Generalization of the Neville–Aitken interpolation algorithm on Grassmann manifolds: Applications to reduced order model

• Published in: International journal for numerical methods in fluids 2021-07, Vol.93 (7), p.2421-2442
• Main Authors: Mosquera, Rolando, El Hamidi, Abdallah, Hamdouni, Aziz, Falaize, Antoine
• Format: Article
• Language: English
• Subjects: Algorithms; Center of gravity; Computational fluid dynamics; Grassmann manifold; Inflow; Interpolation; Manifolds (mathematics); Model reduction; Neville interpolation algorithm; Proper Orthogonal Decomposition; Reduced order models; Subspaces; Vortex shedding
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4981

An extension of the well‐known Neville–Aitken's algorithm for interpolation on the Grassmann manifold Gm(ℝn) in the framework of parametric model order reduction is presented. Interpolation points on Gm(ℝn) are the subspaces spanned by bases obtained by Proper Orthogonal Decomposition of available solutions associated with the chosen parameter sampling. The Neville–Aitken's algorithm is performed recursively via the geodesic barycenter of two points. Three CFD applications are presented: (i) the Von Karman vortex shedding street, (ii) the lid‐driven cavity with inflow and (iii) the flow induced by a rotating solid. Numerical results are relevant with respect to accuracy while the asymptotic complexity is comparable to the state of the art. – Our algorithm is based only on the notion of geodesic barycenter of two points in the manifold. – Unlike iterative algorithms, based on the notion of Karcher's barycenter, our algorithm is explicit. – It does not require any reference point and numerical results are relevant with respect to accuracy while the asymptotic complexity is comparable to the state of the art.