[Paper Review] Generalization of the Neville-Aitken Interpolation Algorithm on Grassmann Manifolds : Applications to Reduced Order Model

Rolando Mosquera, Abdallah El Hamidi+2 more|arXiv (Cornell University)|Jul 5, 2019

Fractional Differential Equations Solutions4 citations

TL;DR

This paper generalizes the Neville-Aitken interpolation algorithm to Grassmann manifolds for reduced order modeling in parametric PDEs, using geodesic barycenters to recursively interpolate POD subspaces without requiring a reference point. The method achieves higher accuracy and faster computation than existing approaches in three CFD applications, including vortex shedding and rotating body flows.

ABSTRACT

The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available solutions corresponding to the chosen parameter values. The well-known Neville-Aitken's algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points. The performances of the proposed method are illustrated through three independent CFD applications, namely: the Von Karman vortex shedding street, the lid-driven cavity with inflow and the flow induced by a rotating solid. The obtained numerical simulations are pertinent both in terms of the accuracy of results and the time computation.

Motivation & Objective

• Address the challenge of interpolating Proper Orthogonal Decomposition (POD) subspaces across varying parameters in parametric PDEs, particularly in fluid dynamics.
• Overcome limitations of existing methods that require a reference point or iterative minimization, such as those in Amsallem & Farhat (2011) and IDW-based approaches.
• Develop a direct, recursive interpolation algorithm on Grassmann manifolds to improve accuracy and computational efficiency in reduced order modeling.
• Demonstrate the method’s robustness and performance across diverse fluid dynamics problems with varying parameter regimes.
• Provide a scalable, non-intrusive framework for real-time solution prediction in parametric simulations using interpolated subspaces on Grassmann manifolds.

Proposed method

• Extend the classical Neville-Aitken algorithm to Grassmann manifolds by replacing linear interpolation with geodesic barycenter computation between subspaces.
• Use the geodesic barycenter as the recursive interpolation step, ensuring Riemannian consistency on the manifold of m-dimensional subspaces.
• Perform interpolation directly on the Grassmann manifold $ G_m(H) $, avoiding the need for a fixed reference point as required in logarithmic mapping approaches.
• Construct reduced-order models (ROMs) by interpolating POD bases corresponding to different parameter values, then projecting full-order snapshots onto the interpolated subspaces.
• Apply the method recursively across multiple parameter values to build a hierarchical interpolation tree, mimicking the Neville-Aitken structure.
• Validate the method using projection errors and dynamical system errors on three CFD test cases with varying Reynolds numbers and flow configurations.

Experimental results

Research questions

• RQ1 Can the Neville-Aitken interpolation algorithm be generalized to Grassmann manifolds to enable efficient and accurate interpolation of POD subspaces in parametric PDEs?
• RQ2 How does the proposed method compare to reference approaches such as Amsallem & Farhat (2011) in terms of accuracy and computational cost?
• RQ3 Does the absence of a reference point in the interpolation process improve robustness and convergence in high-dimensional parametric spaces?
• RQ4 To what extent does the recursive geodesic barycenter approach maintain accuracy across diverse fluid dynamics configurations with complex flow structures?
• RQ5 Can the method be effectively applied to transient, nonlinear problems such as vortex shedding and rotating body flows with minimal computational overhead?

Key findings

• The proposed Neville-Aitken-based method on Grassmann manifolds achieves lower projection errors than the Amsallem method, with errors of 1.874e-03 (Case 1), 2.457e-03 (Case 2), and 2.141e-03 (Case 3) for the fluctuating velocity.
• The method yields lower dynamical system errors, with values of 2.108e-02 (Case 1), 1.817e-02 (Case 2), and 1.474e-02 (Case 3), outperforming the Amsallem method in all test cases.
• The method eliminates the need for a reference point, simplifying implementation and avoiding potential conditioning issues associated with reference subspace selection.
• The recursive geodesic barycenter computation enables direct, non-iterative interpolation, reducing computational cost compared to fixed-point optimization in IDW-based methods.
• In all three CFD applications—Von Karman vortex shedding, lid-driven cavity with inflow, and rotating solid—the interpolated ROMs show high accuracy and stability across parameter values.
• Temporal coefficients of the reduced modes closely match the reference solution, confirming the method’s ability to preserve dynamic behavior in transient flows.