TY - JOUR
T1 - Convergent numerical method for the reflector antenna problem via optimal transport on the sphere
AU - Hamfeldt, Brittany Froese
AU - Turnquist, Axel G.R.
N1 - Funding Information:
Funding. Directorate for Mathematical and Physical Sciences (1619807, 1751996, GRFP).
Funding Information:
Acknowledgment. Brittany Froese Hamfeldt was partially supported by NSF DMS-1619807 and NSF DMS-1751996. Axel G. R. Turnquist was partially supported by NSF DMS-1751996 and an NSF GRFP-1849508.
Publisher Copyright:
© 2021 Optical Society of America
PY - 2021/11
Y1 - 2021/11
N2 - We consider a partial differential equation (PDE) approach to numerically solve the reflector antenna problem by solving an optimal transport problem on the unit sphere with cost function c(x, y)= −2 log ||x − y||. At each point on the sphere, we replace the surface PDE with a generalized Monge–Ampère type equation posed on the local tangent plane. We then use a provably convergent finite difference scheme to approximate the solution and construct the reflector. The method is easily adapted to take into account highly nonsmooth data and solutions, which makes it particularly well adapted to real-world optics problems. Computational examples demonstrate the success of this method in computing reflectors for a range of challenging problems including discontinuous intensities and intensities supported on complicated geometries.
AB - We consider a partial differential equation (PDE) approach to numerically solve the reflector antenna problem by solving an optimal transport problem on the unit sphere with cost function c(x, y)= −2 log ||x − y||. At each point on the sphere, we replace the surface PDE with a generalized Monge–Ampère type equation posed on the local tangent plane. We then use a provably convergent finite difference scheme to approximate the solution and construct the reflector. The method is easily adapted to take into account highly nonsmooth data and solutions, which makes it particularly well adapted to real-world optics problems. Computational examples demonstrate the success of this method in computing reflectors for a range of challenging problems including discontinuous intensities and intensities supported on complicated geometries.
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U2 - 10.1364/JOSAA.439679
DO - 10.1364/JOSAA.439679
M3 - Article
C2 - 34807032
AN - SCOPUS:85118681470
SN - 1084-7529
VL - 38
SP - 1704
EP - 1713
JO - Journal of the Optical Society of America A: Optics and Image Science, and Vision
JF - Journal of the Optical Society of America A: Optics and Image Science, and Vision
IS - 11
ER -