Convergence of a damped Newton's method for discrete Monge-Ampere functions with a prescribed asymptotic cone

11/01/2019
by   Gerard Awanou, et al.
0

We prove the convergence of a damped Newton's method for the nonlinear system resulting from a discretization of the second boundary value problem for the Monge-Ampere equation. The boundary condition is enforced through the use of the notion of asymptotic cone. For the differential operator we use a discretization which is based on a partial discrete analogue of the subdifferential.

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1. Introduction

In this paper we prove the convergence of a damped Newton’s method for a finite difference approximation of the second boundary value problem for Monge-Ampère type equations. The method was introduced in [1] as an improvement to the one introduced in [6], and has the particularity among available discretizations that the discrete problem has a unique solution. The damped Newton’s method allows the use of an initial guess which may be far from the solution of the discrete problem. We establish the global convergence of the algorithm for the discretization proposed in [1]. The convergence of the damped Newton’s method for the discretization proposed by Benamou and Duval in [6] is open.

Monge-Ampère type equations with the second boundary value condition arise in geometric optics and optimal transport. In [12] a discretization of the Dirichlet problem which is based on a partial discrete analogue of the subdifferential was proposed. We consider in this paper the generalization to the second boundary value condition proposed in [1]. The approach in [1] is to interpret the boundary condition as the prescription of the asymptotic cone of the convex solution of the Monge-Ampère equation. The convergence analysis of the damped Newton’s method given here generalizes the one given in [12] and is similar in some aspects to the convergence analysis of a damped Newton’s method for a dual problem given in [10].

We organize the paper as follows. In the next section, we start with some preliminaries and recall the discretization introduced in [1] and its properties. The damped Newton’s method is introduced in section 3 and in the last section we give its convergence analysis for our discretization.

2. Preliminaries

Let be a bounded convex domain of and let be a bounded convex polygonal domain of . Let be an integrable function on and a locally integrable function on . We assume that on and that the compatibility condition

(2.1)

holds.

The coordinates of a vector

are said to be co-prime if their great common divisor is equal to 1. A subset of is symmetric with respect to the origin if .

Let be a (finite) subset of of vectors with co-prime coordinates which span and which is symmetric with respect to the origin. Furthermore, assume that contains the elements of the canonical basis of and that contains a normal to each side of the target polygonal domain .

We denote by the boundary vertices of . To the polygonal domain , we associate a cone defined as follows. For each one associates the half-space . The cone is the intersection of the half-spaces , i.e.

Let be a small positive parameter and let denote the orthogonal lattice with mesh length with an offset . The offset may make it easier to choose the decomposition of the domain used for the discrete Monge-Ampère equation (2.3) below. Put and denote by the canonical basis of . Let

The unknown in the discrete scheme is a mesh function (not necessarily convex) on which is extended to using the extension formula

(2.2)

We will refer to such mesh functions as having asymptotic cone . We define for a function on , and

We are particularly interested in those mesh functions with asymptotic cone which are discrete convex in the sense that for all and . We denote by the cone of discrete convex mesh functions.

For and such that , we define

A priori, is multi-valued. We assume that for the implementation a unique choice is made for pairs such that . If , we put .

We consider the following analogue of the subdifferential of a function. For we defined

and consider the following discrete version of the R-curvature which may be referred to as the asymmetrical version of the discrete R-curvature:

In [1], we used a symmetric version of the subdifferential. However, that seemed to require extra assumptions on the domain for the proof of uniqueness and extra assumptions on the target density for the existence of a solution. While a symmetrization of the subdifferential appeared to be inevitable for the Dirichlet problem [12], it is clearly not necessary for the second boundary value problem. This gives rise to the question of what is the proper analogue of the lattice basis reduction scheme used in [5] for the 2D problem. We wish to discuss this in a separate work.

We can now describe our discretization of the second boundary value problem: find with asymptotic cone such that

(2.3)

where form a partition of , i.e. , and is a set of measure 0 for . In the interior of one may choose as the cube centered at with . The requirement that the sets form a partition is essential to assure the mass conservation (2.1) at the discrete level, i.e.

(2.4)

The unknowns in the above equation are the mesh values . For , the value needed for the evaluation of is obtained from the extension formula (2.2). Existence, uniqueness and stability of solutions were given in [1] for a symmetric version of the set function . We outline the corresponding proofs in the next lemmas for the set function .

Recall that the support function of the closed convex set is given by

Lemma 2.1.

Assume that in and let be a solution of (2.3). Then for integers and such that and and are in

(2.5)

Moreover

(2.6)

for and for a constant independent of and .

Proof.

Since in , for , and hence is a set with a non zero Lebesgue measure. For and , we have and

This implies that and hence for all . The remaining part of the proof is identical to the one of [1, Lemma 4.1]. ∎

Lemma 2.2.

Assume that in and let be a solution of (2.3). Then for , .

Proof.

With in Lemma 2.1, we obtain for

Thus, for , . So . Since for and contains a normal to each face of , we conclude that . ∎

It follows from Lemma 2.1 that for in , solutions of (2.3) with for an arbitrary number and , are bounded independently of . From Lemma 2.2, as in the proof of [1, Theorem 4.5], solutions of (2.3) are unique up to an additive constant. Finally as stated in [1] existence of a solution and the convergence of the discretization are established along the same lines as in [1]. In fact, owing to the following lemma, the proof of existence of a solution to (2.3) is identical to the one for the case of convex polygonal approximations [4, Theorem 17.2].

Lemma 2.3.

For , is contained in a set of measure zero.

Proof.

For we define and the partial discrete Legendre transform of as the function defined by

Let . We have . For , . Hence both and are in the subdifferential of . As a supremum of affine functions, is convex and hence is differentiable almost everywhere, c.f. [9, Lemma 1.1.8]. Thus if were differentiable at we would have . Impossible. Thus is contained in the set of points at which is not differentiable, a set of measure zero. ∎

Remark 2.4.

It follows from Lemmas 2.2 and 2.3 that if solves (2.3) or more generally, if is a discrete convex mesh function with asymptotic cone we have and thus where we used (2.4). Therefore, since on , up to a set of measure 0, we have .

3. The damped Newton’s method

We first give a general convergence result, the assumptions of which are then verified in the next section for our discretization. We are interested in the zeros of a mapping with . We let denote a norm in and put

for a parameter to be specified later. We assume that for all . The current iterate is denoted and the following iterate is sought along the path

We make the assumption that there exists such that for all . Let and choose , e.g. .

1:Choose and set
2:If stop
3:Let be the smallest non-negative integer such that and
Set
4: and go to 2
Algorithm 1 A damped Newton’s method

The general convergence result for damped Newton’s methods is analogous to [10, Proposition 6.1]

where maps with values probability measures are considered. Therein, the map

is assumed to be in . For to be merely , as in certain geometric optics problems, and with for all , one has linear convergence [8]. For completeness we adapt the proof of [8] to the case where the domain of the mapping is an open set of . As with [12, Proposition 2.10] we will assume that the mapping is proper, i.e. the preimage of any compact set is a compact set.

Theorem 3.1.

Let and assume that is a proper map with a unique zero in and for all . Assume that there exists in such that for all . Then the iterate of the damped Newton’s method is well defined, converges to and

for if for and sufficiently large. For sufficiently large, with step 3 replaced with , i.e. a full Newton’s step, the convergence is guaranteed to be at least linear.

Proof.

Since , we can find a constant (depending on ) such that

Part 1: The damped Newton’s method is well defined. We first note that it follows from the definitions that for all we have

(3.1)

Assume that . We claim that there exists such that

If such a does not exist, there would exist a sequence converging to 0 such that

(3.2)

Since is , c.f. for example [14, $ 3.2.10], we have

Thus, by (3.1), we have

(3.3)

and by (3.2)

Taking the limit as , we obtain , a contradiction as .

We now show that there exists , with such that for all when .

If for some it must be that at some time . Thus for some we have . Let us assume that is chosen so that for all and all .

We have

since as by assumption. Therefore, by the mean value theorem, c.f. for example [14, $ 3.2.10],

We conclude that . And by construction for all .

Part 2: The sequence converges to . Let

Since is proper, is compact. By construction and for all . Thus up to a subsequence, the sequence converges to and since the sequence is strictly decreasing, there exists such that .

If , then and since has a unique zero in , the whole sequence converges to .

Now we show that it is not possible to have . Assume that .

If the sequence of Step 3 of the algorithm is bounded, there would exist a constant such that for all . This would imply that and , i.e. . We therefore have and consequently . Put

By definition of and (3.3), we have

Recall that the sequence is bounded. Moreover and thus . Since it follows that for sufficiently large for a constant , c.f. [14, $ 2.3.3]. We conclude that

This implies that , a contradiction.

Part 3: We first prove the rate , for if for and sufficiently large. We have with

Moreover . Thus . Since is uniformly bounded for sufficiently large, the claim follows when .

The linear convergence of Newton’s method for mappings is classical but is nethertheless reviewed here. Let now and put .

Let such that for we have . If necessary by taking smaller, we may assume that for and we have using the local uniform continuity of on . We have

from which we get . This shows that when we also have and the linear convergence rate. ∎

4. Convergence of the damped Newton’s method for the discretization

Let denote the cardinality of and denote the points of by .

The set of discrete convex mesh functions on which satisfy (2.2), and with for an arbitrary number and , can be identified with a subset of . We consider a map defined by

Note that for , we have and this value is used for computing in the expression of . Also, by (2.4) we have

Hence

when . In other words, the above equation is automatically satisfied when .

We now make the assumption that

We consider the set of mesh functions with , extended to using (2.2) and which satisfy and for all . We have . Our iterates will be shown to be in with an accumulation point which satisfies . This implies that as well. We then have . Therefore is a set of non zero measure. In particular, it is non empty. There exists such that for each direction , as is symmetric with respect to the origin, . Thus as well and the accumulation point is the unique solution of (2.3).

Define as in the previous section . The set is non empty since the solution of (2.3) with solves for all . We note that for we have for all . Hence for all and .

Theorem 4.1.

Assume that is differentiable at and . There exists in such that for all .

Proof.

Since , for all and . Let and let . We note that if , then . Indeed, for , when

If , we have for some integer and . We have