  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.
