Iterative Scheme for Solving Optimal Transportation Problems Arising in Reflector Design

Table 2

Results of an iterative run with the “Iterative Refinement” Scheme 6. Memory was insufficient to handle the sixth iteration due to matrix size limitations in MATLAB. Note the run times (last two rows). The bulk of the algorithm run times is taken in setting up the linear programming problem, that is, deciding which constraints to include. Solving the LP once it is set up is fairly fast (last column). The data used was that of Section 4.1, the number of nearest neighbors searched over 𝜅_nn=6, and the quasi-activity threshold was 𝜀=0.01ℎ². 𝑘: iteration number; 𝑀: number of mesh points in 𝐷; 𝑁: number of mesh points in 𝑇; constr.: number of constraints; con. dens.: constraint density, that is, used constraints as a percentage of all possible constraints 𝑀⋅𝑁; time: computation time for iterative step in seconds, including generating meshes, building, and solving the LP problem; LP time: time in seconds to solve the LP problem.

𝑘	ℎ	𝑀	𝑁	Max error 𝑅1	𝐿2 error 𝑅1	Max error 𝑅2	𝐿2 error 𝑅2	Constr.	Con. dens.	Time (s)	LP time (s)
0	0.064	1,059	1,059	1.2884𝐸−3	3.77819𝐸−4	1.88783𝐸−3	4.09084𝐸−4	1,121,481	100%	15.962	12.3146
1	0.0512	1,675	1,678	9.40425𝐸−4	3.36918𝐸−4	1.26944𝐸−3	3.63526𝐸−4	56,080	2.00%	208.533	0.938877
2	0.04096	2,647	2,647	7.56437𝐸−4	1.67174𝐸−4	7.53459𝐸−4	1.85922𝐸−4	89,724	1.28%	392.982	1.59831
3	0.032768	4,171	4,175	3.64902𝐸−4	9.63559𝐸−5	5.25941𝐸−4	1.21839𝐸−4	141,550	0.81%	969.239	2.54405
4	0.0262144	6,582	6,567	3.03932𝐸−4	8.8103𝐸−5	5.30343𝐸−4	1.41908𝐸−4	224,327	0.52%	2363.79	4.32524
5	0.02097152	10,313	10,318	2.49854𝐸−4	9.9045𝐸−5	3.9372𝐸−4	1.51851𝐸−4	351,514	0.33%	5909.25	9.21891