Mirrors, lenses and Monge–Ampère equations

1 Anidolic optics and Monge Ampère type equations

In anidolic optics, or non-imaging optics, one studies the design of devices that transfer light energy between a source and a target. The general problem is to design the shape of a mirror (or a lens) that reflects (or refracts) the light emitted from a given source towards a target whose geometry and intensity distributions are prescribed (see Figures ⁠1⁠ and ⁠2⁠). Applications of anidolic optics include the design of solar ovens, public lighting, car headlights, and more generally the optimization of the use of light energy and the reduction of light pollution.

Near-field and far-field light sources

There exists many different problems in anidolic optics, depending for instance on the geometry of the light source, the type of optical component, and the target to be illuminated. These problems are distinguished in particular by the spatial position of the target illumination. A problem is called near-field when the target is located within a finite distance, i.e., when one wishes to illuminate an area of space such as a screen. In Figures ⁠1⁠ and ⁠2⁠, the target illumination is on a wall, making the problem near-field. Most of the illustrations in this article correspond near-field targets. However, we will first consider the far-field case, which is mathematically simpler, and in which the target lies “at infinity”, in the space of directions. In practice, this means that when light is reflected or refracted from a point of the optical component, we can forget the spatial position of the reflected or refracted ray, keeping only its direction, which can then be encoded by a unit vector. Note that if the near-field target illumination is far away from the optical component, each point of the target almost corresponds to a direction, so that the far-field problem is a good approximation of the near-field one in this situation. We will see in Section ⁠4.2⁠ that one can solve a problem involving a near-field target by iteratively solving problems with far-field targets.

We will first present two far-field mirror problems in their continuous form, as illustrated in Figure ⁠1⁠. Then, we will explain how these continuous problems can be approached by discrete problems, following the so-called supporting quadric method introduced by Luis Caffarelli and Vladimir Oliker. This method can be traced back to work of Hermann Minkowski and Aleksandr Aleksandrov in convex geometry.

1.1 Mirror for a point light source

In this first problem, light is emitted from a point $O$, which we assume to be located at the origin of the space $\mathbb{R}^{3}$. The intensity of the light source is modeled by a probability density $\mu$ on the sphere of directions $\mathbb{S}^{2}$. Let $X\subset\mathbb{S}^{2}$ denote the support of the measure $\mu$. For example, if the light is emitted in a solid cone, then $X$ is a disc on the sphere. The quantity of light emanating from a measurable set of directions $A\subset X$ is given by $\mu(A)$. For the far-field problem, the target is described by a probability measure $\nu$ on the sphere of directions $\mathbb{S}^{2}$, which then represents the directions “at infinity”, i.e., after reflection. Let $Y\subset\mathbb{S}^{2}$ denote the support of the measure $\nu$.

The inverse problem considered here consists in constructing the surface $\mathcal{R}$ of a mirror which will transport the intensity $\mu$ of the light source to the desired light distribution $\nu$ at infinity using Snell’s law of reflection. For example, if the target measure is a Dirac mass $\delta_{y}$, meaning that we want to reflect all the light in a single direction $y$, then the shape of the mirror is given by a paraboloid of revolution.

Let $\langle\cdot\,|\,\cdot\rangle$ denote the Euclidean scalar product on $\mathbb{R}^{3}$. An incident ray $x\in\mathbb{S}^{2}$ is reflected by a surface $\mathcal{R}$ in the direction $R(x)=x-\langle x\,|\,n(x)\rangle n(x)$, where $n(x)$ is the unit vector normal to the surface $\mathcal{R}$ at the point touched by the direction $x$ and oriented so that $\langle x\,|\,n(x)\rangle\leq 0$. The surface $\mathcal{R}$ solves the inverse mirror problem if $R$ transports the source measure $\mu$ to the target measure $\nu$, in the sense that for any measurable subset $B$ of the sphere one has

Note that the preservation of overall light quantity was already ensured by having chosen probability measures, i.e., $\mu(\mathbb{S}^{2})=\nu(\mathbb{S}^{2})=1$. Now assume that $\mu$ and $\nu$ are absolutely continuous measures with respect to the area measure on the sphere. Let $\mu(x)=\rho(x)\mathrm{d}x$ and $\nu(x)=\sigma(x)\mathrm{d}x$, where $\rho$ and $\sigma$ are the densities of $\mu$ and $\nu$ respectively. The previous equation then reads

Suppose furthermore that the densities $\rho$ et $\sigma$ are continuous and that $R$ is a diffeomorphism from $X$ to $Y$. By the change of variable $y=R(x)$, the last equation is then equivalent to $\sigma(R(x))\det(DR(x))=\rho(x)$ for any $x\in X$.

Since the mirror reflects rays emitted from the origin, we will assume that the surface $\mathcal{R}$ is radially parametrized by $x\in\mathbb{S}^{2}\mapsto u(x)x$, where $u:\mathbb{S}^{2}\to\mathbb{R}^{+}$ is a positive function that must be determined. The unit normal to the surface $\mathcal{R}$ at the point $xu(x)$ and the direction of the reflected ray can both be expressed as a function of $x$ and of the gradient $\nabla u(x)\in\mathrm{T}_{x}\mathbb{S}^{2}$:

This allows us to formulate the problem as a system of partial differential equations, i.e., the problem of finding a positive function $u:\mathbb{S}^{2}\to\mathbb{R}^{+}$ of class $\mathcal{C}^{2}$ which satisfies

The first line of equation (⁠Mir-Ponc-C⁠) involves the determinant of a quantity which depends on the second derivatives of $u$. This equation belongs to the family of Monge–Ampère equations. Note that the requirement that $R_{u}$ is a diffeomorphism is non-standard and difficult to handle. In practice, it is replaced by a condition on $u$ which is akin to convexity, and by the so-called second boundary condition $R_{u}(X)=Y$. These two conditions ensure the ellipticity of the problem. Caffarelli and Oliker proved in 1994 [⁠1⁠ L. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics. Journal of Mathematical Sciences154, 39–49 (2008) ] the existence of weak solutions to this equation, i.e., the existence of a locally Lipschitz function $u$ such that the application $R$ defined by the last two lines of (⁠Mir-Ponc-C⁠) satisfies (⁠1⁠). The existence of regular solutions to the problem (⁠Mir-Ponc-C⁠) is due to Wang and Guan [⁠6⁠ X.-J. Wang, On the design of a reflector antenna. Inverse Problems12, 351–375 (1996) , ⁠2⁠ P. Guan and X.-J. Wang, On a monge-ampere equation arising in geometric optics. J. Differential Geom48, 205–223 (1998) ].

1.2 Mirror for a collimated light source

We now present a second inverse problem arising in anidolic optics. This time the light source is collimated, which means that all the rays of light emitted by the source are parallel. We furthermore assume that they are positively collinear to the vertical vector $e_{z}=(0,0,1)$ and emitted from a domain of the horizontal plane $X\subset\mathbb{R}^{2}\times\{0\}$. For convenience, we will identify $\mathbb{R}^{2}$ and $\mathbb{R}^{2}\times\{0\}$. We assume that the surface of the optical component is smooth and parametrized by a height function $u:X\to\mathbb{R}$. The intensity of the light source is modeled by a probability measure $\mu$ on $X$. As in the previous case, the intensity of the target illumination is modeled by a probability measure $\nu$ on the sphere of directions at infinity. At each point $(x,u(x))$ of the optical component, the gradient $\nabla u(x)$ encodes the direction of the normal to the surface and we denote by $F(\nabla u(x))\in\mathbb{S}^{2}$ the direction of the ray reflected by Snell’s law. The reflector defined by $u$ solves the inverse mirror problem between $\mu$ and $\nu$ if for any measurable set $B\subset\mathbb{S}^{2}$ one has

Let us introduce the measure $\tilde{\nu}$ defined by $\tilde{\nu}(B)=\nu(F(B))$, which is supported on $\mathbb{R}^{2}$. We assume that $\mu$ and $\tilde{\nu}$ are absolutely continuous with respect to the Lebesgue measure, with continuous densities $\rho$ and $\sigma$, and that $x\mapsto\nabla u(x)$ is a diffeomorphism on its image. Then, with the change of variable $y=\nabla u(x)$, the inverse mirror problem for a collimated source can also be phrased as a partial differential equation:

We finally note that if $u$ is smooth and strongly convex (or strongly concave), the application $\nabla u$ is a diffeomorphism on its image.

1.3 Lenses

The construction of lenses that transform a light source into a target illumination prescribed at infinity are similar and are also described by Monge–Ampère type equations. As with mirrors, when the light is emitted from a point, the equation to be solved is on the sphere and when the light source is collimated, it is on the plane. In these problems, the light source passes through the surface of one side of the lens, either flat or spherical, and the aim is to construct the surface of the other side of the lens such that it refracts the light onto a prescribed target illumination at infinity. We do not detail the modelling of these problems here, but we will show results with lenses at the end.

2 Geometric discretization of Monge–Ampère equations

The two inverse problems in anidolic optics described in the previous section each involve two sets $X$ and $Y$, on which we have two probability measures $\mu$ and $\nu$ respectively, representing the light source and the desired target illumination. We saw that when these measures are absolutely continuous, the problems of construction of optical components correspond to partial differential equations of Monge–Ampère type.

The most direct method to solve a partial differential equation numerically is to approximate the domain $X$ with a discrete grid and to replace the partial derivatives with differences of the values of the function at the points of the grid divided by the grid step. In the case of Monge–Ampère equations, the application of these methods is made difficult by the non-linearity of the Monge–Ampère operator and by the diffeomorphism condition. We refer to the work of Adam Oberman, Brittany Froese, Jean-David Benamou and Jean-Marie Mirebeau for this line of research.

In recent years, alternative methods, called semi-discrete methods, have been used to discretize and numerically solve Monge–Ampère type equations arising from optimal transport. In order to apply this method, one assumes that one of the two measures $\mu$ or $\nu$ is absolutely continuous, while the other is finitely supported. Here we assume that $\mu$ has a density $\mu(x)=\rho(x)\mathrm{d}x$ on the space $X$ and that $\nu$ is a discrete measure on the space $Y=\{y_{1},\ldots,y_{N}\}$, i.e., $\nu=\sum_{1\leq i\leq N}\delta_{y_{i}}\nu_{i}$ where $\delta_{y_{i}}$ is the Dirac mass in $y_{i}$.

In this section, we describe the semi-discrete variant of the two far-field mirror problems seen in the previous section, leaving aside the problem of convergence of the solutions to the discrete problems towards those to the continuous problems. These constructions give rise to equations which can naturally be seen as discrete Monge–Ampère equations. We also propose an economic interpretation by addressing the bakeries problem.

2.1 Mirror for a point light source

Let us go back to the problem of constructing mirrors that transform the light emitted by a point light source (see Section ⁠1.1⁠). As in the previous section, the light source is modeled by a continuous probability density $\rho$ on the sphere of directions $\mathbb{S}^{2}$, whose support $X_{\rho}:=\{x\in\mathbb{S}^{2},\ \rho(x)>0\}$ corresponds to the set of directions in which light is emitted. This time we assume that the desired target illumination is described by a probability measure $\nu=\sum_{1\leq i\leq N}\delta_{y_{i}}\nu_{i}$ supported on a set $Y=\{y_{1},\ldots,y_{N}\}\subset\mathbb{S}^{2}$ of distinct directions. The problem is still to find the mirror surface $\mathcal{R}$ that will reflect the measure $\mu$ onto the measure $\nu$ under Snell’s law, but this time the target measure $\nu$ is discrete.

Mirror composed of paraboloid pieces

We use the method of supporting paraboloids proposed by Caffarelli and Oliker in 1994 [⁠1⁠ L. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics. Journal of Mathematical Sciences154, 39–49 (2008) ], which was originally developed to show the existence of weak solutions in the case where both measures are absolutely continuous. Caffarelli and Oliker’s idea is based on a well-known property of paraboloids of revolution: a paraboloid of revolution with focal point $O$ and direction $y$ reflects any ray coming from point $O$ to the direction $y$. It is thus natural to seek to construct a mirror whose surface is composed of pieces of paraboloids, each paraboloid illuminating a direction $y_{i}$.

More precisely, we take $\psi=(\psi_{1},\ldots,\psi_{N})\in\mathbb{R}^{N}$ and denote by $P(y_{i},\psi_{i})$ the solid (i.e. filled) paraboloid of direction $y_{i}$, with focal point at the origin $O$ and focal distance $\psi_{i}$. This means that $\frac{1}{2}\psi_{i}$ is the distance between $O$ and the paraboloid’s closest point to $O$. We define by $\mathcal{R}_{\psi}$ the surface bordering the intersection of the solid paraboloids $P(y_{i},\psi_{i})$:

For each $i\in\{1,\ldots,N\}$ we denote by $\mathrm{V}_{i}(\psi)$ the set of rays $x\in\mathbb{S}^{2}$ emitted by the light source and reflected by Snell’s law in the direction $y_{i}$. This set is called the $i$-th visibility cell of the mirror $\mathcal{R}_{\psi}$. By construction, it corresponds to the radial projection of $\mathcal{R}_{\psi}\cap\partial P(y_{i},\psi_{i})$ onto the sphere (see Figure ⁠2.1⁠).

A simple calculation shows that the intersection of two confocal paraboloids $\partial P(y_{i},\psi_{i})$ and $\partial P(y_{j},\psi_{j})$ is included in a plane curve. Projecting radially onto the unit sphere, this implies that the intersection of two visibility cells $\mathrm{V}_{i}(\psi)\cap\mathrm{V}_{j}(\psi)$ is included in a curve on the sphere. We deduce that the set of visibility cells forms a partition of the sphere $\mathbb{S}^{2}$, up to a set of measure zero.

The paraboloid of revolution $\partial P(y_{k},\psi_{k})$ can be parametrized radially by the function $x\in\mathbb{S}^{2}\mapsto x\rho_{k}(x)$, where $\rho_{k}(x)=\psi_{k}/(1-\langle x\,|\,y_{i}\rangle)\in\mathbb{R}$. We deduce that $x$ belongs to the visibility cell $\mathrm{V}_{i}(\psi)$ if and only if the distance $\rho_{i}(x)$ is smaller than the distances $\rho_{j}(x)$ for $j\in\{1,\ldots,N\}$. Composing with the logarithm to linearize the expression in $\psi$, we obtain an explicit expression for the visibility cells

where $c(x,y)=-\ln(1-\langle x\,|\,y\rangle)$.

By construction, each ray emitted by the point source and belonging to the cell $\mathrm{V}_{i}(\psi)$ hits the mirror $\mathcal{R}_{\psi}$ at a point which belongs to the paraboloid $\partial P(y_{i},\psi_{i})$ and which is reflected in the direction $y_{i}$. The quantity of light received in the direction $y_{i}$ is therefore exactly the quantity of light emanating from the visibility cell $\mathrm{V}_{i}(\psi)$, i.e., $\mu(\mathrm{V}_{i}(\psi))$. The desired quantity of light in the direction $y_{i}$ is $\nu_{i}$. The equation to be solved is therefore $\mu(\mathrm{V}_{i}(\psi))=\nu_{i}$ for any $i\in\{1,\ldots,N\}$. Moreover, note that a paraboloid of revolution is only determined by its focal point, its direction and its focal distance. The free parameter remaining for each paraboloid $\partial P(y_{i},\psi_{i})$ is the focal distance $\psi_{i}$.

Formulation of the problem

The semi-discrete far-field mirror problem for a point source can be formulated as the problem of finding focal distances $\psi=(\psi_{1},\ldots,\psi_{N})\in\mathbb{R}^{N}$ that satisfy

where $c(x,y)=-\ln(1-\langle x\,|\,y\rangle)$ and where

We will see in Section ⁠3⁠ how to solve such systems of equations. Note that if $\psi\in\mathbb{R}^{N}$ is a vector of focal distances solving the mirror problem for a point-like source, then the surface of the mirror is parametrized by

In the numerical experiments, we assume that the target illumination $\nu$ is included in the half-sphere $\mathbb{S}^{2}_{-}:=\{x\in\mathbb{S}^{2},\ \langle x\,|\,e_{z}\rangle\leq 0\}$, that the support $X_{\rho}$ of $\rho$ is included in the half-sphere $\mathbb{S}^{2}_{+}:=\{x\in\mathbb{S}^{2},\ \langle x\,|\,e_{z}\rangle\geq 0\}$, and that the mirror is parametrized above the domain $X_{\rho}$.

2.2 Mirror for a collimated light source

Let us now consider the mirror problem for a collimated light source, already seen in Section ⁠1.2⁠. As before, the probability measure modelling the light source has a density $\rho$ with respect to the Lebesgue measure on the plane. However the probability measure modelling the target illumination intensity is discrete $\nu=\sum_{1\leq i\leq N}\delta_{y_{i}}\nu_{i}$, supported on a finite set $Y=\{y_{1},\ldots,y_{N}\}\subset\mathbb{S}^{2}$ of distinct directions. The problem is, again, to find the surface $\mathcal{R}$ of a mirror which reflects the measure $\mu$ to the measure $\nu$.

Mirror with planar faces

We choose to construct the mirror surface $\mathcal{R}$ as the graph of affine height functions of the form $x\in\mathbb{R}^{2}\mapsto\max_{i}\langle x\,|\,p_{i}\rangle-\psi_{i}$ (see Figure ⁠2.2⁠). The vector $p_{i}$ is chosen so that the plane $P_{i}=\{(x,\langle x\,|\,p_{i}\rangle)\mid x\in\mathbb{R}^{2}\}$ reflects vertical rays, i.e., with direction $e_{z}$, into the direction $y_{i}\in\mathbb{S}^{2}$. We need to determine the heights $\psi_{i}$ of those planes. Given a family of heights $\psi\in\mathbb{R}^{N}$, we define the $i$-th visibility cell as

By construction, for each $i\in\{1,\ldots,N\}$, any vertical ray emitted from a point $x\in V_{i}(\psi)$ hits the mirror $\mathcal{R}$ at a height $\langle x\,|\,p_{i}\rangle-\psi_{i}$ and is reflected in the direction $y_{i}$. Thus, the amount of light reflected in the direction $y_{i}$ is equal to $\mu(\mathrm{V}_{i}(\psi))$.

Formulation of the problem

Solving the semi-discrete far-field mirror problem for a collimated light source amounts to finding the heights $\psi\in\mathbb{R}^{N}$ that satisfies

where $c(x,y)=-\langle x\,|\,y\rangle$ and

A solution of the equation (⁠Mir-Colli-SD⁠) induces a parametrization of the convex mirror $\mathcal{R}$ that reflects $\mu$ onto $\nu$:

In practice, we only consider the part of the mirror located above the domain $X_{\rho}:=\{x\in\mathbb{R}^{2}\times\{0\},~{}\rho(x)\neq 0\}$.

2.3 The bakeries problem

We now present an economic analogy which leads to an equation having the same structure as in the two optical problems presented above. We assume that $X$ represents a city whose population density is described by a probability density $\rho$, that the finite set $Y=\{y_{1},\ldots,y_{N}\}$ represents the locations of the city’s bakeries and that $\nu_{i}$ represents the quantity of bread available in bakery $y_{i}$. Customers living at a location $x$ in $X$ naturally will look for the bakery minimizing the cost of walking from $x$ to $y_{i}$, denoted $c(x,y_{i})$. This leads to a decomposition of the city space into Voronoi cells,

The number of customers going to a bakery $y_{i}$ is equal to the integral of the density $\rho$ over $\operatorname{Vor}_{i}$. Suppose that a bakery $y_{i}$ receives too many customers in comparison to its bread’s production capacity $\nu_{i}$ – this could be the case in Figure ⁠6⁠ for the downtown bakery $y_{1}$ where the population density is high. This means we have $\mu(\operatorname{Vor}_{1})>\nu_{1}$, where we denote $\mu(x)=\rho(x)\mathrm{d}x$. The baker $y_{1}$ will then seek to increase the price of his bread. This will reduce the number of potential customers, but will increase the baker’s profit as long as he manages to sell all his stock. We write $\nu_{i}\geq 0$ for the proportion of the population that the bakery $y_{i}$ is able to serve, and $\psi_{i}$ the price of the bread in the bakery $y_{i}$. If we assume that the customers living at point $x$ make a compromise between walking cost and price of bread by minimizing the sum ($c(x,y_{i})+\psi_{i}$), the city is then decomposed into Laguerre cells

Note that we do not necessarily have $y_{i}\in\operatorname{Lag}_{i}(\psi)$, and that it is even possible to have $\operatorname{Lag}_{i}(\psi)=\emptyset$: indeed, if the bread is very expensive in a certain bakery, even people living next door may prefer going to a more distant one.

Problem formulation

The bakeries problem therefore boils down to finding a price vector $\psi\in\mathbb{R}^{N}$ such that each bakery sells all its stock of bread $\nu_{i}$. This is described by the system of equations

This equation has exactly the same structure as (⁠Mir-Ponc-SD⁠) and (⁠Mir-Colli-SD⁠). We will see in the next section how to solve this class of equations.

3 Numerical resolution

The discrete problems mentioned in the previous section all show the same structure; our focus will now be on their numerical resolution. We start by introducing the semi-discrete Monge–Ampère equation, and show that its solution is equivalent to finding the maximum of a concave function. Subsequently, we present a Newton method that allows us to solve these equations efficiently.

3.1 Semi-discrete Monge–Ampère equation

Let $X$ be a compact subset of the space $\mathbb{R}^{2}$ or of the sphere $\mathbb{S}^{2}$, let $Y=\{y_{1},\ldots,y_{N}\}$, and let $c\in\mathcal{C}^{1}(X\times Y)$ be a cost function. The Laguerre cell (which corresponds to a visibility cell in optics) associated with a family of real numbers $\psi=(\psi_{1},\ldots,\psi_{N})\in\mathbb{R}^{N}$ is given by

Suppose that the cost function satisfies the Twist condition

which ensures that the Laguerre cells form a partition of the domain $X$ up to a negligible set.

Semi-discrete Monge–Ampère equation

Let $\mu$ be a probability measure on $X$ with density $\rho$ with respect to the area measure, and let $\nu=\sum_{i}\nu_{i}\delta_{y_{i}}$ be a probability measure on $Y$. In the following equation, the discrete probability measure $\nu$ is conflated with the vector $\nu=(\nu_{i})_{1\leq i\leq N}$. We are seeking $\psi\in\mathbb{R}^{N}$ satisfying

where the function $G:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is defined by

3.2 Variational formulation

The following theorem shows that the function $G$ in the semi-discrete Monge–Ampère equation is the gradient of a concave function.

As we will see in the next paragraph, the function $\mathcal{K}$ is related to the Kantorovitch duality in optimal transport theory, and we will therefore call it the Kantorovitch functional. Moreover, since a concave function of class $\mathcal{C}^{1}$ reaches its maximum exactly at its critical points, we obtain the following corollary:

Since the function $\mathcal{K}$ is invariant under addition of a constant, one can choose to work on the set $\mathcal{M}_{0}$ of vectors whose coordinates sum to zero. It can be shown that the function $\mathcal{K}$ is proper on $\mathcal{M}_{0}$, i.e., $\lim_{\left\|\psi\right\|\to+\infty,\psi\in\mathcal{M}_{0}}\mathcal{K}(\psi)=-\infty$, which ensures that it reaches its maximum: the problem (⁠MA⁠) thus admits a solution.

3.3 Relation to optimal transport

The variational formulation of the Monge–Ampère equation, i.e., the search for a maximizer of the Kantorovitch functional, corresponds in fact to the dual of the Monge–Kantorovitch problem in optimal transport theory. We discuss this link in detail below in the semi-discrete case. The reader interested in the proofs may refer for instance to the book chapter [⁠4⁠ Q. Mérigot and B. Thibert, Optimal transport: Discretization and algorithms. In Handbook of Numerical Analysis, Geometric PDEs 22, Elsevier, 621–679 (2020) ].

Monge’s problem

The image of a probability measure $\mu$ on $X$ under a measurable application $T:X\to Y$ is the measure $T_{\#}\mu$ on $Y$ defined by $T_{\#}\mu(B)=\mu(T^{-1}(B))$. If $T_{\#}\mu=\nu$, we say that $T$ transports $\mu$ to $\nu$. Since the set $Y$ is finite, we have $T_{\#}\mu=\sum_{1\leq i\leq N}\mu(T^{-1}(y_{i}))\delta_{y_{i}}$. Monge’s optimal transport problem consists in finding a transport map $T$ that transports $\mu$ to $\nu$ and that minimizes the total cost $\int_{X}c(x,T(x))d\mu(x)$. If the cost function $c$ satisfies the Twist condition, Brenier and Gangbo–McCann, relying on Kantorovich duality, proved the existence of a minimizer for this problem when the source $\mu$ is absolutely continuous. For example, one can state the following:

3.4 Laguerre cells and derivatives

Before applying Newton’s method to solve the equation $G(\psi)=\nu$, we need to show that the function $G$ is of class $\mathcal{C}^{1}$ (or equivalently that $\mathcal{K}$ is of class $\mathcal{C}^{2}$), calculate its partial derivatives and study the (strict) concavity of $\mathcal{K}$. To do this, we need a genericity assumption which is somewhat technical, but which is natural and not restrictive in practice. In the optical cases mentioned in this paper, this assumption is satisfied if the intersection of three distinct Laguerre cells is finite and if the intersection of two Laguerre cells with the boundary of $X$ is also finite. For more details, the reader may refer to the book chapter [⁠4⁠ Q. Mérigot and B. Thibert, Optimal transport: Discretization and algorithms. In Handbook of Numerical Analysis, Geometric PDEs 22, Elsevier, 621–679 (2020) ].

The formula for the partial derivatives of $G$ has a geometric interpretation. In the following two figures, which are obtained for the cost $c(x,y)=\left\|x-y\right\|^{2}$ on $\mathbb{R}^{2}$, we explain why the formula for partial derivatives involves integrals over the interfaces between Laguerre cells and how the singularities of $DG$ may occur depending on the geometry of the points $y_{i}$.

Figure ⁠3.4⁠ illustrates that the partial derivative $\partial G_{i}/\partial\psi_{j}(\psi)$ is an integral over the interface $\operatorname{Lag}_{ij}(\psi)$: the value $G_{i}(\psi)$ is an integral over the Laguerre cell $\operatorname{Lag}_{i}(\psi)$ (in grey on the left); we increase the value $\psi_{j}$ by $\varepsilon>0$ considering $\psi+\varepsilon e_{j}$; the rate of increase $(G_{i}(\psi)-G_{i}(\psi+\varepsilon e_{j}))/\varepsilon$ is proportional to an integral over the symmetric difference between two Laguerre cells (in grey in the middle); passing to the limit we obtain an integral over the green segment $\operatorname{Lag}_{ij}(\psi)$. The signs that occurs in the formula for the partial derivatives can also be interpreted with the bakeries metaphor: when the price of bread $\psi_{i}$ increases, the number of customers of the bakery $y_{i}$ decreases (i.e., the Laguerre cell $\operatorname{Lag}_{i}(\psi)$ shrinks) and the number of customers for the other bakeries increases, so that $\partial G_{i}/\partial\psi_{i}(\psi)\leq 0$ and $\partial G_{i}/\partial\psi_{j}(\psi)\geq 0$ for $j\neq i$.

In Figure ⁠3.4⁠, the genericity condition is not satisfied because $y_{1}$, $y_{2}$ and $y_{3}$ are aligned, and there exists $\psi\in\mathbb{R}^{N}$ for which $\operatorname{Lag}_{1}(\psi)\cap\operatorname{Lag}_{2}(\psi)\cap\operatorname{Lag}_{3}(\psi)$ is a line segment. The partial derivative $\partial G_{2}/\partial\psi_{3}(\psi)$ is an integral on the (green) segment $\operatorname{Lag}_{23}(\psi)$. If we simultaneously decrease $\psi_{1}$ and $\psi_{2}$ by the same amount, we can see that the segment $\operatorname{Lag}_{23}(\psi)$ varies continuously and then suddenly becomes empty when the cell $\operatorname{Lag}_{2}(\psi)$ gets empty (bottom right of Figure ⁠3.4⁠). Thus, $\partial G_{2}/\partial\psi_{3}(\psi)$ is not continuous. Newton’s method requires a certain regularity, and we will see below that it converges under the above genericity assumptions.

To establish the convergence of Newton’s method, we also need to study the concavity of the Kantorovitch functional $\mathcal{K}$, or equivalently the monotonicity of its gradient $\nabla\mathcal{K}=G-\nu$. The functions $\mathcal{K}$ and $G$ are invariant by addition of a constant vector (i.e., $\mathcal{K}(\psi+C(1,\ldots,1))=\mathcal{K}(\psi)$), which can be seen in the definition of Laguerre cells. Thus, we can only hope to establish strong concavity of $\mathcal{K}$ in the directions orthogonal to constant vectors, i.e., belonging to the set

Another reason for the lack of strong concavity of $\mathcal{K}$ is that if $\psi_{i}$ is very large, then $\operatorname{Lag}_{i}(\psi)$ is empty and remains empty in a neighborhood of $\psi$. In this case, $G_{i}(\psi)$ is constant equal to zero, and the Hessian matrix $D^{2}G(\psi)=DG$ has a row of zeros. We can therefore hope to establish strong concavity only if $\psi$ belongs to the set

The next theorem shows, in a nutshell, that these are the only two obstructions to strong concavity.

We will see in the next paragraph how these results of regularity and monotonicity allow us to iteratively construct a sequence $(\psi^{(k)})_{k\geq 0}$ converging to the unique zero-average $\psi^{*}$ satisfying $G(\psi^{*})=\nu$.

3.5 Newton’s method

Newton’s method in 1D

We begin by recalling Newton’s method for solving the equation $g(x)=0$, where $g:\mathbb{R}\to\mathbb{R}$ is a real function. Newton’s method starts from $x_{0}\in\mathbb{R}$ and constructs the sequence $x^{k+1}=x^{k}-g(x^{k})/g^{\prime}(x^{k})$ by induction. If we assume that $g$ is of class $\mathcal{C}^{1}$ and that there exists $a\in\mathbb{R}$ such that $g(a)=0$ and $g^{\prime}(a)\neq 0$, then one can show, using Taylor–Lagrange formulas, that for $x^{0}$ sufficiently close to $a$, the sequence $(x^{k})_{k\geq 0}$ converges to $a$. The convergence is then said to be local. Thus, under a regularity hypothesis ($g\in\mathcal{C}^{1}$) and monotonicity ($g^{\prime}$ has constant sign in a neighborhood of $a$), Newton’s method converges locally.

Newton’s method (local)

Assume that we are given a zero-average vector $\psi^{0}\in\mathcal{M}_{0}$ such that the mass of all Laguerre cells is strictly positive:

We define $\psi^{k+1}$ in the following way: we start by calculating the Newton direction $d^{k}$, i.e., the vector $d^{k}$ satisfying

which exists and is unique by according to Theorem ⁠3.5⁠. The second equation enables us to overcome the invariance of $G$ and thus the non-invertibility of $DG(\psi^{k})$. We then define $\psi^{k+1}=\psi^{k}+d^{k}$. As in the 1D case, it can be shown that the method converges locally: if $\psi^{0}$ is chosen close enough to the $\psi^{*}$ solution, then the sequence $(\psi^{k})$ converges to $\psi^{*}$.