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The second boundary value problem for a discrete Monge-Ampere equation with symmetrization

10/31/2019
by   Gerard Awanou, et al.
0

In this work we propose a natural discretization of the second boundary condition for the Monge-Ampere equation of geometric optics and optimal transport. For the discretization of the differential operator, we use a recently proposed scheme which is based on a partial discrete analogue of a symmetrization of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.

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1. Introduction

In this paper we propose a natural discretization of the second boundary condition for the Monge-Ampère equation. Let and be bounded convex domains of . Let be an integrable function on and a locally integrable function on . We are interested in discrete approximations of convex weak solutions in the sense of Aleksandrov of the model problem

(1.1)

where denotes the subdifferential of the function .

Crucial to our analysis is the point of view that the unknown in (1.1) is a function defined on whose behavior at infinity is prescribed by the second boundary condition. In other words, the equation gives the asymptotic cone of the convex body prescribed by the convex function on . We review the notion of asymptotic cone in section 2. We approximate by convex closed polyhedra and give an explicit formula for the extension of a mesh function on which guarantees that the latter has an asymptotic cone associated with so that . One then only need to apply the discrete Monge-Ampère operator in this class of mesh functions, c.f. (3.11) below. It was thought [22, p. 24] that ”dealing with an asymptotic cone as the boundary condition is inconvenient”.

In this paper we consider Cartesian grids and a generalization of the discretization of the Monge-Ampère operator proposed by Mirebeau [20] for the Dirichlet problem. The left hand side of (1.1) is to be interpreted as the density of a measure associated to the convex function and the mapping c.f. section 3.1. It is defined through the subdifferential of . We define a discrete analogue based on a symmetrization of a discrete version of the subdifferential. There is no explicit approximation of the gradient in our scheme.

Equations of the type (1.1) appear for example in optimal transport and geometric optics. While there have been previous numerical simulations of the second boundary value problem (1.1), c.f. [13, 10, 25, 19], advances on theoretical guarantees are very recent [19, 9, 17]. The approach in [19, 17] is to enforce the constraint at the discrete level at all mesh points of the computational domain. Open questions include uniqueness of solutions to the discrete problem obtained in [17] and existence of a solution to the discrete problem obtained in [19] for a target density only assumed to be locally integrable.

Our work is closer to the one by Benamou and Duval [9] who proposed a convergence analysis based on the notion of minimal Brenier solution. Our analysis relies exclusively on the notion of Aleksandrov solution with guarantees on existence and uniqueness of a solution to the discrete problem. Unlike the approaches in [19, 9, 17], we do not use a discretization of the gradient in the first equation of (1.1). Perhaps the main difference of this work with [9] is that we do not view the second boundary condition as an equation to be discretized. Analogous to methods based on power diagrams [14, 18], the unknown is sought as a function over only the domain with the second boundary condition enforced implicitly. That feature is lacking in [9] leading to an artificial treatment of the nonlinear discrete problem.

The discrete 2D Monge-Ampère operator based on lattice basis reduction used in [9] and our discrete Monge-Ampère operator , are both consistent for (a class of) quadratic polynomials in the case with at a mesh point .

We note that the approaches in [14, 18] for example which are related to the Oliker-Prussner discretization [23], do not discretize the first equation in (1.1) but solve a related optimal transport problem which may not be available for some generalizations of (1.1) which appear in geometric optics.

The symmetrization of a discrete version of the subdifferential, c.f. section 3.2, is not necessary for the second boundary condition. For uniqueness of a solution to the discrete problem, the symmetrization requires us to assume that , the Minkowski sum of and , is contained in . This holds for example when , an assumption fulfilled in all numerical experiments in [9].

The uniqueness of a solution of the discrete problem is important for the use of globally convergent Newton’s methods. The assumption can be easily avoided with the use of the non symmetric discrete version of the subdifferential [5]. This leads to a new closely related scheme based on lattice basis reduction method which we consider in [2]. We believe that the discretizations of the Monge-Ampère operator based on an integration on a set related to the subdifferential, such as the one analyzed in this paper, are fundamental in the sense that they allow a natural geometric proof of uniqueness for the second boundary value problem. Uniqueness for other schemes, such as the one we consider in [5, 2], can then be obtained from the former through a perturbation argument.

Existence of a solution and the convergence of the discretization are established for constant densities and when is homogeneous of degree , with even. The arguments rely on a comparison between set functions associated with the symmetric and non symmetric discrete subdifferentials. Again, these restrictions seem due to the symmetrization. We choose to present the arguments for the symmetric version because of its connection, mentioned above, with the numerical experiments in [9]. Convergence of the discretization does not assume any regularity on solutions of (1.1).

The implementation of the method we introduce in the case may require a numerical integration, the effect of which we study in [3]. Obviously the discretization of the second boundary value problem proposed in this paper can also be applied to the Oliker-Prussner discretization of the Monge-Ampère operator [23]. We wish to discuss this case as well as numerical experiments in a separate work [4].

We organize the paper as follows: In the next section we review the notion of asymptotic cone of convex bodies. This leads to the extension formula. In section 3 we introduce some notation and recall the interpretation of (1.1) as [22] ” the second boundary value problem for Monge-Ampère equations arising in the geometry of convex hypersurfaces [8] and mappings with a convex potential [11].” We then describe the numerical scheme. Existence, uniqueness and stability of solutions are given in section 4. In section 5 we give several convergence results for the approximations.

2. Asymptotic cone of convex bodies

The purpose of this section is to review the geometric notion of asymptotic cone and give an analytical formula, with a geometric interpretation, for the extension to of a convex function on a polygon , in such a way that it has a prescribed behavior at infinity, i.e. asymptotic cone a polygon .

Let be a convex function on . For , the normal image of the point (with respect to ) or the subdifferential of at is defined as

For , the local normal image of the point (with respect to ) is defined as

Since we have assumed that is convex and is convex, the local normal image and the normal image coincide for [15, Exercise 1].

2.1. Cones

Following [8], denote by the Euclidean space. We will use the notation

for the vector with initial point

and endpoint . We will often identify with the Euclidean space . In that case, for , denote the vector with initial point and endpoint .

Let be a line in , be some point of , and be a direction vector of . The sets

and

are the rays of with vertex . Any convex set consisting of the union of rays with a common vertex is called a convex cone. The common vertex of all these rays is called the vertex of this cone. Formally

Definition 2.1.

A set is a cone with vertex if for , we have for all .

Let be a set. We denote by the set of points lying on the rays starting from the point and contained in . If there are no such rays, we set . We say that a set is a parallel translation of if for some direction . It is known that when is convex, is a convex cone independent of the point (up to a parallel translation) called asymptotic cone of the convex set [8, Theorem 1.8 and Corollary 1]. Formally

Definition 2.2.

The asymptotic cone of the set is defined for as

We recall the following equivalent characterization of the asymptotic cone [1].

Lemma 2.3.

Let be a closed set and . The following two statements are equivalent

  1. and such that .

Proof.

Assume that and let . Then and .

Conversely suppose and is such that . Put . Then and . Let and choose sufficiently large such that . Since is convex

is in and hence its limit is in as is closed. ∎

Let be a convex polygon with vertices . Define for the convex function

Recall that the epigraph of is the convex set

Lemma 2.4.

The epigraph of is a convex cone in with vertex and hence is equal to its asymptotic cone. Furthermore, the epigraph of can be obtained from the one of by a parallel translation.

Proof.

As the maximum of convex functions, is a convex function and hence is a convex set.

We now show that is a cone with vertex . Let . We show that . Since we have

It follows that

From the definition of we have which proves the claim.

Next, we show that . It is enough to show that if and only if . But this follows immediately from the definition.

Finally it is immediate that a cone is equal to its asymptotic cone since by definition and moreover, by definition of a cone, . ∎

To the convex polygon we associate the cone . Recall that .

2.2. Unbounded polyhedra

Following [8], the points are in general position if the vectors are linearly independent.

A convex closed set M in is called a -convex body, if contains points in general position, but does not contain points in general position.

A -convex body in is called a -convex solid polyhedron, if is the intersection of a finite number of closed half spaces.

A -convex polyhedron is the boundary of a -convex solid polyhedron in . Every -convex polyhedron can be decomposed into the finite union of -solid convex polyhedra (lying in ). These -solid convex polyhedra are called -faces of the convex polyhedron .

Applying this process of decreasing dimensions of the faces of we finally obtain the zero-faces of which are called the vertices of or the vertices of the corresponding -solid convex polyhedron such that .

We call polyhedral angle a convex body formed by three or more planes intersecting at a common point, called the vertex of the angle. The asymptotic cone of an unbounded or infinite solid convex polyhedron is a polyhedral angle.

Theorem 2.5.

[8, Theorem 4.2] Every solid infinite convex polyhedron is the convex hull of its vertices and its asymptotic convex polyhedral angle, which is placed at one of its vertices.

2.3. Minkowski sum and sweeping

Let and be two subsets of and put for

The Minkowski sum of and is defined to be . Then we have

We say that the sum is obtained by sweeping the set over ,

(2.1)

2.4. Convex extensions

Let be a set of points in and assume that the convex hull of these points is a domain with , on the lower part of its boundary. We recall that a point is on the lower part of the boundary of if for all . Let be a polyhedral angle with vertex at and consider the infinite convex solid polyhedron which is the convex hull of and the points . We denote by the convex polyhedron which is the boundary of . It defines a piecewise linear convex function on such that

Our goal is to determine for a formula for . Let us assume that the polyhedral angle has boundary given by the graph of the function

for given vectors .

Lemma 2.6.

Let S denote the convex hull of the points of . And let denote the polyhedral angle with boundary given by the graph of . Then the closure of the convex hull of and the polyhedral angle is given by .

Proof.

Let and put . There exists points for an integer and scalars such that

Since is convex and the origin , . On the other hand . Thus . Since is closed bounded and closed, is closed. Thus .

Let now , i.e. with and . We have . Let and consider the point

The point is a convex combination of a point in and a point in . Thus . As , . This proves that . The proof is complete. ∎

Theorem 2.7.

Let S denote the convex hull of the points of . Assume that the projection on of the lower part of has vertices . Let denote the polyhedral angle with boundary given by the graph of . The convex hull of and the polyhedral angle defines a piecewise linear convex function which is given for by

Proof.

Put for the translate of by . Note that if and only if or .

We have by (2.1)

We conclude that

(2.2)

We need a representation formula for which involves only the points . By construction for , we have . In other words, there exists and such that . If , then

(2.3)

Let and assume that . Then there is a subset such that

In particular, for , the plane is a proper face of the polyhedral angle . We show that . Note that is closed. For a given integer , consider the closed set

If , we choose closest to and . Since is finite, for some integer , we have . Let now .

On , will be given by a face of . That is, on , we have

A priori, as , we have on , , . But as well, so that

Since and using the above representation of on we obtain

and hence by (2.3)

Therefore . We conclude that for

(2.4)

Next, we observe that if with , and linear on the line segment joining to , then

It follows that for

Otherwise, if for example, we would have . In other words, the minimum of the linear function for on the line segment is reached at an endpoint.

Next, we note that the piecewise linear function induces a simplicial decomposition of with linear on each simplex. If is in a simplex such that and we consider a line segment through with endpoints on opposite faces of ,

Continuing this process with simplices of decreasing dimension, we obtain

that is, the minimum is reached at a vertex on the boundary of . The proof is complete. ∎

Theorem 2.8.

Let S denote the convex hull of the points of . And let denote the polyhedral angle with boundary given by the graph of . Then the closure of the convex hull of and the polyhedral angle has asymptotic cone .

Proof.

We prove that .

We first note that if and , then . Indeed if , then there is a direction such that . Since we have .

Let now . Put . Let such that for some and . We show that .

By Lemma 2.3 there exists a sequence and sequences and such that . But is compact and so we may assume that the sequence converges to . This implies that and hence . By Lemma 2.3 again, , where is the origin of . It follows that .

Theorems 2.7 and 2.8 provide the formula for the extension of a convex function, defined by the lower part of the convex hull of a finite set of points, to have a given asymptotic cone.

2.5. Minimal convex extensions

The formula for given by Theorem 2.7 can be interpreted as a minimal convex extension in some sense or a special form of infimal convolution [9, (15)]. Such an extension of a function is given by

In our formula, based on geometric arguments, the supremum is restricted to boundary vertices of and the infimum taken over boundary vertices of in the case that is polygonal.

Another extension formula used in [12, p. 157] is given by

The above formula for was used to make sure that is well defined on . Its discrete version would require a discretization of the gradient.

3. The discrete scheme

In this section, we introduce some notation and recall the interpretation of (1.1) as the second boundary value problem for Monge-Ampère equations arising in the geometry of convex hypersurfaces. We then recall discrete versions of the notion of subdifferential and describe the numerical scheme. We now assume that on .

3.1. R-curvature of convex functions

The presentation of the R-curvature of convex functions given here is essentially taken from [8] to which we refer for further details. Let be a convex function on .

For any subset , the normal image of (with respect to ) is defined as

It can be shown that is Lebesgue measurable when is also Lebesgue measurable. The R-curvature of the convex function is defined as the set function

which can be shown to be a measure on the set of Borel subsets of . For an integrable function on and extended by 0 to , equation (1.1) is the equation in measures

(3.1)

This implies the compatibility condition

(3.2)

Associated to the domain and a point in is a cone whose boundary defines a function on with the property that , [22, p. 22]. For each one associates the half-space . The cone is the intersection of the half-spaces , i.e.

(3.3)

Note that in the case is the convex hull of , depends only on . Indeed if , then for all implies .

Theorem 3.1.

[8] Assume that (3.2) holds. There exists a convex function on with asymptotic cone such that

Such a function is unique up to an additive constant.

Corollary 3.2.

[22, p. 23] The function given by Theorem 3.1 satisfies .

Theorem 3.1 and Corollary 3.2 give existence of a convex solution on which solve (3.1). Its unicity up to a constant follows from Theorem 3.1 and the following lemma.

Lemma 3.3.

Let be a function on such that . Then has asymptotic cone .

Proof.

Let denote the epigraph of and assume that .

We first prove that . Let and put . We show that for all , . Assume by contradiction that this does not hold. Let be the point of intersection with of the line through and with direction . The half line is then not contained in . Choose and put

By construction . Now let . We know that the plane

is a supporting hyperplane to

at with and on opposite sides. Thus

Since and , we obtain . By assumption . Since , by (3.3) we have . Contradiction.

Next, we prove that . Let . The half-line with . That is for all .

For each we can find such that is a supporting hyperplane to at . Thus

This gives . Taking we obtain for all . Thus and the proof is complete. ∎

3.2. Discretizations of the R-curvature

Let be a small positive parameter and let denote the orthogonal lattice with mesh length . Put and denote by the canonical basis of . Let