```
The  Weizmann  Institute  of  Science
Faculty of Mathematics and Computer Science

Mathematical Analysis and Applications Seminar

Seminar Room, Room 261, Ziskind Building
on Tuesday, January 3, 2012
11:00 - noon

Note the unusual location

V. Oliker
Emory University

will speak on

Designing lenses and mirrors
with help from geometry and optimal mass transport

Abstract:
Systems of lenses or mirrors converting an incident plane wave of a given cross
section and intensity distribution into an output plane wave irradiating at a
given target set with prescribed intensity are required in many applications.
Most of the known designs of devices with such capabilities are restricted to
rotationally symmetric mirrors/lenses. In many cases this is a significant
limitation.

In this talk I will discuss designs with free-form mirrors/lenses, that is,
without a priori assumption of rotational symmetry. Assuming the geometrical
optics approximation, it can be shown that  the functions describing such
free-form mirrors/lenses satisfy fully nonlinear Monge-Amp\`{e}re type partial
differential equations (PDE's) derived from the basic laws of geometrical
optics. However, because of strong nonlinearities investigation of such PDE's
is difficult and many basic questions are open. Fortunately, by extending
classical ideas from convexity and of the Legendre transform some of these
problems can also be formulated (in weak form) as problems in calculus of
variations in which instead of solving the nonlinear PDE's one needs to find
extrema of some Fermat-like functionals. Moreover, these functionals arise in a
physically natural way as cost functions in optimal (linear and nonlinear)
transportation problems. This approach was successfully applied in several
design problems and allowed to prove existence, uniqueness and numerically
calculate solutions by methods based on mathematical programming. I will
describe these ideas in the case of the "two-lens" design problem.

---------------------------------------------------------
Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il>
Announcement from: Yaeli Malka   <yaeli.malka@weizmann.ac.il>
```