Model order reduction by convex displacement interpolation

TL;DR

This paper introduces a nonlinear convex displacement interpolation method for parametric fields that improves accuracy in model reduction by leveraging optimal transportation and registration techniques, applicable to complex fluid flow simulations.

Contribution

It generalizes previous nonlinear interpolation methods to multi-dimensional parameters and multiple snapshots, enhancing the accuracy and efficiency of model reduction for complex flows.

Findings

Accurate interpolation of parametric solution fields in fluid dynamics.

Effective data augmentation reduces offline costs in model reduction.

Method performs well on compressible and incompressible, viscous and inviscid flows.

Abstract

We present a nonlinear interpolation technique for parametric fields that exploits optimal transportation of coherent structures of the solution to achieve accurate performance. The approach generalizes the nonlinear interpolation procedure introduced in [Iollo, Taddei, J. Comput. Phys., 2022] to multi-dimensional parameter domains and to datasets of several snapshots. Given a library of high-fidelity simulations, we rely on a scalar testing function and on a point set registration method to identify coherent structures of the solution field in the form of sorted point clouds. Given a new parameter value, we exploit a regression method to predict the new point cloud; then, we resort to a boundary-aware registration technique to define bijective mappings that deform the new point cloud into the point clouds of the neighboring elements of the dataset, while preserving the boundary of the domain; finally, we define the estimate as a weighted combination of modes obtained by composing the neighboring snapshots with the previously-built mappings. We present several numerical examples for compressible and incompressible, viscous and inviscid flows to demonstrate the accuracy of the method. Furthermore, we employ the nonlinear interpolation procedure to augment the dataset of simulations for linear-subspace projection-based model reduction: our data augmentation procedure is designed to reduce offline costs -- which are dominated by snapshot generation -- of model reduction techniques for nonlinear advection-dominated problems.