the Institute for Theoretical and Engineering Science
Department of Mathematics, University of Houston

Scientific Computing Seminar

Professor Roland Glowinski
Department of Mathematics, University of Houston

On the Least-Squares Solution of the Elliptic
Monge-Ampère Equation in Dimension Two$^a$

Thursday, February 2, 2006$^*$
3:00 PM- 4:00 PM
Room 646 PGH
$^*$Note: Different Location

$^a$This is a joint work with Edward J. Dean.

Abstract: In a previous seminar, we discussed an augmented Lagrangian based method for the numerical solution of the Dirichlet problem for the Monge-Ampère equation in two-dimension, namely:

$$\operatorname{det} {\bf D}^2 \psi = f in \Omega, \psi=g on \Gamma,$$ (E-MA-D)

where, in (E-MA-D), $\Omega$ is a bounded domain of $\mathbf{R}^2$, $\Gamma$ is the boundary of $\Omega$, ${\bf D}^2 \psi$ is the Hessian of the unknown function $\psi$, and $f$ and $g$ are given function; if $f > 0$, the above Monge-Ampère equation is elliptic (actually, fully nonlinear elliptic). As shown in refs. [1] and [2], if incompatibilities between $f$ and $g$ imply that (E-MAD-D) has no solution, the augmented Lagrangian based method described there solves it in a least-squares sense (assuming for example that $f \in L^1(\Omega)$ and $g \in H^{3/2}(\Gamma)$) in the Hilbert space $H^2(\Omega) \times {\bf Q}$ where ${\bf Q}=\{{\bf q}\vert{\bf q}=(q_{ij})_{1\le i,j\le 2}, q_{i,j} \in L^2(\Omega), {\bf q}={\bf q}^t\}$. This lead us, smallmore recently, (see ref. [3]) to consider directly the least-squares solution of (E-MA-D) via

$$\operatorname{min} j(\phi,{\bf q}), \{\phi,{\bf q}\} \in V_g \times {\bf Q}_f,$$ (L-S)

where, in (L-S), $V_g=\{ \phi\vert\phi \in H^2(\Omega), \phi=g on \Gamma \}$, ${\bf Q}_f=\{ {\bf q}\vert {\bf q} \in {\bf Q}, \operatorname{det} {\bf q}=f\}$, and $j(\phi,{\bf q})=\int_{\Omega} \vert{\bf D}^2\phi-{\bf q}\vert^2 dx$ (with $\vert{\bf q}\vert=(\vert q_{11}^2+\vert q_{22}\vert^2+2\vert q_{12}\vert^2)^{1/2}$),

In this lecture, we will discuss (following [3]) the solution of problem (LS), the basic idea being to associate to its Euler-Lagrange equation a well-chosen initial-value problem (flow in the Dynamical System terminology) that we solve via an operator-splitting scheme decoupling differential operators from nonlinearity, making the numerical process highly modular. The results of numerical experiments will be presented, some of them concerning situations where (E-MA-D) has no classical solution.

References: 1. E.J. DEAN, R.GLOWINSKI, Numerical solution of the two-dimensional Monge-Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach, C. R. Acad. Sci. Paris, Ser. I, 336, (2003), 779-784. 2. E.J. DEAN, R.GLOWINSKI, An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimension, Electronic Transactions in Numerical Analysis (to appear in 2006). 3. E.J. DEAN, R.GLOWINSKI, Numerical solution of the two-dimensional Monge-Ampère equation with Dirichlet boundary conditions: a least-squares approach, C. R. Acad. Sci. Paris, Ser. I, 339(12), (2004), 887-892.

This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.