A geometrical numerical linear algebra approach to residual‐enhanced parametric model reduction

Abstract

AbstractThe most prominent approaches to model reduction share the general principle, that a given large‐scale system is projected onto a suitable subspace spanned by a low‐dimensional basis. The projection basis essentially determines the approximation quality of the resulting reduced order system. Nonlinear parametric dependencies may be taken into account by applying interpolation techniques on matrix manifolds for computing projection bases and/or reduced systems at arbitrary parameter conditions, see [1] for a recent survey. In this short note, an original approach to enhance the prediction capabilities of the interpolated reduced‐order projection bases for parametrized linear systems is proposed. To this end, a suitable residual‐based goal function is introduced, which measures the approximation quality of a given projection subspace. As this function is invariant under changes of basis, it is to be considered as a mapping on the Grassmann‐Manifold Gr(n, p) of p‐dimensional subspaces of ℝn. As such, it may be optimized using the conjugate gradient method specifically adapted to the manifold structure of the Grassmannian Gr(n, p), [2,3]. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)