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8 JEAN-DAVID BENAMOU, BRITTANY D. FROESE, AND ADAM M. OBERMANsingle Poisson solve. Each iteration of [9, 15, 7, 8, 10] involves a Poisson solve andadditional calculations.Feng and Neilan [14, 12, 13] do not give details about the computation time oriteration count of their numerical method.3.2. Comparison of Method 1 and Method 2. The computation time for (M1)was roughly independent of the regularity of the solution. On the other hand, thecomputation times for (M2) were faster (by more than one order of magnitude) onvery smooth solutions, about equal on mildly singular solutions, and much slower(by a couple of orders of magnitude) on the more singular solutions.Naturally, for any finite resolution, the discretised solutions are not singular, sowe cannot conclude that the method diverges. But since the relative computationalcosts increase dramatically as the solutions become less regular, this behavior isconsistent with divergence.We find (M2) performs extremely rapidly when solutions are smooth, but isvery slow when solutions are not smooth or not strictly convex. The explicitmethod (M1), on the other hand, requires approximately the same amount of CPUtime to converge without regard for the regularity of the solutions.To show this more concretely, we consider a representative example from eachcategory: the smooth radial solution of §4.1, the example with noisy data of §5.1,and the non-smooth cone of §6.1. We compare the absolute computation timerequired for these three examples to demonstrate our observation that the speedof (M1) is independent of regularity while the speed of (M2) is very sensitive toregularity; see Figure 1. We compare the ratio of CPU times required by the twomethods for a representative example, see Table 1.(a)fig:Meth1(b)fig:Meth2Figure 1. CPU time required to converge for the smooth radialsolution of §4.1, the noisy solution of §5.1, and the non-smoothcone of §6.1. (a) Results for (M1). (b) Results for (M2).fig:times3.3. Computational Details. The solutions are computed on an N × N gridand iteration is continued until the maximum difference between two subsequentiterates is less than 10 −14 . By convention, h is the spatial discretisation parameterh = L/N where L is the length of one side of the square domain.
TWO NUMERICAL METHODS FOR THE ELLIPTIC MONGE-AMPÈRE EQUATION 9table:ratiosN §4.1 §5.1 §6.121 6.3 1.3 .1241 4.1 2.1 .0761 7.7 1.9 .0681 11.7 1.5 .05101 14.1 1.2 .04121 23.5 1.0 .07141 29.9 .7 .05Table 1. Ratio of computation times required for (M1) and (M2)for three representative examples: the smooth radial solution of§4.1, the noisy solution of §5.1, and the non-smooth cone of §6.1.A reasonable choice for the initial data is the solution of u xx + u yy = √ 2f,which was used when measuring the solution time and iteration count. However,the methods are robust: the iterations converge to a fixed point for all initial dataattempted. The initial data attempted included: a constant (which didn’t respectthe boundary conditions), and both non-convex and random initial data.The implementation of (M2) can be performed using the method of choice. Wefocus on the simple finite difference method which arises from discretising (2.3)using central differences (as with the first method). We also briefly discuss in §7.2the use of an adaptive finite element method.sec:smoothsec:radialsec:blowupsec:moderate4. Smooth or Mildly Singular ExamplesIn the following examples, the solutions are smooth, or mildly singular, and (M2)is much faster than (M1). We observed convergence rates consistent with of O(h 2 )for the smooth solution, and O(h 1.5 ) for the mildly singular solution.4.1. An exact radial solution. An exact radial solution is( x 2 + y 2 )u(x, y) = exp, f(x, y) = (1 + x 2 + y 2 ) exp(x 2 + y 2 ).2Both methods converge to the same numerical solution, which is accurate to secondorder in h, O(h 2 ), see Figure 2(a). For this smooth example, (M2) is by far thefastest method: see Table 2 and Figure 2(b).4.2. Blow-up at boundary. Another exact solution isu(x, y) = 2√ 23 (x2 + y 2 ) 3/4 1, f(x, y) = √x2 + y 2on the square [0, 1] × [0, 1]; the function f blows up at the boundary. Again, bothmethods converge to the exact solution, although the order of convergence is nowonly 1.5. For this example (M2) was significantly faster than (M1). See Tables 3and 4 and Figure 3.5. Moderately Singular ExamplesIn the following examples, the solutions are moderately singular, and the twomethods perform about equally.
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