H2-optimal model reduction
http://orcid.org/0000-0002-8251-6275
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ABSTRACT
In this contribution, a new framework for -optimal reduction is presented, motivated by the local nature of both (tangential) interpolation and
-optimal approximations. The main advantage is given by a decoupling of the cost of reduction from the cost of optimization, resulting in a significant speedup in
-optimal reduction. In addition, a middle-sized surrogate model is produced at no additional cost and can be used e.g. for error estimation. Numerical examples illustrate the new framework, showing its effectiveness in producing
-optimal reduced models at a far lower cost than conventional algorithms. Detailed discussions and optimality proofs are presented for applying this framework to the reduction of multiple-input, multiple-output linear dynamical systems. The paper ends with a brief discussion on how this framework could be extended to other system classes, thus indicating how this truly is a general framework for interpolatory
reduction.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1. For dense matrices, this can be motivated by simple asymptotic operation counts. The QR decomposition of a matrix via Householder requires
flops and is hence linear in
. The flops involved in the product
are
for a dense
and hence quadratic in
. Note however that for a diagonal
matrix – an ideally sparse invertible matrix – the flops become at most
, hence being linear in
[Citation23].
2. For the results of this contribution, we have used sparse LU decompositions, which are still feasible for models of relative large size even on standard machines (cf. ). This approach bears the advantage of allowing the recycling of LU factors, e.g. to solve LSEs with same left-hand sides. If other methods are used instead, some of the values and statements in this contribution may vary. Note, in addition, that complex conjugated pairs of shifts yield complex conjugated directions
. Therefore, the actual number of LSE to be solved may actually vary, making
a worst-case estimate.
3. All numerical examples presented in this contribution were generated using the sss and sssMOR toolboxes in MATLAB® [Citation24].