[Paper Review] Minimal geodesics along volume preserving maps, through semi-discrete optimal transport

Quentin Mérigot, Jean‐Marie Mirebeau|arXiv (Cornell University)|May 13, 2015

Geometric Analysis and Curvature Flows2 references4 citations

TL;DR

This paper introduces a numerical method to compute minimal geodesics in the group of volume-preserving maps using semi-discrete optimal transport, enabling the first numerical reconstruction of non-classical, multi-valued solutions to Euler's equations for incompressible fluids. The approach leverages Brenier's generalized polar decomposition and converges to solutions in the relaxed setting of generalized flows, revealing fractal-like support structures and non-deterministic particle paths beyond classical flow regimes.

ABSTRACT

We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L2 metric, which as observed by Arnold solve Euler's equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier, numerically implemented through semi-discrete optimal transport. It is robust enough to extract non-classical, multi-valued solutions of Euler's equations, for which the flow dimension is higher than the domain dimension, a striking and unavoidable consequence of this model. Our convergence results encompass this generalized model, and our numerical experiments illustrate it for the first time in two space dimensions.

Motivation & Objective

• To develop a robust numerical method for computing minimal geodesics in the group of volume-preserving diffeomorphisms under the $L^2$ metric, which correspond to solutions of Euler's equations for incompressible fluids.
• To address the failure of classical minimizers in dimensions $d \geq 3$ by working in the relaxed setting of generalized flows, where particle paths may split and cross.
• To numerically implement Brenier's generalized polar decomposition using semi-discrete optimal transport to extract minimal geodesics even when classical solutions do not exist.
• To provide the first numerical illustrations of non-classical, multi-valued fluid flows with fractal-like support structures in two spatial dimensions.

Proposed method

• The method formulates the geodesic problem as a convex optimization over probability measures on continuous paths $\Omega = C^0([0,1], X)$, minimizing the action $\int_0^1 |\dot{\omega}(t)|^2 dt$ subject to incompressibility and boundary constraints.
• It uses the generalized polar decomposition of Brenier, which decomposes a volume-preserving map into a composition of a gradient map and a measure-preserving map, enabling numerical computation via semi-discrete optimal transport.
• The semi-discrete optimal transport framework discretizes the target measure (final fluid configuration) while keeping the source (initial configuration) continuous, allowing efficient computation of transport plans.
• The algorithm computes generalized flows by solving a dual problem involving pressure as a Lagrange multiplier, with convergence established in the relaxed, generalized flow setting.
• Particle paths are reconstructed by sampling from the optimal transport plan, and non-deterministic behavior is visualized through clustering and box-counting dimension estimation.
• The method is implemented in open-source software available at https://github.com/mrgt/EulerSemidiscrete, enabling reproducibility and further experimentation.

Experimental results

Research questions

• RQ1 Can minimal geodesics in the group of volume-preserving maps be numerically computed even when classical solutions do not exist due to oscillatory behavior in higher dimensions?
• RQ2 How do non-classical, multi-valued fluid flows—where particles split and paths cross—behave in two spatial dimensions, and can they be captured numerically?
• RQ3 What is the geometric structure of the support of such generalized flows, and does it exhibit fractal-like properties as suggested by theoretical analysis?
• RQ4 How does the regularity of the pressure field relate to the non-deterministic nature of the reconstructed fluid paths?
• RQ5 To what extent do numerical solutions remain deterministic near boundaries even when global non-determinism emerges in the interior?

Key findings

• For $t_{\text{max}} < 1$, the computed solutions recover the classical Beltrami flow in the unit square, with deterministic particle paths and smooth pressure fields.
• For $t_{\text{max}} \geq 1$, the solutions become non-classical and non-deterministic, with particle paths crossing and splitting, indicating the onset of generalized flow behavior.
• The box-counting dimension of the solution support increases from $D=2$ (for $t_{\text{max}}=0.9$) to values strictly between 2 and 3 for $t_{\text{max}} \in \{1.1, 1.3, 1.5\}$, suggesting a fractal structure.
• Near the boundary of the domain, particle motion remains approximately deterministic and clockwise, while the interior exhibits highly non-deterministic, counter-clockwise mixing.
• The pressure gradient field shows signs of reduced regularity near initial and final times, consistent with the theoretical $L^2_{\text{loc}}(]0,T[ , \text{BV}(X))$ regularity result.
• The algorithm successfully reconstructs generalized flows for both the disk inversion and Beltrami flow, with the latter showing stronger non-deterministic behavior at $t_{\text{max}} = 1.5$.