Parametric vector flows for registration fields in bounded domains with applications to nonlinear interpolation of shock-dominated flows

We present a registration procedure for parametric model order reduction (MOR) in two- and three-dimensional bounded domains. In the MOR framework, registration methods exploit solution snapshots to identify a parametric coordinate transformation that improves the approximation of the solution set through linear subspaces. For each training parameter, optimization-based (or variational) registration methods minimize a target function that measures the alignment of the coherent structures of interest (e.g., shocks, shear layers, cracks) for different parameter values, over a family of bijections of the computational domain $Ω$. We consider diffeomorphisms $Φ$ that are vector flows of given velocity fields $v$ with vanishing normal component on $\partial Ω$; we rely on a sensor to extract appropriate point clouds from the solution snapshots and we develop an expectation-maximization procedure to simultaneously solve the point cloud matching problem and to determine the velocity $v$ (and thus the bijection $Φ$); finally, we combine our registration method with the nonlinear interpolation technique of [Iollo, Taddei, J. Comput. Phys., 2022] to perform accurate interpolations of fluid dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past a NACA airfoil and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the many elements of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated fields.

💡 Research Summary

This paper addresses a fundamental challenge in parametric model order reduction (MOR) for problems featuring sharp, parameter‑dependent coherent structures such as shocks, shear layers, or cracks. Classical linear subspace MOR struggles when the Kolmogorov n‑width decays slowly, prompting the development of nonlinear strategies based on parametric coordinate transformations. The authors propose a registration framework that constructs a family of bijective mappings Φ(·;μ):Ω→Ω for a bounded Lipschitz domain Ω⊂ℝ^d (d=2 or 3).

The core idea is to represent each mapping as the endpoint of a vector‑flow generated by a time‑dependent velocity field v(x,t) that satisfies a no‑normal‑flux condition on the boundary (v·n=0 on ∂Ω). By discretizing the velocity space with a finite‑element basis {ϕ_i}⊂V₀(Ω) and writing v(x,t)=∑_i v_i ϕ_i(x,t), the flow map Φ=N(v) is obtained by integrating the ordinary differential equation ∂_t X(ξ,t)=v(X(ξ,t),t) from t=0 to t=1. Theoretical results guarantee that for any coefficient vector v the resulting map is a diffeomorphism (bijective, orientation‑preserving, with positive Jacobian determinant), thus eliminating the need for explicit bijectivity constraints in the optimization.

To align the coherent structures across snapshots, the authors introduce a sensor that extracts a point cloud Y_μ from each solution U(·;μ). A reference point cloud {ξ_i} is defined once for all parameters. The alignment quality is measured by a weighted squared distance f_tg(Φ;P,μ)=½∑{i,j} P{ij}‖Φ(ξ_i)−y_j(μ)‖², where P∈