MADRID LECTURE # 6 On the Numerical Solution of the Elliptic Monge-Ampère Equation in 2-D A Least Squares Approach

1. Introduction Our goal here is to discuss the least-squares solution in H 2 (Ω) of some fully nonlinear elliptic equations of the Monge-Ampère type in 2-D. Why H 2 (Ω) and why least-squares? ● Because from a computational point of view there is always advantage at solving a given problem in a Hilbert space and, here H 2 (Ω) is a natural choice. ● Least-squares methods are well suited to Hilbert spaces and provide apparently an alternative to viscosity solution based methods.

Introduction (2) We will focus only on the solution of the Dirichlet problem for the canonical Monge-Ampère equation (E-MA-D) det D 2 ψ = f in Ω, ψ = g on Г, with Ω  R 2 and f > 0, but “our” methodology applies also (among other problems) to the Pucci-Dirichlet problem (PUC-D)  λ + + λ – = 0 in Ω, ψ = g on Г,

Introduction (3) with λ + (resp., λ – ) the largest (resp., the smallest ) eigenvalue of the matrix-valued function (Hessian) D 2 ψ and   (1, +∞) (if  = 1, one recovers the linear Poisson-Dirichlet problem). The Gaussian curvature equation det D 2 ψ = f (1 + |  ψ| 2 ) 2 in Ω is also in our agenda.

Introduction (4) The Mathematics of Monge-Ampère type equations has generated a large literature (Th. Aubin, L.A.Caffarelli, …). On the other hand (cf. Google Scholar) one can not say the same of their Numerics, with some notable exceptions such as Olicker- Prussner and Benamou-Brenier, and more recently A. Oberman; indeed (from B-B): “ It follows from this theoretical result that a natural computational solution of the L2 MKP is the numerical resolution of the Monge-Ampère equation (6). Unfortunately, this fully nonlinear second-order elliptic equation has not received much attention from numerical analysts and, to the best of our knowledge, there is no efficient finite-difference or finite-element methods, comparable to those developed for linear second-order elliptic equations (such as fast Poisson solvers, multigrid methods, preconditioned conjugate gradient methods,….).” Our goal is to show that several of the tools mentioned in the above statement concerning the solution of linear second order elliptic problems still apply for these fully nonlinear elliptic equations.

2. A least-squares method for the elliptic Monge-Ampère equation in dimension 2 The Dirichlet problem for the prototypical Monge-Ampère equation reads as follows: detD 2 ψ = f in , ψ = g on . (MA-D) If f is positive the above equation is elliptic (E-MA-D). This equation is somewhat tricky. Take  = (0, 1) 2 and consider the particular case of (E-MA-D) defined by  2 ψ/  x 1 2  2 ψ/  x 2 2 – |  2 ψ/  x 1  x 2 | 2 = 1 in , ψ = 0 on . (2.1) Clearly, (2.1) can not have smooth solutions, despite the smoothness of its data. Trouble lies with the non-strict convexity of .

Section 2 (2) From now on, we suppose that f > 0 and that {f, g}  {L 1 (  ), H 3/2 (  )}, implying that the following space and set are non empty: V g = {  |   H 2 (  ),  = g on  }, Q f = {q| q  Q, det q = f }, with Q = {q| q  (L 2 (  )) 2  2, q = q t }. Solving the Monge-Ampère equation in H 2 (  ) is equivalent to looking at the intersection in Q of D 2 V g and Q f.

Section 2 (3) (E-MA-D) has a solution in H 2 (  )

Section 2 (3) (E-MA-D) has no solution in H 2 (  )

Section 2 (4) In order to handle those situations where (E-MA-D) has no solution in H 2 (  ) despite the fact that neither V g nor Q f are empty we suggest to solve the above problem via the following least squares formulation Min { ,q} { ½ ∫  |D 2  – q| 2 dx}, { , q}  V g ×Q f (LSQ) with |q| = (q q q 12 2 ) ½.

Section 2 (5) In order to solve (LSQ) by operator-splitting techniques we observe that (LSQ) is equivalent to Min { ,q} { ½ ∫  |D 2  – q| 2 dx + I f (q)}, { , q}  V g × Q (LSQ-P) 0 if q  Q f, with I f (q) = i.e., I f is the indicator functional +  if q  Q \ Q f, of Q f.

Section 2(6) We can now solve (LSQ-P) by a block relaxation method operating alternatively between V g and Q f. A closely related algorithm is obtained as follows: (i) Derive the Euler-Lagrange equation of (LSQ-P), namely {ψ, p}  V g  Q, ∫ Ω D 2 ψ:D 2 φ dx = ∫ Ω p:D 2 φ dx,  φ  V 0, ∫ Ω p:q dx + = ∫ Ω D 2 ψ:q dx,  q  Q,

Section 2(7) with V 0 = H 2 (  )  H 0 1 (  ) and ∂I f (p) a generalized differential of I f at p. (ii) Associate to this E-L equation an initial value problem (flow) in V g ×Q. (iii) Use operator-splitting to time discretize the above flow problem.

Section 2(8) The above program leads to the following algorithm:

Section 2 (9) (1){ψ 0, p 0 } = {ψ 0, p 0 }; for n  0, {ψ n, p n } being known, solve for {ψ n+1, p n+1 } (2)(p n+1 – p n )/  + p n+1 +  I f (p n+1 )  D 2 ψ n, ψ n+1  V g, (3) ∫  Δ[(ψ n+1 – ψ n )/  ] Δ  dx + ∫  D 2 ψ n+1 : D 2  dx = ∫  p n+1 : D 2  dx,    V 0,

Section 2 (10) with r:s = r 11 s 11 + r 22 s r 12 s 12 if r = r t, s = s t. Problem (2) can be solved point-wise, while problem (3 ) can be solved by a conjugate gradient algorithm operating in V g and V 0 equipped with the scalar product {v, w} → ∫  Δv Δw dx. Each iteration of the above c.g. algorithm requires the solution of 2 Poisson-Dirichlet problems.

3. Finite Element Approximation of (E-MA-D). Numerical Experiments 3.1. Finite Element Approximation Suppose that T h is a finite element triangulation of Ω; we approximate H 2 (Ω),V g, V 0 (=H 2 (Ω)  H 1 0 (Ω)), Q and Q f by: (3.1) V h = {φ|φ  C 0 (  Γ), φ| T  P 1,  T  T h }, (3.2) V gh = {φ|φ  V h, φ(P) = g(P),  P  Г and vertex of T h }, (3.3) V 0h = {φ|φ  V h, φ = 0 on Г }, (3.4) Q h = {q|q  (V 0h ) 4, q = q t }, (3.5) Q fh = {q|q  Q h, (q 11 q 22 – q 2 12 )(P) = f h (P),  P vertex of T h,P  Г }, with f h a continuous approximation of f.

Section 3 (2) Next, we approximate ∂ 2 φ/∂x i ∂x j by D ijh (φ) defined as follows for 1 ≤ i, j ≤ 2: D ijh (φ)  V 0h, ∫ Ω D ijh (φ) v dx = – ½ ∫ Ω [∂φ/∂x i ∂v/∂x j + ∂φ/∂x j ∂v/∂x i ]dx,  v  V 0h,  φ  V h. This is a mixed finite element approximation of the second order derivatives, classically used for solving linear and nonlinear bi-harmonic problems (Cahn-Hilliard, Von Kármán equations for plates, Navier-Stokes equations in their {ψ, ω} formulation, etc…).

Section 3 (3) Deriving a discrete analogue of the above least squares formulation of (E-MA-D) is pretty obvious now Numerical Experiments The first test problem is defined as follows: (i) Ω = (0, 1) × (0, 1). (ii) f(x) = 1/|x|,  x  Ω. (iii) g(x) = (2|x|) 3/2 /3,  x  Γ. With these data one can easily show that the function ψ defined by ψ(x) = (2|x|) 3/2 /3,  x  Ω, is solution of the corresponding (E-MA-D) problem. The above function does not belong to C 2 (Ω  Г) but belongs to W 2, p (Ω) for p  [1, 4); it has in principle enough regularity to be handled by our approach. We have used a uniform mesh like the one below.

Section 3 (4) A uniform triangulation of Ω (h= 1/4).

Section 3 (5) h  nit ||D 2 h ψ c h – p c h || Q || ψ c h – ψ|| L 2 (Ω) _____________________________________________________________________ 1/ × 10 – × 10 – 4 1/ × 10 – × 10 – 4 1/ × 10 – × 10 – 4 1/ 32 1, × 10 – × 10 – 4 1/ × 10 – × 10 – 4 1/ × 10 – × 10 – 4 1/ × 10 – × 10 – 4 1/ 64 1, × 10 – × 10 – 4 First Test Problem The above results suggest an approximation error in O(h 2 ) for the L 2 (Ω)- norm.

First test problem: Graph of f

First test problem: Graph of Ψ h c

Data:  = (0,1)×(0,1), f = 1, g = 0. Results: Section 3(8) Second Test Problem

Section 3 (9) Second Test Problem

Section 3 (10) Second Test Problem

4. Other Fully Nonlinear Elliptic Equations With some subtle differences the methodology we applied to the solution of the Monge-Ampère equation applies also to the solution of the following Pucci’s Equation (PE) αλ + + λ - = 0 in Ω, Ψ = g on ∂Ω, where: (i) α  (1, + ∞). (ii) λ + and λ - are the largest and smallest eigenvalues of the Hessian matrix D 2 Ψ of the function Ψ. (iii) Ω  R 2.

Section 4 (2) (PE) is equivalent to the following system α|  Ψ| 2 + (α – 1) 2 detD 2 Ψ = 0 in Ω, Ψ = g on ∂Ω, (PE)’  Ψ ≤ 0 in Ω. A note which appeared in the CRAS, Paris (2005) describes the LS/OS solution of (PE)’.

5. Final Observations Solving (E-MA-D) by a mixed method is really solving it via its equivalent PFAFF System, namely completed by: ψ = g on Г. The “burden of nonlinearity” has been transferred from ψ to p.

Section 5 (2) A natural question is the following: Is our approach a (kind of) viscosity method ? The answer is “yes” as shown below. Let us show it: The Flow associated to the Least-Squares optimality conditions reads as follows: Find {ψ(t), p(t)}  V g  Q,  t > 0, such that ∫ Ω ∂(Δψ)/∂t Δφ dx + ∫ Ω D 2 ψ:D 2 φ dx = ∫ Ω p:D 2 φ dx,  φ  V 0, ∫ Ω ∂p/∂t : q dx + ∫ Ω p:q dx + = ∫ Ω D 2 ψ:q dx,  q  Q, (FE) {ψ(0), p (0)} = {ψ 0, p 0 }.

Section 5 (3) Assuming that Ω is simply connected, introduce: u = {u 1, u 2 } = {∂ψ/∂x 2, – ∂ψ/∂x 1 }, v = {v 1, v 2 } = {∂φ/∂x 2, – ∂φ/∂x 1 }, ω = ∂u 2 /∂x 1 – ∂u 1 /∂x 2, θ = ∂v 2 /∂x 1 – ∂v 1 /∂x 2, V g = {v| v  (H 1 (Ω)) 2, .v = 0, v.n = dg/ds on Γ}, V 0 = {v| v  (H 1 (Ω)) 2, .v = 0, v.n = 0 on Γ}, L = ( – ). The formulation (FE) is equivalent to

Section 5 (3) Find u(t)  V g,  t > 0, such that ∫ Ω ∂ω/∂t θ dx + ∫ Ω  u:  v dx = ∫ Ω Lp:  v dx,  v  V 0, ∂p/∂t + p + ∂I f (p) + L  u = 0, (FE)* {u(0), p (0), ω(0)} = {u 0, p 0, ω 0 }.. Problem (FE)* has a visco-elasticity flavor, – L p playing here the role of the so-called extra-stress tensor. As t → +∞, we obtain thus at the limit a viscosity solution, but in a sense different from M.Crandall- P.L. Lions’.