  Approximating solutions of non-linear parametrized physical problems by
interpolation presents a major challenge in terms of accuracy. In fact,
pointwise interpolation of such solutions is rarely efficient and leads
generally to incorrect results. However, instead of using a straight forward
interpolation on solutions, reduced order models can be interpolated. More
particularly, Amsallem and Farhat proposed an efficient POD reduced order model
interpolation technique based on differential geometry tools. This approach,
named in this paper ITSGM (Interpolation On a Tangent Space of the Grassmann
Manifold), allows through the passage to the tangent space of the Grassmann
manifold, to approximate accurately the reduced order basis associated to a new
untrained parameter. This basis is used afterwards to build the interpolated
ROM describing the temporal dynamics by performing the Galerkin projection on
the high fidelity model. Such Galerkin ROMs require to access to the underlying
high fidelity model, leading by that to intrusive ROMs. In this paper, and
contrary to the ITSGM/Galerkin approach, we propose a non-intrusive reduced
order modeling method which is independent of the governing equations. This
method is named through this paper Bi-CITSGM (Bi-Calibrated ITSGM). It consists
first to interpolate the spatial and temporal POD sampling bases considered as
representatives of points on Grassmann manifolds, by the ITSGM method. Then,
the resulting bases modes are reclassified by introducing two orthogonal
matrices. These calibration matrices are determined as analytical solutions of
two optimization problems. Results on the flow problem past a circular cylinder
where the parameter of interpolation is the Reynolds number, show that for new
untrained Reynolds number values, the developed approach produces satisfyingly
accurate solutions in a real computational time.
