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>