Transport-based interpolation of dyadic meshes for problems with variable conditions
Abstract
Mesh adaptation reduces the cost of simulations with localized solution features, but repeated adaptation can become prohibitive in parametric studies. Selecting a precomputed library mesh may be insufficient when the refined region moves with the operating condition. We introduce transport-based mesh interpolation (TMI), which constructs an adapted mesh for a queried parameter by interpolating nearby meshes from an offline library. Each dyadic mesh is represented by the empirical measure of its cell centers, encoding refinement patterns through spatial point density. The interpolated representative is defined as the transport barycenter of neighboring mesh measures, using parametric barycentric weights. Geometric rules then reconstruct an admissible dyadic mesh from this representative. The proof-of-concept implementation uses a sliced Wasserstein approximation for simplicity and efficiency. Projection experiments on moving hyperbolictangent cuts isolate interpolation and reconstruction behavior, while a reaction-diffusion problem with a moving triangular source evaluates PDE-adapted meshes. In the tested configurations, the depth-informed and max-one-point reconstructions produce meshes with errors consistent with direct adaptation, with a favorable accuracy-size compromise for the depth-informed variant. Moreover, amortized online TMI becomes faster than full adaptation for the finest tested tolerances.
Related
- Stability of Edelstein Hidden-Variable Fractal Interpolation Functions
- Space-time finite element interpolation of nonsmooth functions satisfying boundary conditions
- Interpolation Conditions for Instant Data Consistency with Port-Hamiltonian Structure
- Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems
- Which modelling choices shape your interpolation map? An Optimal Transport sensitivity analysis for groundwater sulfate concentration
BibTeX
@misc{dornier2026transportbased,
title = {{Transport-based interpolation of dyadic meshes for problems with variable conditions}},
author = {Hugo Dornier and Olivier Le Maitre and Pietro Marco Congedo and Théotime Niault},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05802486v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Transport-based interpolation of dyadic meshes for problems with variable conditions
AU - Hugo Dornier
AU - Olivier Le Maitre
AU - Pietro Marco Congedo
AU - Théotime Niault
PY - 2026
DA - 2026/10/15
PB - HAL
UR - https://hal.science/hal-05802486v1
AB - Mesh adaptation reduces the cost of simulations with localized solution features, but repeated adaptation can become prohibitive in parametric studies. Selecting a precomputed library mesh may be insufficient when the refined region moves with the operating condition. We introduce transport-based mesh interpolation (TMI), which constructs an adapted mesh for a queried parameter by interpolating nearby meshes from an offline library. Each dyadic mesh is represented by the empirical measure of its cell centers, encoding refinement patterns through spatial point density. The interpolated representative is defined as the transport barycenter of neighboring mesh measures, using parametric barycentric weights. Geometric rules then reconstruct an admissible dyadic mesh from this representative. The proof-of-concept implementation uses a sliced Wasserstein approximation for simplicity and efficiency. Projection experiments on moving hyperbolictangent cuts isolate interpolation and reconstruction behavior, while a reaction-diffusion problem with a moving triangular source evaluates PDE-adapted meshes. In the tested configurations, the depth-informed and max-one-point reconstructions produce meshes with errors consistent with direct adaptation, with a favorable accuracy-size compromise for the depth-informed variant. Moreover, amortized online TMI becomes faster than full adaptation for the finest tested tolerances.
ER -