Solving the Monge-Ampère equation on triangle-meshes for use in optical freeform design

Designing freeform optical surfaces with a large number of degrees of freedom has been a field of extensive research and development. Several design methods have been proposed. Starting point in the design process often is an idealized light source that has zero étendue (e.g. point source or collimated light). With this assumption the solution is unique and corresponds to the solution of an equation of Monge-Ampère type. We propose a method to solve the Monge-Ampère equation on convex bounded domains by using triangle meshes and by minimizing the difference between prescribed and actual target light distribution which is computed by tracing rays through the optical surface. The mathematical solution has to comply with two conditions: the boundary of the source domain has to be mapped onto the boundary of the target domain and the solution has to be convex. The boundary condition problem is solved using a signed distance function that is computed in advance by a fast marching algorithm. The actual light distribution is computed by tracing rays along the triangle nodes and computing the light irradiance on the target by dividing the light flux through a triangle by its mapped area on the target. Under certain conditions this is also an approximate solution to the Optimal Transportation Problem with quadratic cost.