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

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 $Ω$ be a bounded convex domain of $R^{d}$ and let $Ω^{*}$ be a bounded convex polygonal domain of $R^{d}$ . Let $f \geq 0$ be an integrable function on $Ω$ and $R > 0$ a locally integrable function on $Ω^{*}$ . We assume that $R = 0$ on $R^{d} ∖ Ω$ and that the compatibility condition

(2.1)

\int_{Ω} f (x) d x = \int_{Ω^{*}} R (p) d p,

holds.

The coordinates of a vector

e \in Z^{d}

are said to be co-prime if their great common divisor is equal to 1. A subset

W

Z^{d}

is symmetric with respect to the origin if

\forall y \in W, - y \in W

Let $V$ be a (finite) subset of $Z^{d} ∖ {0}$ of vectors with co-prime coordinates which span $R^{d}$ and which is symmetric with respect to the origin. Furthermore, assume that $V$ contains the elements of the canonical basis of $R^{d}$ and that $V$ contains a normal to each side of the target polygonal domain $Ω^{*}$ .

We denote by $a_{j}^{*}, j = 1, \dots, N$ the boundary vertices of $Ω^{*}$ . To the polygonal domain $K^{*}$ , we associate a cone $K_{Ω^{*}}$ defined as follows. For each $p \in^{*}$ one associates the half-space $H (p) = {(x, z), z \geq p \cdot x}$ . The cone $K_{Ω^{*}}$ is the intersection of the half-spaces $H (p), p \in^{*}$ , i.e.

K_{Ω^{*}} = \cap_{p \in^{*}} H (p) .

Let $h$ be a small positive parameter and let $Z_{h}^{d} = a + {m h, m \in Z^{d}}$ denote the orthogonal lattice with mesh length $h$ with an offset $a \in Z^{d}$ . The offset $a$ may make it easier to choose the decomposition of the domain used for the discrete Monge-Ampère equation (2.3) below. Put $Ω_{h} = Ω \cap Z_{h}^{d}$ and denote by $(r_{1}, \dots, r_{d})$ the canonical basis of $R^{d}$ . Let

\partial Ω_{h} = {x \in Ω_{h} such that for some i = 1, \dots, d, x + h r_{i} \notin Ω_{h} or x - h r_{i} \notin Ω_{h}} .

The unknown in the discrete scheme is a mesh function (not necessarily convex) on $Ω_{h}$ which is extended to $Z_{h}^{d}$ using the extension formula

(2.2)

v_{h} (x) = min y \in \partial Ω_{h} max 1 \leq j \leq N (x - y) \cdot a_{j}^{*} + v_{h} (y) .

We will refer to such mesh functions as having asymptotic cone $K_{Ω^{*}}$ . We define for a function $v_{h}$ on $Z_{h}^{d}$ , $e \in Z^{d}$ and $x \in Ω_{h}$

Δ_{h e} v_{h} (x) = v_{h} (x + h e) - 2 v_{h} (x) + v_{h} (x - h e) .

We are particularly interested in those mesh functions $v_{h}$ with asymptotic cone $K_{Ω^{*}}$ which are discrete convex in the sense that $Δ_{h e} v_{h} (x) \geq 0$ for all $x \in Ω_{h}$ and $e \in V$ . We denote by $C_{h}$ the cone of discrete convex mesh functions.

For $x \in Ω_{h}$ and $e \in V$ such that $x + h e \notin Ω_{h}$ , we define

Γ (x + h e) = arg min y \in \partial Ω_{h} max 1 \leq j \leq N (x - y) \cdot a_{j}^{*} + v_{h} (y) .

A priori, $Γ (x + h e)$ is multi-valued. We assume that for the implementation a unique choice is made for pairs $(x, e)$ such that $x + h e \notin Ω_{h}$ . If $x + h e \in Ω_{h}$ , we put $Γ (x + h e) = x + h e$ .

We consider the following analogue of the subdifferential of a function. For $y \in Z_{h}^{d}$ we defined

\partial_{V} v_{h} (y) = {p \in R^{d}, p \cdot (h e) \geq v_{h} (y) - v_{h} (y - h e) \forall e \in V},

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

ω_{a} (R, v_{h}, E) = \int_{\partial_{V} v_{h} (E)} R (p) d p .

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 $R$ 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 $u_{h} \in C_{h}$ with asymptotic cone $K_{Ω^{*}}$ such that

(2.3)

\begin{matrix} ω_{a} (R, u_{h}, {x}) & = \int_{C_{x}} f (t) d t, x \in Ω_{h}, \end{matrix}

where $(C_{x})_{x \in Ω_{h}}$ form a partition of $Ω$ , i.e. $C_{x} \cap Ω_{h} = {x}, \cup_{x \in Ω_{h}} C_{x} = Ω$ , and $C_{x} \cap C_{y}$ is a set of measure 0 for $x \neq y$ . In the interior of $Ω$ one may choose as $C_{x} = x + [- h / 2, h / 2]^{d}$ the cube centered at $x$ with $E_{x} \cap Ω_{h} = {x}$ . The requirement that the sets $C_{x}$ form a partition is essential to assure the mass conservation (2.1) at the discrete level, i.e.

(2.4)

\sum x \in Ω_{h} ω_{a} (R, u_{h}, {x}) = \sum x \in Ω_{h} \int_{C_{x}} f (t) d t, x \in Ω_{h} = \int_{Ω} f (t) d t = \int_{Ω^{*}} R (p) d p .

The unknowns in the above equation are the mesh values $u_{h} (x), x \in Ω_{h}$ . For $x \notin Ω_{h}$ , the value $u_{h} (x)$ needed for the evaluation of $\partial_{V} v_{h} (x)$ 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 $ω_{a} (R, u_{h}, .)$ . We outline the corresponding proofs in the next lemmas for the set function $ω_{a} (R, u_{h}, .)$ .

Recall that the support function $σ_{Y}$ of the closed convex set $Y$ is given by

σ_{Y} (p) = sup z \in Y p \cdot z .

Lemma 2.1.

Assume that $f > 0$ in $Ω$ and let $v_{h}$ be a solution of (2.3). Then for integers $k$ and $l$ such that $k \geq l$ and $x + k h e$ and $x + l h e$ are in $Ω_{h}$

(2.5)

- σ_{Ω^{*}} (- h e) \leq v_{h} (x + l h e) - v_{h} (x + (l - 1) h e) < v_{h} (x + (k + 1) h e) - v_{h} (x + k h e) \leq σ_{Ω^{*}} (h e) .

Moreover

(2.6)

| v_{h} (x) - v_{h} (y) | \leq C | | x - y | |_{1},

for $x, y \in ¯ ¯¯ ¯ Ω \cap Z_{h}^{d}$ and for a constant $C$ independent of $h$ and $v_{h}$ .

Proof.

Since $f > 0$ in $Ω$ , for $x \in Ω_{h}$ , $ω_{a} (R, u_{h}, {x}) > 0$ and hence $\partial_{V} v_{h} (x)$ is a set with a non zero Lebesgue measure. For $p \in \partial_{V} v_{h} (x)$ and $e \in V$ , we have $- e \in V$ and

v_{h} (x) - v_{h} (x - h e) \leq p \cdot (h e) \leq v_{h} (x + h e) - v_{h} (x) .

This implies that $v_{h} (x) - v_{h} (x - h e) \leq v_{h} (x + h e) - v_{h} (x)$ and hence $Δ_{h e} v_{h} (x) \geq 0$ for all $e \in V$ . The remaining part of the proof is identical to the one of [1, Lemma 4.1]. ∎

Lemma 2.2.

Assume that $f > 0$ in $Ω$ and let $v_{h}$ be a solution of (2.3). Then for $x \in Ω_{h}$ , $\partial_{V} v_{h} (x) \subset Ω^{*}$ .

Proof.

With $k = l = 0$ in Lemma 2.1, we obtain for $e \in V$

- σ_{Ω^{*}} (- h e) \leq v_{h} (x) - v_{h} (x - h e) < v_{h} (x + h e) - v_{h} (x) \leq σ_{Ω^{*}} (h e) .

Thus, for $p \in \partial_{V} v_{h} (x)$ , $p \cdot (h e) \geq v_{h} (x) - v_{h} (x - h e) \geq - σ_{Ω^{*}} (- h e)$ . So $p \cdot (- e) \leq σ_{Ω^{*}} (- e)$ . Since for $e \in V, - e \in V$ and $V$ contains a normal to each face of $Ω^{*}$ , we conclude that $\partial_{V} v_{h} (x) \subset Ω^{*}$ . ∎

It follows from Lemma 2.1 that for $f > 0$ in $Ω$ , solutions of (2.3) with $v_{h} (x^{1}) = α$ for an arbitrary number $α$ and $x^{1} \in Ω_{h}$ , are bounded independently of $h$ . 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 $x^{1}, x^{2} \in Ω_{h}, x^{1} \neq x^{2}$ , $\partial_{V} v_{h} (x^{1}) \cap \partial_{V} v_{h} (x^{2})$ is contained in a set of measure zero.

Proof.

For $x \in Ω_{h}$ we define $V (x) = {x - e, e \in V}$ and the partial discrete Legendre transform of $v_{h}$ as the function $v_{h}^{*} : R^{d} \to R$ defined by

v_{h}^{*} (p) = sup x \in V (x^{1}) \cup V (x^{2}) (x \cdot p - v_{h} (x)) .

Let $p \in \partial_{V} v_{h} (x^{1}) \cap \partial_{V} v_{h} (x^{2})$ . We have $v_{h}^{*} (p) = x^{i} \cdot p - v_{h} (x^{i}), i = 1, 2$ . For $z \in R^{d}$ , $v_{h}^{*} (z) \geq x^{i} \cdot z - v_{h} (x^{i}) = x^{i} \cdot z + (v_{h}^{*} (p) - x^{i} \cdot p) = v_{h}^{*} (p) + x^{i} \cdot (z - p), i = 1, 2$ . Hence both $x^{1}$ and $x^{2}$ are in the subdifferential of $v_{h}^{*}$ . As a supremum of affine functions, $v_{h}^{*}$ is convex and hence is differentiable almost everywhere, c.f. [9, Lemma 1.1.8]. Thus if $v_{h}^{*}$ were differentiable at $p$ we would have $D v_{h}^{*} (p) = x^{i}, i = 1, 2$ . Impossible. Thus $p$ is contained in the set of points at which $v_{h}^{*}$ is not differentiable, a set of measure zero. ∎

Remark 2.4.

It follows from Lemmas 2.2 and 2.3 that if $v_{h}$ solves (2.3) or more generally, if $v_{h}$ is a discrete convex mesh function with asymptotic cone $K_{Ω^{*}}$ we have $\cup_{x \in Ω_{h}} \partial_{V} v_{h} (x) \subset Ω^{*}$ and thus $\int_{\cup_{x \in Ω_{h}} \partial_{V} v_{h} (x)} R (p) d p = \sum_{x \in Ω_{h}} ω_{a} (R, u_{h}, {x}) = \int_{Ω^{*}} R (p) d p$ where we used (2.4). Therefore, since $R > 0$ on $Ω^{*}$ , up to a set of measure 0, we have $\cup_{x \in Ω_{h}} \partial_{V} v_{h} (x) = Ω^{*}$ .

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 $G : U \to R^{M}$ with $G (v_{h}) = (G_{i} (v_{h}))_{i = 1, \dots, M}$ . We let $| | . | |$ denote a norm in $R^{M}$ and put

U_{ϵ} = {v_{h} \in U, G_{i} (v_{h}) > - ϵ, i = 1, \dots, M},

for a parameter $ϵ > 0$ to be specified later. We assume that $G \in C^{1} (U_{ϵ}, R^{M})$ for all $ϵ > 0$ . The current iterate is denoted $v_{h}^{k}$ and the following iterate is sought along the path

p^{k} (τ) = v_{h}^{k} - τ G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) .

We make the assumption that there exists $τ_{k} \in (0, 1)$ such that $p^{k} (τ) \in U$ for all $τ \in [0, τ_{k}]$ . Let $δ \in (0, 1)$ and choose $ρ \in (0, 1)$ , e.g. $ρ = 1 / 2$ .

1:Choose

v_{h}^{0} \in U_{ϵ}

and set

k = 0

2:If

G (v_{h}^{k}) = 0

stop

3:Let

i_{k}

be the smallest non-negative integer

i

such that

p^{k} (ρ^{i}) \in U_{ϵ}

and

| | G (p^{k} (ρ^{i})) | | \leq (1 - δ ρ^{i}) | | G (v_{h}^{k}) | | .

Set

v_{h}^{k + 1} = p^{k} (ρ^{i_{k}})

k \leftarrow k + 1

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

G

is assumed to be in

C^{1, α}, 0 < α \leq 1

. For

G

to be merely

C^{1}

, as in certain geometric optics problems, and with

det G^{'} (x) \neq 0

for all

x \in U_{ϵ}

, one has linear convergence [8]. For completeness we adapt the proof of [8] to the case where the domain of the mapping

G

is an open set of

R^{M}

. As with [12, Proposition 2.10] we will assume that the mapping

G

is proper, i.e. the preimage of any compact set is a compact set.

Theorem 3.1.

Let $G \in C^{1} (U_{2 ϵ}, R^{M})$ and assume that $G$ is a proper map with a unique zero $u_{h}$ in $_{ϵ}$ and $det G^{'} (x) \neq 0$ for all $x \in U_{2 ϵ}$ . Assume that there exists ${ˇ τ}_{k}$ in $(0, 1]$ such that for all $0 < τ \leq {ˇ τ}_{k}, p_{k} (τ) \in U$ . Then the iterate $v_{h}^{k}$ of the damped Newton’s method is well defined, converges to $u_{h}$ and

| | v_{h}^{k + 1} - u_{h} | | \leq C o (| | v_{h}^{k} - u_{h} | |),

for $k \geq k_{0}$ if $i_{k} = 0$ for $k \geq k_{0}$ and $k_{0}$ sufficiently large. For $k$ sufficiently large, with step 3 replaced with $i_{k} = 0$ , i.e. a full Newton’s step, the convergence is guaranteed to be at least linear.

Proof.

Since $G \in C^{1} (U_{2 ϵ}, R^{M})$ , we can find a constant $L$ (depending on $ϵ$ ) such that

| | G^{'} (x) | | \leq L, \forall x \in U_{ϵ} .

Part 1: The damped Newton’s method is well defined. We first note that it follows from the definitions that for all $τ \in [0, 1]$ we have

(3.1)

G (v_{h}^{k}) + G^{'} (v_{h}^{k}) (p^{k} (τ) - v_{h}^{k}) = (1 - τ) G (v_{h}^{k}) .

Assume that $G (v_{h}^{k}) \neq 0$ . We claim that there exists $τ_{k}^{'} \in (0, 1)$ such that

| | G (p (τ)) | | \leq (1 - δ τ) | | G (v_{h}^{k}) | |, \forall τ \in [0, τ_{k}^{'}] .

If such a $τ_{k}^{'}$ does not exist, there would exist a sequence $τ_{l}$ converging to 0 such that

(3.2)

| | G (p (τ_{l})) | | > (1 - δ τ_{l}) | | G (v_{h}^{k}) | |, \forall k .

Since $G$ is $C^{1}$ , c.f. for example [14, $ 3.2.10], we have

G (p (τ_{l})) = G (v_{h}^{k}) + G^{'} (v_{h}^{k}) (p (τ_{l}) - v_{h}^{k}) + o (| | p (τ_{l}) - v_{h}^{k} | |) .

Thus, by (3.1), we have

(3.3)

G (p (τ_{l})) = (1 - τ_{l}) G (v_{h}^{k}) + o (| | p (τ_{l}) - v_{h}^{k} | |),

and by (3.2)

	$(1 - δ τ_{l}) \| \| G (v_{h}^{k}) \| \|$	$< (1 - τ_{l}) \| \| G (v_{h}^{k}) \| \| + o (\| \| p (τ_{l}) - v_{h}^{k} \| \|)$
	$(1 - δ) \| \| G (v_{h}^{k}) \| \|$	$< \frac{1}{τ_{l}} o (τ_{l} \| \| G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) \| \|) .$

Taking the limit as $τ_{l} \to 0$ , we obtain $(1 - δ) | | G (v_{h}^{k}) | | \leq 0$ , a contradiction as $G (v_{h}^{k}) \neq 0$ .

We now show that there exists ${¯ ¯ ¯ τ}_{k} \leq min {τ_{k}^{'}, {ˇ τ}_{k}, τ_{k}}$ , with ${¯ ¯ ¯ τ}_{k} > 0$ such that for all $τ \in [0, {¯ ¯ ¯ τ}_{k}], p (τ) \in U_{ϵ}$ when $v_{h}^{k} \in U_{ϵ}$ .

If $p (τ) \notin U_{ϵ}$ for some $τ \leq min {τ_{k}^{'}, {ˇ τ}_{k}, τ_{k}}$ it must be that at some time ${¯ ¯ ¯ τ}_{k} \leq τ, p ({¯ ¯ ¯ τ}_{k}) \in \partial U_{ϵ}$ . Thus for some $i_{k} \in {1, \dots, M}$ we have $G_{i_{k}} (p ({¯ ¯ ¯ τ}_{k})) = - ϵ$ . Let us assume that ${¯ ¯ ¯ τ}_{k}$ is chosen so that $G_{i} (p (τ)) > - ϵ$ for all $τ \in [0, {¯ ¯ ¯ τ}_{k})$ and all $i \in {1, \dots, M}$ .

We have

| | G (p ({¯ ¯ ¯ τ}_{k})) - G (v_{h}^{k}) | | \geq | G_{i_{k}} (p ({¯ ¯ ¯ τ}_{k})) - G_{i_{k}} (v_{h}^{k}) | = | - ϵ - G_{i_{k}} (v_{h}^{k}) | > 0,

since $G_{i_{k}} (v_{h}^{k}) > - ϵ$ as $v_{h}^{k} \in U_{ϵ}$ by assumption. Therefore, by the mean value theorem, c.f. for example [14, $ 3.2.10],

0 < | | G (p ({¯ ¯ ¯ τ}_{k})) - G (v_{h}^{k}) | | \leq L | | p ({¯ ¯ ¯ τ}_{k}) - v_{h}^{k} | | \leq L {¯ ¯ ¯ τ}_{k} | | G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) | | .

We conclude that ${¯ ¯ ¯ τ}_{k} > 0$ . And by construction $p (τ) \in U_{ϵ}$ for all $τ \in [0, {¯ ¯ ¯ τ}_{k})$ .

Part 2: The sequence $v_{h}^{k}$ converges to $u_{h}$ . Let

K = {v \in U_{ϵ}, | | G (v) | | \leq | | G (v_{h}^{0}) | |} .

Since $G$ is proper, $K$ is compact. By construction $| | G (v_{h}^{k + 1}) | | < | | G (v_{h}^{k}) | |$ and $v_{h}^{k} \in U_{ϵ}$ for all $k$ . Thus up to a subsequence, the sequence $v_{h}^{k}$ converges to $v^{*} \in_{ϵ}$ and since the sequence $| | G (v_{h}^{k}) | |$ is strictly decreasing, there exists $η \geq 0$ such that $| | G (v_{h}^{k}) | | \to η$ .

If $η = 0$ , then $G (v^{*}) = 0$ and since $G$ has a unique zero $u_{h}$ in $_{ϵ}$ , the whole sequence converges to $v^{*} = u_{h}$ .

Now we show that it is not possible to have $η \neq 0$ . Assume that $η > 0$ .

If the sequence $i_{k}$ of Step 3 of the algorithm is bounded, there would exist a constant $ξ > 0$ such that $ρ^{i_{k}} \geq ξ$ for all $k$ . This would imply that $1 - δ ρ^{i_{k}} \leq 1 - δ ξ$ and $η \leq (1 - δ ξ) η$ , i.e. $η = 0$ . We therefore have ${lim}_{k \to \infty} i_{k} = \infty$ and consequently ${lim}_{k \to \infty} ρ^{i_{k} - 1} = 0$ . Put

{^τ}_{k} = ρ^{i_{k} - 1} .

By definition of $i_{k}$ and (3.3), we have

	$(1 - δ {^τ}_{k}) \| \| G (v_{h}^{k}) \| \|$	$< \| \| G (p ({^τ}_{k})) \| \| \leq (1 - {^τ}_{k}) \| \| G (v_{h}^{k}) \| \| + \| \| o ({^τ}_{k} \| \| G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) \| \|) \| \|$
	$(1 - δ) \| \| G (v_{h}^{k}) \| \|$	$< \frac{1}{{^τ}_{k}} \| \| o ({^τ}_{k} \| \| G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) \| \|) \| \| .$

Recall that the sequence $| | G (v_{h}^{k}) | |$ is bounded. Moreover $v^{*} \in U_{2 ϵ}$ and thus $det G^{'} (v^{*}) \neq 0$ . Since $G \in C^{1} (U_{2 ϵ}, R^{M})$ it follows that for $k$ sufficiently large $| | G^{'} (v_{h}^{k})^{- 1} | | \leq C$ for a constant $C$ , c.f. [14, $ 2.3.3]. We conclude that

(1 - δ) | | G (v_{h}^{k}) | | < \frac{1}{{^τ}_{k}} o ({^τ}_{k}) .

This implies that $η = 0$ , a contradiction.

Part 3: We first prove the rate $| | v_{h}^{k + 1} - u_{h} | | \leq C o (| | v_{h}^{k} - u_{h} | |)$ , for $k \geq k_{0}$ if $i_{k} = 0$ for $k \geq k_{0}$ and $k_{0}$ sufficiently large. We have with ${~ τ}_{k} = ρ^{i_{k}}$

	$G^{'} (v_{h}^{k}) (v_{h}^{k + 1} - u_{h})$	$= G^{'} (v_{h}^{k}) (v_{h}^{k} - u_{h} - {~ τ}_{k} G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}))$
		$= G^{'} (v_{h}^{k}) (v_{h}^{k} - u_{h}) - {~ τ}_{k} G (v_{h}^{k}) .$

Moreover $0 = G (u_{h}) = G (v_{h}^{k}) - G^{'} (v_{h}^{k}) (v_{h}^{k} - u_{h}) + o (| | v_{h}^{k} - u_{h} | |)$ . Thus $G^{'} (v_{h}^{k}) (v_{h}^{k + 1} - u_{h}) = (1 - {~ τ}_{k}) G (v_{h}^{k}) + o (| | v_{h}^{k} - u_{h} | |)$ . Since $G^{'} (v_{h}^{k})^{- 1}$ is uniformly bounded for $k$ sufficiently large, the claim follows when ${~ τ}_{k} = 1$ .

The linear convergence of Newton’s method for $C^{1}$ mappings is classical but is nethertheless reviewed here. Let now $v_{h}^{k + 1} = v_{h}^{k} - G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k})$ and put $e_{h}^{k} = v_{h}^{k} - u_{h}$ .

Let $β > 0$ such that for $| | v_{h}^{k} - u_{h} | | \leq β$ we have $| | G^{'} (v_{h}^{k})^{- 1} | | \leq C_{1}$ . If necessary by taking $β$ smaller, we may assume that for $| | v - u_{h} | | \leq β$ and $| | w - u_{h} | | \leq β$ we have $| | G^{'} (v) - G^{'} (w) | | \leq 1 / C_{1}$ using the local uniform continuity of $G^{'}$ on $U_{ϵ}$ . We have

e_{h}^{k + 1} = e_{h}^{k} - G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) = G^{'} (v_{h}^{k})^{- 1} \int_{0}^{1} (G^{'} (v_{h}^{k}) - G^{'} (u_{h} + t e_{h}^{k})) e_{h}^{k} d t,

from which we get $| | e_{h}^{k + 1} | | \leq | | e_{h}^{k} | |$ . This shows that when $| | v_{h}^{k} - u_{h} | | \leq β$ we also have $| | v_{h}^{k + 1} - u_{h} | | \leq β$ and the linear convergence rate. ∎

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

Let $M$ denote the cardinality of $Ω_{h}$ and denote the points of $Ω_{h}$ by $x^{i}, i = 1, \dots, M$ .

The set of discrete convex mesh functions on $Ω_{h}$ which satisfy (2.2), and with $v_{h} (x^{1}) = α$ for an arbitrary number $α$ and $x^{1} \in Ω_{h}$ , can be identified with a subset $U$ of $R^{M - 1}$ . We consider a map $G : U \to R^{M - 1}$ defined by

G_{i} (v_{h})

= ω_{a} (R, v_{h}, {x^{i}}) - \int_{C_{x^{i}}} f (t) d t % for i = 2, \dots, M .

Note that for $v_{h} \in U$ , we have $v_{h} (x^{1}) = α$ and this value is used for computing $ω_{a} (R, v_{h}, {x^{i}})$ in the expression of $G_{i} (v_{h})$ . Also, by (2.4) we have

ω_{a} (R, v_{h}, {x^{1}}) = \int_{Ω^{*}} R (p) d p - M \sum i = 2 ω_{a} (R, v_{h}, {x^{i}}) .

Hence

ω_{a} (R, v_{h}, {x^{1}}) = \int_{C_{x^{1}}} f (t) d t,

when $G_{i} (v_{h}) = 0, i = 2, \dots, M$ . In other words, the above equation is automatically satisfied when $G_{i} (v_{h}) = 0, i = 2, \dots, M$ .

We now make the assumption that

there exists a constant ϵ > 0 such that f > 2 ϵ / h^{d} on Ω .

We consider the set $V$ of mesh functions $v_{h}$ with $v_{h} (x^{1}) = α$ , $v_{h}$ extended to $Z_{h}^{d}$ using (2.2) and which satisfy $Δ_{h e} v_{h} (x^{i}) \geq 0, i = 2, \dots, M$ and for all $e \in V$ . We have $U \subset V$ . Our iterates will be shown to be in $V$ with an accumulation point which satisfies $G_{i} (v_{h}) = 0, i = 2, \dots, M$ . This implies that $G_{1} (v_{h}) = 0$ as well. We then have $ω_{a} (R, v_{h}, {x^{1}}) > ϵ > 0$ . Therefore $\partial_{V} v_{h} (x^{1})$ is a set of non zero measure. In particular, it is non empty. There exists $p$ such that for each direction $e$ , as $V$ is symmetric with respect to the origin, $v_{h} (x^{1}) - v_{h} (x^{1} - e) \leq p \cdot e \leq v_{h} (x^{1} + e) - v_{h} (x^{1})$ . Thus $Δ_{h e} v_{h} (x^{1}) \geq 0$ as well and the accumulation point is the unique solution of (2.3).

Define as in the previous section $V_{ϵ} = {v_{h} \in V, G_{i} (v_{h}) > - ϵ, i = 2, \dots, M}$ . The set $V_{ϵ}$ is non empty since the solution $u_{h}$ of (2.3) with $u_{h} (x^{1}) = α$ solves $G_{i} (u_{h}) = 0 > - ϵ$ for all $i$ . We note that for $v_{h} \in V_{ϵ}$ we have $ω_{a} (R, v_{h}, {x^{i}}) > ϵ > 0$ for all $i = 2, \dots, M$ . Hence $Δ_{h e} v_{h} (x^{i}) > 0$ for all $i = 2, \dots, M$ and $e \in V$ .

Theorem 4.1.

Assume that $G$ is differentiable at $v \in V_{ϵ}$ and $det G^{'} (v) \neq 0$ . There exists $ˇ τ$ in $(0, 1]$ such that for all $0 < τ \leq ˇ τ, p (τ) = v - τ G^{'} (v)^{- 1} G (v) \in V$ .

Proof.

Since $v \in V_{ϵ}$ , $Δ_{h e} v (x^{i}) > 0$ for all $i = 2, \dots, M$ and $e \in V$ . Let $c_{0} = min {Δ_{h e} v_{h} (x^{i}), i = 2, \dots, M, e \in V}$ and let $ζ = c_{0} / 8$ . We note that if $| w - v |_{\infty} := max {| w (x) - v (x) |, x \in Ω_{h}} \leq ζ$ , then $Δ_{h e} w (x^{i}) \geq 0, i = 2, \dots, M, e \in V$ . Indeed, for $x \in Ω_{h}, x \neq x^{1}$ , when $x \pm e \in Ω_{h}$

	$Δ_{h e} w (x)$	$= w (x + h e) - 2 w (x) + w (x - h e)$
		$= Δ_{h e} v (x) + ((w - v) (x + h e) - 2 (w - v) (x) + (w - v) (x - h e))$
		$\geq Δ_{h e} v (x) - 4 ζ \geq c_{0} - 4 ζ = \frac{c_{0}}{2} > 0.$

If $x + e \notin Ω_{h}$ , we have $w (x + e) = w (y) + (x + e - y) \cdot a_{l}^{*}$ for some integer $l, 1 \leq l \leq N$ and $y \in \partial Ω_{h}$ . We have $v (x$

	$(1 - δ τ_{l}) \| \| G (v_{h}^{k}) \| \|$	$< (1 - τ_{l}) \| \| G (v_{h}^{k}) \| \| + o (\| \| p (τ_{l}) - v_{h}^{k} \| \|)$
	$(1 - δ) \| \| G (v_{h}^{k}) \| \|$	$< \frac{1}{τ_{l}} o (τ_{l} \| \| G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) \| \|) .$

	$(1 - δ {^τ}_{k}) \| \| G (v_{h}^{k}) \| \|$	$< \| \| G (p ({^τ}_{k})) \| \| \leq (1 - {^τ}_{k}) \| \| G (v_{h}^{k}) \| \| + \| \| o ({^τ}_{k} \| \| G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) \| \|) \| \|$
	$(1 - δ) \| \| G (v_{h}^{k}) \| \|$	$< \frac{1}{{^τ}_{k}} \| \| o ({^τ}_{k} \| \| G^{'} (v_{h}^{k})^{- 1} G (v_{h}^{k}) \| \|) \| \| .$

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

Authors

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Weighted estimates of the Cayley transform method for boundary value problems in a Banach space

1. Introduction

2. Preliminaries

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

Remark 2.4.

3. The damped Newton’s method

Theorem 3.1.

Proof.

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

Theorem 4.1.

Proof.

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

Authors

Related Research

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Convergence Analysis of the Algorithm in "Efficient and Robust Discrete Conformal Equivalence with Boundary"

A comparison between pre-Newton and post-Newton approaches for solving a physical singular second-order boundary problem in the semi-infinite interval

Weak convergence of Monge-Ampere measures for discrete convex mesh functions

A Dual-Mixed Approximation for a Huber Regularization of the Herschel-Bulkey Flow Problem

On the solution of contact problems with Tresca friction by the semismooth* Newton method

Weighted estimates of the Cayley transform method for boundary value problems in a Banach space

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1. Introduction

2. Preliminaries

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

Remark 2.4.

3. The damped Newton’s method

Theorem 3.1.

Proof.

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

Theorem 4.1.

Proof.