1. A least-squares method for the Monge-Ampère equation
Corien Prins

2. Free-form illumination optics: street lighting

3. Free-form illumination optics: lowbeam

4. How to speed up the design of free-form lenses

5. The lens design problem
Xi en eta in plaatje

6. The Monge-Ampère equation

7. The Monge-Ampère equation

8. The Monge-Ampère equation

9. New numerical method
Least squares method

10. Least squares method

11. Minimization procedure

12. Minimization procedure

13. Minimize b

14. Minimization procedure

15. Minimize P

16. Minimization procedure

17. Minimize m

18. Minimize m

19. Summary Solve by iterative minimization of functional J
Minimize b: distance to boundary
Minimize P: analytical
Minimize m: two Poisson equations
Next: calculate surface u from m

20. Find the lens surface

21. Numerical examples
Example

22. Iteration 1

23. Iteration 2

24. Iteration 3

25. Iteration 10

26. Iteration 20

27. Iteration 30

28. Iteration 40

29. Iteration 50

30. Target and progress
Target
Progress

31. Output simulation

33. Conclusions
New numerical method to solve Monge Ampère equation/Optimal Mass Transport
Works better if the target distribution is smooth, but also works if it is not smooth
Compared to the other numerical method we know:
Slower
Uses less memory
Also usable if target domain not convex
Easier to adapt for other variants of the Monge-Ampère equation

34. Future work Saddle-shaped lenses Other Monge-Ampère equations:
Light emitted from point in space
Drop assumption that screen is far away and big compared to lens
Further study of connection lens/reflector design and Optimal Mass Transport
/ name of department

35. Thank you!