10:40
MS19: Numerical methods for Monge-Ampere equations (Part 2)
Chair: Jan ten Thije Boonkamp
10:40
25 mins
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A least-squares Galerkin gradient recovery method for fully nonlinear elliptic equations
Omar Lakkis, Amireh Mousavi
Abstract: Least squares recovery methods provide a simple and practical way to
approximate equations in nondivergence form where standard variational
approach either fails or requires technically complex modifications.
In this talk I will explain a new method based on the recovery of the
Hessian and the gradient.
The technique consists in introducing both at the continuum and
discrete stages two extra unknowns for the gradient and the Hessian of the unknown. Correct functional spaces and penalties in the cost functional
must be crafted in order to ensure stability and convergence of the
scheme with a good approximation of the gradient and Hessian which is
useful, for example, for Newton--Raphson approximation of the
Monge--Ampère equation. I will cover also variations including a
Hessianless version (when no precise Hessian representation is
required).
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11:05
25 mins
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A conforming C^1 finite element method for PDEs of Monge-Ampère type with optimal transport boundary conditions
Elisa Friebel, Wolfgang Dahmen, Siegfried Müller
Abstract: The mathematical model of an optical surface is closely related to optimal transport (OT) problems.
Similarly, if their solution is regular, it also solves a partial differential equation (PDE) of Monge-Ampère (MA) type equipped with so-called transport boundary conditions
\begin{align*}
\det(D^2 u + A(u,\nabla u)) - \frac f {g(\nabla u)} J(u,\nabla u) &= 0 \qquad \text{ in }\Omega, \\
\nabla u(\partial \Omega) &= \partial \Sigma,
\end{align*}
where $\Omega$, $\Sigma$ are bounded convex domains, $f:\Omega \rightarrow \R_>0$ and $g:\Sigma\rightarrow \R_>0$ with $\int f = \int g$.
The most obvious $L^2$-projection is to find $u$ such that
\begin{align}
\int_{\Omega} \left( \det(D^2 u+ A(u,\nabla u)) - \frac f {g(\nabla u)} J(u,\nabla u)\right) v =0 \;\; \text{ for all } v \in L^2(\Omega). \label{eq: projection}
\end{align}
To directly embed this in a finite element method, $C^1$ ansatz functions are necessary \cite{Boehmer2008}. Although the construction of such elements is rather cumbersome, the strategy utilising the projection \eqref{eq: projection} is applicable to a wide range of PDEs.
We discuss the implementation of such a FE method and how to incorporate the nonlinear transport boundary conditions in the solution procedure. The resulting algorithm yields numerical accuracy, geometric flexibility and high order of approximation for smooth solutions. \\
In order to solve practically relevant OT problems, we identify the critical parts of the input data and introduce stabilisation techniques within a nested iteration process.
We conclude by employing the method to design refractive and reflecting surfaces in illumination systems.
\newcommand{\etalchar}[1]{$^{#1}$}
\begin{thebibliography}{BBW{\etalchar{+}}13}
\bibitem{Boehmer2008}
Klaus B{\"{o}}hmer.
\newblock {On Finite Element Methods for Fully Nonlinear Elliptic Equations of
Second Order}.
\newblock {\em SIAM J. Numer. Anal.}, 46(3):1212--1249, 2008.
\end{thebibliography}
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11:30
25 mins
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Monotone and second order consistent scheme for the two dimensional Pucci equation
Joseph Frederic Bonnans, Guillaume Bonnet, Jean-Marie Mirebeau
Abstract: I will describe Voronoi's first reduction, a tool coming from the field of additive lattice geometry, which turns out to be particularly efficient for the discretization of anisotropic PDEs on cartesian grids.
This approach is versatile, and yields monotone and second order consistent finite difference schemes for various PDEs, ranging from anisotropic elliptic PDEs to the Monge-Ampere equation, and more. In turn, these applications raise new questions on Voronoi's reduction, related to its continuity or its extension to inhomogeneous forms. Numerical results illustrate the method's robustness and accuracy.
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