efficient, inexact matrix-vector products in the partial SVD computation.
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