A Riemannian Barycentric Interpolation : Derivation of the Parametric Unsteady Navier-Stokes Reduced Order Model

TL;DR

This paper introduces a novel Riemannian barycentric interpolation method for constructing nonlinear parametric reduced order models, offering easier implementation and flexibility, with competitive accuracy and reduced computational cost demonstrated through fluid dynamics experiments.

Contribution

It proposes a new barycentric PROM approach based on Riemannian geometry of subspace manifolds, improving flexibility and simplicity over existing methods.

Findings

Achieves competitive accuracy in flow simulations.

Reduces computational cost significantly.

Demonstrates effectiveness on fluid flow problems.

Abstract

A new application of subspaces interpolation for the construction of nonlinear Parametric Reduced Order Models (PROMs) is proposed. This approach is based upon the Riemannian geometry of the manifold formed by the quotient of the set of full-rank N-by-q matrices by the orthogonal group of dimension q. By using a set of untrained parametrized Proper Orthogonal Decomposition (POD) subspaces of dimension q, the subspace for a new untrained parameter is obtained as the generalized Karcher barycenter which solution is sought after by solving a simple fixed point problem. Contrary to existing PROM approaches, the proposed barycentric PROM is by construction easier to implement and more flexible with respect to change in parameter values. To assess the potential of the barycentric PROM, numerical experiments are conducted on the parametric flow past a circular cylinder and the flow in a lid driven cavity when the value of Reynolds number varies. It is demonstrated that the proposed barycentric PROM approach achieves competitive results with considerably reduced computational cost.