I'm reading this paper on a numerical algorithm to solve the optimal transport problem in the semi-discrete case. Let me put down some notation (all the paper works on a subset of $\mathbb{R}^2$):
- $\mu$ is a finite measure with density $\rho$
- $\nu = \sum\limits_{i=1}^n \nu_i \delta_{x_i}$ is a discrete measure
At the start of the problem we know $\rho$ and $n$ and we want to find the discrete measure $\nu$ that minimize the 2-Wasserstein between $\mu$ and $\nu$. It is know that all the optimal transport map can be defined through a partition of $\mathbb{R}^2$ given by an additively weighted Voronoi diagram. So given $X = \{x_1, \dots, x_n \}$ a set of point and $w = (w_1, \dots, w_n)$ a vector of weights the voronoi tassellation is given by these sets: $$Vor(x_i, w) = \{x \in \mathbb{R}^2 \mid |x - x_i|^2 -w_i \leq |x - x_j| ^2 - w_j \ \ \forall \ j \neq i\}$$ Clearly to solve the minimization problem we need to find the best $X$ and $w$. As a constrain we want that $\nu_i = \frac{m}{n}$ where $m$ is the total mass. Now define $$\mathcal{E}(X,w) = \sum_{i=1}^n\int_{Vor(x_i,w)}\rho(x)|x-x_i|^2 \ dx$$ $$m_i = \int_{Vor(x_i,w)} \rho(x) \ dx$$ The problem is now find a stationary point of: $$ \mathcal{E}(X,w) + \sum_{i=1}^n \lambda_i(m_i - m)$$ where $\lambda_i$ is the lagrangian multiplier. The paper shows that this problem is equivalent to find the stationary point of: $$\mathcal{F}(X,w) = \mathcal{E}(X,w) - \sum_{i=1}^n w_i(m_i - m)$$ In the appendix the author proceed to calculate the gradient of $\mathcal{F}$ and this is where the things are not clear for me. On the paper we can read
We denote by $e_{ij}$ the regular edge between two adjacent points $x_i$ and $x_j$ , and by $e^\ast_{ij}$ the dual edge separating the partition regions $Vor(x_i,w)$ and $Vor(x_j,w)$. We also refer to the average value of the field $\rho$ over $e^\ast_{ij}$ as $\bar\rho_{ij}$ , and to the one-ring of $x_i$ in the regular triangulation of $(X,w)$ as $\Omega_i$
First of all I don't have a clear intuition of what is $\Omega_i$ but then the author state that applying the Reynold theorem to $\nabla_{w_i} m_j$ we can obtain:
$$- \dfrac{\bar\rho_{ij}}{2}\dfrac{|e^\ast_{ij}|}{|e_{ij}|}$$
and I have no clue of how we can obtain this result. If anyone of you know how to apply Reynolds theorem here I will be thankful.