Gallouët-Mérigot's Partial OT Scheme

• Gallouët-Mérigot's scheme is an analytical and algorithmic framework for partial optimal transport, enabling precise computation of cell volumes by intersecting generalized Laguerre cells with spheres.
• It utilizes a variational formulation with Newton iterations and analytic decompositions to compute volumes and facets accurately, bypassing complex mesh discretizations.
• The framework significantly enhances both simulation accuracy and computational efficiency in applications like free-surface fluid simulation and deformation mechanics.

Gallouët-Mérigot's scheme refers to an analytic and algorithmic framework for solving partial optimal transport (OT) problems, with a focus on applications such as free-surface fluid simulation and deformation mechanics. This scheme is centered on a variational formulation that enables the exact computation of particle-based fluid geometry by intersecting generalized Laguerre cells (or power diagrams) with spheres, as motivated by the requirements of partial OT. Its algorithmic innovations yield substantial improvements over traditional polygonal discretization approaches both in accuracy and computational efficiency, providing precise volumetric and area quantities needed for physics-based simulation (Plateau--Holleville et al., 9 Jan 2026).

1. Mathematical Formulation of Partial Optimal Transport

Gallouët-Mérigot's scheme is defined on a fixed container domain $\Omega \subset \mathbb{R}^d$ of known total volume $|\Omega|$. The problem starts with $n$ fluid sites $x_1, \ldots, x_n \in \Omega$ with prescribed masses $m_1, \ldots, m_n > 0$, and a background (air) mass reservoir $m_0 := |\Omega| - \sum_{i=1}^n m_i$. The underlying primal (semi-discrete) partial OT problem is to minimize the quadratic transport cost between the continuum (Lebesgue measure on $\Omega$) and a set of $n+1$ points $\{x_0, x_1, \ldots, x_n\}$, where $x_0$ formally represents the air. The minimization is over transport plans $\gamma$ from $\Omega$ to $\{0,\ldots,n\}$:

$$W^2 = \min_{\gamma} \int_{\Omega \times \{0,\ldots,n\}} \|x - x_i\|^2 \, d\gamma(x, i)$$

subject to per-mass constraints:

$$\forall i=0,\ldots,n : \int_{\Omega} d\gamma(x, i) = m_i$$

$$\gamma \text{ has first marginal equal to Lebesgue on } \Omega.$$

The dual (Kantorovich) formulation involves potentials $\psi = (\psi_0, \ldots, \psi_n)$, with $\psi_0 = 0$. The core geometric entity is the Laguerre cell:

$$L_i(\psi) = \{ x \in \Omega \mid \|x-x_i\|^2 - \psi_i \leq \|x-x_j\|^2 - \psi_j, \;\; \forall j \}$$

For the fluid portion, the relevant cell is $$V_i(\psi) = L_i(\psi) \cap \{ x \mid \|x-x_i\|^2 - \psi_i \leq 0 \}$$, i.e., the intersection with the ball of squared radius $\psi_i$. The dual objective is

$$K(\psi) = \sum_{i=0}^n m_i \psi_i + \int_\Omega \min_{0 \leq i \leq n} (\|x - x_i\|^2 - \psi_i) \, dx$$

The optimal $\psi^*$ (unique up to translation) equates prescribed masses to the volumes of the associated cells: $|V_i(\psi^*)| = m_i$.

2. Structure and Properties of (Generalized) Laguerre Cells

Given the sites $\{x_i\}$ and weights $\{\psi_i\}$, the unrestricted Laguerre cell $L_i(\psi)$ generalizes the concept of a Voronoi cell, with the "power" $\psi_i$ controlling the effective size of the region around $x_i$. For partial OT, $L_i(\psi)$ is further intersected with a ball $S_i = \{\|x-x_i\| \leq \sqrt{\psi_i}\}$, yielding the fluid-support cell $V_i(\psi)$. The boundary structure of $V_i(\psi)$ involves both planar facets (from the Laguerre diagram) and spherical caps (from the intersection with $S_i$).

3. Analytic Construction of Volumes and Facets

The computation of cell metrics (volumes and facets) uses a decomposition of each $V_i$ into pyramids $P_{ij}$ with apex $x_i$ and base $B_{ij}$, where:

• $B_{ij}$ is the restricted planar facet $L_i \cap L_j \cap S_i$,
• For the background ($j=0$), $B_{i0}$ is a spherical patch $K_i$ on $\partial S_i$.

Pyramid volumes are given by

$|P_{ij}| = h_{ij} \cdot |B_{ij}|$

with

$$h_{ij} = \begin{cases} \frac{(\|x_j - x_i\|^2 + \psi_i - \psi_j)}{2\|x_j - x_i\|} & \text{if } 1 \leq j \leq n \ \sqrt{\psi_i} & \text{if } j = 0 \end{cases}$$

Total cell volume is

$$|V_i| = \sum_{j=1}^n h_{ij}|B_{ij}| + \sqrt{\psi_i}|K_i|$$

To compute $|K_i|$ (the area of the spherical patch), the approach avoids costly Gauss-Bonnet curvature integrals, instead using an analytic method:

$$|K_i| = |\partial S_i| - \sum_{j=1}^n |\text{Proj}(B_{ij}, \partial S_i, c)|$$

Here, $\text{Proj}(\cdot)$ is the central projection of each planar facet onto the sphere from a chosen interior point $c \in V_i$. This reduces geometric complexity to analytic expressions in terms of base facet areas and projections.

4. Algorithmic Pipeline: Iterative Solution and Physical Update

At each simulation time-step, the Gallouët-Mérigot pipeline follows an alternating geometry-optimization and physical update routine:

1. Initialization: Set positions $x_i$, masses $m_i$, and initialize weights $\psi_i$.
2. Newton Iteration (Per Geometry Update):
   • Construct unrestricted Laguerre diagram $L_i$.
   • Compute all $B_{ij}$, $|K_i|$, $|B_{ij}|$, and $h_{ij}$ for each cell $i$.
   • Evaluate volumes $|V_i|$ and gradients $g_i = m_i - |V_i|$.
3. Hessian Computation:
   • For neighbors $j \neq i$, $$H_{ij} = \frac{1}{2} \frac{|B_{ij}|}{\|x_j - x_i\|}$$.
   • Diagonal $$H_{ii} = -\sum_{j \neq i} H_{ij} - \frac{1}{2} \frac{|K_i|}{\sqrt{\psi_i}}$$.
4. Newton Step:
   • Solve $Hu = -g$ for update direction $u$.
   • Backtracking line-search to maintain $|V_i| > 0$, updating $\psi \leftarrow \psi + \alpha u$.
5. Convergence: Iterate steps 2–4 until $\max_i |g_i| \leq \epsilon$.
6. Physical Time Integration: With optimal $\psi^*$, update Lagrangian sites $x_i$ via the fluid solver, integrating pressure, viscosity, and surface tension forces cell-wise.

5. Comparison with Classical Convex-Cell Clipping

Conventional implementations of OT-based fluid schemes employ a mesh-based intersection of convex Laguerre cells and bounding spheres, followed by polygonalization:

• Facets are subdivided into numerous triangles,
• Maintaining adjacency data for vertices/edges is nontrivial,
• Volumes and areas are estimated numerically over the mesh,
• Resolution control is mediated by mesh fineness.

The analytic Gallouët-Mérigot scheme, in contrast:

• Provides closed-form (floating-point) evaluations of volumes and areas,
• Eliminates combinatorial complexity (no intersection mesh, only polygons and spherical caps),
• Has no user-set discretization parameter,
• Reduces the number of arithmetic operations substantially,
• Allows direct parallelization over independent cells or facets,
• Yields speed-ups of 10–20% even against coarse mesh discretizations, and eliminates volumetric estimation errors caused by piecewise-linear surface representations.

These characteristics enable simulations that are both more accurate (cell volumes to machine precision) and computationally more efficient.

6. Applications and Significance in Fluid Simulation

Gallouët-Mérigot's framework is particularly well-suited for free-surface fluid simulations and deformation mechanics, where accurate and efficient computation of cell geometry directly governs the integration of pressure, viscous, and surface tension forces at each time step. The analytic construction of the underlying geometry facilitates cell-wise computations as required for particle-based simulation approaches, and supports both rendering and numerical integration purely from the cell structure (Plateau--Holleville et al., 9 Jan 2026).

A plausible implication is that this scheme provides a foundation for more robust and scalable methods in computational fluid dynamics, especially when complex interfaces or background reservoirs (e.g., air) must be handled consistently within the optimal transport formulation.