European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS19: Numerical methods for Monge-Ampere equations (Part 1)
Chair: Jan ten Thije Boonkamp
25 mins
A second order method for the approximation of a parabolic 2D Monge-Ampère equation
Alexandre Caboussat, Dimitrios Gourzoulidis
Abstract: Parabolic fully nonlinear equations may be found in various applications, for instance in optimal portfolio management strategy. A numerical method for the approximation of a canonical parabolic Monge-Ampère equation is investigated in this work. A second order semi-implicit time-stepping method is presented, coupled to safeguarded Newton iterations A low order finite element method is used for space discretization. Numerical experiments exhibit appropriate convergence orders and a robust behavior.
25 mins
Monge-Ampère equations for complex optical systems
Jan ten Thije Boonkkamp, Lotte Romijn, Wilbert IJzerman
Abstract: We present the generalized Monge-Ampère equation for several optical systems, containing one or two optical surfaces. We outline a least-squares solution method and demonstrate the method for several examples.
25 mins
A Monge-Ampère least-squares solver for the design of a freeform lens
Lotte Romijn, Jan ten Thije-Boonkkamp, Wilbert IJzerman
Abstract: Designing freeform optical surfaces that control the redistribution of light from a particular source distribution to a target irradiance poses challenging problems in the field of illumination optics. There exists a wide variety of strategies in academia and industry, and there is an interesting link with optimal transport theory. Many freeform optical design problems can be formulated as a generalized Monge-Ampère equation. In this paper, we consider the design of a single freeform lens that converts the light from an ideal (zero-étendue) point source into a far-field target. We derive the generalized Monge-Ampère equation and numerically solve it using a generalized least-squares algorithm. The algorithm first computes the optical map and subsequently constructs the optical surface. We show that the numerical algorithm is capable of computing optical surfaces that produces a projection of a painting on a screen in the far field.
25 mins
Semi-discrete methods for nonimaging optics
Boris Mr Thibert
Abstract: Nonimaging optics is a field of optics where one is interested in the transfer of light energy between a source and a target. Unlike traditional optics, or imaging-optics, the goal is not to reproduce an image of the input light, but to design optical components that transfer a given source light to a prescribed target light. Several inverse problems arising in this field amount to solve Monge-Ampère type equations. In this talk, I will present a geometric discretization of these equations that is called semi-discrete. It corresponds to the case where the source light is represented by a continuous measure, whereas the target light is represented by a discrete one. I will show how several Monge Ampère equations arising in optics correspond to optimal transport problems and can then be recast as concave maximization problems leading to efficient algorithms. I will then present the design of different kinds of mirrors or lenses that allow to transfer any punctual or collimated source to any target. Joint work with Quentin Mérigot and Jocelyn Meyron.