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12 JEAN-DAVID BENAMOU, BRITTANY D. FROESE, AND ADAM M. OBERMANso that the time required for the two methods is comparable; compare Tables 2and 5.(a)fig:f1(b)fig:f14(c)fig:sol14(d)fig:err14Figure 4. Results for u(x, y) = exp(x 2 +y 22)with noisy, nonconvexdata on an 81 × 81 grid. (a) f without noise. (b) f with∣ ∣ ∣∣noise. (c) Surface plot of the solution with noise. (d) ∣u − u noisyfig:ex14table:ex14N Iterations CPU Time (sec)(M1) (M2) (M1) (M2)21 1016 58 .2 0.141 3866 99 2.4 1.261 8432 185 10.7 5.681 14672 409 32.0 21.0101 22516 695 77.6 63.4121 31965 1205 153.0 156.0141 43008 2162 291.0 401.0Table 5. Computation times for u(x, y) = expnoisy data on an N × N grid.(x 2 +y 22)with
TWO NUMERICAL METHODS FOR THE ELLIPTIC MONGE-AMPÈRE EQUATION 13sec:discts5.2. Discontinuous source term. We next consider an example source term fthat is discontinuous on a circle{1 (x − 0.5) 2 + (y − 0.5) 2 > 0.16 2f =0 otherwise.The solution is{ (x − 0.5) 2 + (y − 0.5) 2 }u = max, 0.08 .2The solution is flat where f vanishes and parabolic elsewhere. See the surface plotin Figure 5(a). Both methods require about the same amount of time for thisexample as is seen in Figure 5(b) and Table 6.fig:ex15(a)fig:sol15(b)fig:time15Figure 5. Results for §5.2. (a)Surface plot of solution. (b) TotalCPU time.table:ex15N Iterations CPU Time (sec)(M1) (M2) (M1) (M2)21 950 90 0.2 0.341 3651 259 2.0 3.461 7944 363 9.4 12.181 13810 783 21.5 42.5101 21162 893 68.0 85.0121 29966 1678 132.0 221.0141 40196 2076 244.0 480.0Table 6. Computation times for §5.2 on an N × N grid.sec:nonsmooth5.3. An example which is singular at the boundary. This example was takenfrom [8] We consider the solution of <strong>Monge</strong>-Ampère when f = 1 with constantboundary values 1 on the square [−1, 1]×[−1, 1]. There is no exact solution availablefor this example, however the solution is not smooth: since u xx = 0 on part of theboundary, u yy → ∞ there. Method (M1) converges to a solution that is convexexcept near the corners of the domain (Figure 6(b)). This is comparable to the
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