Optimization approach for the Monge-Ampère equation

Optimization approach for the Monge-Ampère equation

Abstract.

This paper studies the numerical approximation of solution of the Dirichlet problem for the fully nonlinear Monge-Ampère equation. In this approach, we take the advantage of reformulation the Monge-Ampère problem as an optimization problem, to which we associate a well defined functional whose minimum provides us with the solution to the Monge-Ampère problem after resolving a Poisson problem by the finite element Galerkin method. We present some numerical examples, for which a good approximation is obtained in 68 iterations.

Key words. elliptic Monge-Ampère equation, gradient conjugate method, finite element Galerkin method.

AMS subject classifications. 35J60, 65K10, 65N30.

fethi ben belgacem 111 Department of Mathematics, Higher Institute of Computer Sciences and Mathematics of Monastir (ISIMM), Avenue de la Korniche - BP 223 - Monastir - 5000, TUNISIA. (fethi.benbelgacem@isimm.rnu.tn).

1. Introduction

In this paper, we give a numerical solution for the following Monge-Ampère problem

(1.1)

where is a smooth convex and bounded domain in is the Hessian of and

Equation (1.1) belongs to the class of fully nonlinear elliptic equation. The mathematical analysis of real Monge-Ampère and related equations has been a source of intense investigations in the last decades; let us mention the following references ( among many others and in addition to [7], [9], [15]): [10], [8], [17, chapter 4], [2], [28], [11]-[14]. Applications to Mechanics and Physics can be found in [27], [4], [5], [18], [24], [26],[31], (see also the references therein).

The numerical approximations of the Monge-Ampère equation as well as related equations have recently been reported in the literature. Let us mention the references [4], [29], [39], [26 ], [11], [32], [25],[28], [33]; the method discussed in [11], [32],[25] is very geometrical in nature. In contrast with the method introduced by Dean and Glowinski in [19 ] [20] [21], which is of the variational type.

On the existence of smooth solution for (1.1), we recall that if equation (1.1) has a unique strictly convex solution (see [14]).

To obtain a numerical solution for (1.1), we propose a least-square formulation of (1.1). In this approach, we take the advantage of reformulation of the Monge-Ampère problem as a well defined optimization problem, to which we associate a well functional whose minimum provides us with the solution to the Monge-Ampère problem after resolving a Poisson problem by the finite element Galerkin method. The minimum is computed by the conjugate gradient method.

The remainder of this article is organized as follows. In section 2, We introduce the optimization problem. In section 3, we discuss a conjugate gradient algorithm for the resolution of the optimization problem. The finite element implementation of the above algorithm is discussed in section 4. Finally, in section 5, we show some numerical results.

2. Formulation of the Dirichlet problem for the elliptic Monge-Ampère equation

Let be the solution of (1.1). Let and be the eigenvalues of the matrix We have

Then and are the solutions of the equation

So

Then

Let us set

We conclude that is solution of the following Dirichlet Poisson problem

To compute we consider the least-squares functional defined on

as follows:

where is the solution of the Dirichlet Poisson problem

The minimization problem

(2.1)

is thus a least-squares formulation of (1.1).

Theorem 1.

is the strictly convex solution of (1.1) if and only if there exist a unique solution of (2.1) such that

Proof.

Since is solution of we have So and is a unique solution of (2.1).

Conversely, let be a solution of (2.1). Since (1.1) has a solution we can deduce immediatly that and so, It follows that

We have and we can deduce that is strictly convex and from the uniqueness of solution for (1.1) we get

3. Iterative solution for the minimisation problem

3.1. Description of the algorithm.

The algorithm we consider to solve the problem (2.1) which is based on the PRP (Polak-Ribière-Polyak [36,37]) conjugate gradient method reads:

Given

then, for being known in , solve

Compute, ,

If


and update by

Where is computed with the Armijo-type line search.

3.2. Solution of sub-problem

We consider first the variational formulation of

(3.1)

where

(3.2)

and

(3.3)

in (3.2) is coercive on For we have Since is bounded and for in (3.3) is continuous, then by the Lax-Milgram theorem has a unique solution

4. Finite element approximation of the minimization problem

For simplicity, we assume that is a bounded polygonal domain of Let a finite triangulation of (like those discussed in e.g, [16]).

We introduce a

with the space of the two-variable polynomials of degree A function being given in we denote by

(4.1)
(4.2)

Let taking advantage of relations (4.1) and (4.2) we define the discrete analogues of the differential operators by

(4.3)
(4.4)

To compute the above discrete second order partial derivatives we will use the trapezoidal rule to evaluate the inegrals in the left hand sides of (4.3) and (4.4). We consider the set of the vertices of and = We define the integers and by and So dim and dim

For we associate the function uniquely defined by

It is well known (e.g., [16]) that the sets and are vector bases of and respectively.

We denote by the area of the polygonal which is the union of those triangles of which have as a common vertex. By applying the trapezoidal rule to the integrals in the left hand side of relations (4.3) and (4.4) we obtain:

(4.5)
(4.6)

Computing the integrals in the right hand sides of (4.5) and (4.6) is quite simple since the first order derivatives of and are piecewise constant.

Taking the above relations into account. We approximate the space by

and then the minimization problem (2.1) by

Where

and , are respectively a continuous approximations of functions and is the solution of the discret variant of the Dirichlet Poisson problem .

4.1. Discrete variant of the algorithm

We will discuss now the solution of (2.1) by a discrete variant of algorithm 3.1.

Given

then, for being known in , solve

Compute, ,

If


and update by

Remark 2.

There are many approches for finding an avaible step size . Among them the exact line search is an ideal one, but is cost-consuming or even impossible to use to find the step size. Some inexact line searches are sometimes useful and effective in practical computation, such as Armijo line search [1], Goldstein line search and Wolfe line search [24,38].

The Armijo line search is commonly used and easy to implement in practical computation.

Armijo line search

Let be a constant, and Choose to be the largest in such that

However, this line search cannot guarantee the global convergence of the PRP method and even cannot guarantee to be descent direction of at

4.1.1. Solution of the sub problem

Any sub-problem is equivalent to a finite dimensional variational linear problem which reads as follows: Find such that

(4.7)

By the Lax-milgram theorem we can easily show that (4.7) has a unique solution

5. Numerical experiments

In this section we are going to apply the method discussed in the previous section to the solution of some test problems. For all these test problems we shall assume that is the unit disk. We first approximate by a polygonal domain We consider a finite triangulation of

The first test problem is expressed as follows

(5.1)

with The exact solution to problem (5.1) is given by

Remark 3.

When computing the approximate solutions of these problems, we stopped the iterations of the algorithm as soon as

We have discretized the optimization problem associated to the problem (5.1). We solved the Poisson problem encountred at each iteration of the algorithm by a fast Poisson solvers.

We have used as initial guess three different constant values for The results obtained after 68 iterations are summarized in Table 1 (where denotes the computed approximate solution and ).

The graph of and its contour plot obtained, for has been respectively visualized on Figure 2 and Figure 3.

Table 1. First test problem : Convergence of the approximate solution.
Figure 5.1. First test problem : Graph of
Figure 5.2. First test problem : Contour plot of

We conclude from the results in Table 1 that the value is optimal and quite accurate approximations of the exact solutions are obtained.

Remark 4.

We did not try to find the optimal value of (it seems that is a difficult problem).

In the second test problem we take

The solution to the corresponding Monge-Ampère problem is the function defined by

The method provides after 64 iterations the results summurized in Table 2.

The value is again optimal.

Table 2. second test problem : Convergence of the approximate solution.
Figure 5.3. Second test problem : Graph of
Figure 5.4. Second test problem : Contour plot of

The third test problem is defined as follows

(5.2)

The function given by

is the solution of (5.2) and

Table 3. Third test problem : Convergence of the approximate solution.
Figure 5.5. Third test problem : Graph of
Figure 5.6. Third test problem : Contour plot of

We deduce from Table 3 that is an optimal value.

Unfortunaly I did not find any other initial value that gives more accurate results. Even for the results are not satisfied.

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