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I am working with a numerical application where some known techniques solve $\Delta u=\rho$ for $\rho(x,y)\geq0$ defined on a planar rectangle (given as the density of a large pointset). Since the application is related to transportation problems, I suspect that solving Monge-Ampere $\det(D^2 u)=\rho$ (with the same boundary conditions) may work better. Hence, the following questions.

  1. What are the most striking differences between the two PDEs ? It would be useful to have examples of $\rho$ leading to very different solutions, and to identify general properties of individual solutions that differ between the two equations. For example, do solutions of Poisson's equation with nonzero $\rho$ minimize some useful functionals ?

  2. Are the best numerical techniques for these equations essentially the same ? The answers may differ if best is interpreted in terms of speed/convergence, robustness, ease of programming, support in Matlab, or availability of open-source solvers.

  3. Should I expect the (nonlinear) Monge-Ampere equation to be more difficult to solve (numerically) than the Poisson's equation with the same $\rho$ and the same boundary conditions ? Is it possible/practical to solve Monge-Ampere by somehow solving/resolving Poisson's equation ?

  4. What are good references on these topics ?

Just in case, my $\rho$ can be heavily concentrated in small regions, often vanishes in other regions, and may experience sharp steps-like discontinuities. But this can be papered over by Gaussian smoothing (at some cost).

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Thinking aloud: for #1, we may want to consider two cases (1) $\rho$ with well-articulated saddle points and (2) $\rho$ without saddle points. –  Igor Markov Feb 6 '12 at 9:56
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Your reasons for wanting to consider the Monge-Ampere equation are rather vaguely stated, and your questions about the equation are quite broad. The differences are sufficiently big that you should not pursue this unless the application really does call for the Monge-Ampere equation, which is nonlinear, not necessarily elliptic, and therefore much more difficult to work with numerically. I also see that you are at Michigan, so I encourage you to consult colleagues in the math department and maybe even others in your own department about this. –  Deane Yang Feb 6 '12 at 10:26
    
A basic difference is that the Laplacian is invariant under Euclidean motions, while the Hessian determinant is invariant under unimodular affine motions. –  Dan Fox Feb 6 '12 at 14:40
    
Hmm... I am not sure how to exploit such invariance in my case, since the domain is bounded and $\rho$ seems unlikely to be invariant. –  Igor Markov Feb 6 '12 at 23:22
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Igor, my point was that the local geometry naturally associated to the two operators is different, so already in the case where the domain is the entire space and $\rho$ is a constant, or for the Dirichlet problem, the equations/solutions behave quite differently. The Monge-Ampere equation is more difficult than the Laplacian. As Deane Yang has pointed out above, the nonlinearity/linearity is one reason. The local geometry is another. On the other hand, by the arithmetic/geometric mean inequality there is some relation. In this regard see papers of Dean/Glowinski about the M-A case. –  Dan Fox Feb 7 '12 at 8:22

1 Answer 1

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I agree with Deane Yang's comment and will try to expand on it in this answer.

First, the standard formulation of the Monge-Ampere equation is $ \det(D^2u)=\rho $.

Second, the Monge-Ampere equation and the Poisson equation are completely different, and the numerical techniques for the two problems are very different. The Monge-Ampere equation is fully nonlinear and will in general be much more difficult to solve numerically.

Finally, a recent paper on numerical methods for Monge-Ampere is S.C. Brenner, T. Gudi, M. Neilan, and L.-Y. Sung, C0 penalty methods for the fully nonlinear Monge-Ampere equation, Mathematics of Computation, 80:1979-1995, 2011. The references inside should give you a starting point for your questions.

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Omitting $det$ in the question was a typo (sorry!) - I fixed that. –  Igor Markov Feb 6 '12 at 17:16
    
@Andrew, @Deane - I am looking at ``A gradient descent solution to the Monge-Kantorovich problem,'' Chartrand et al (Applied Math Sci 2009, 3(22)) math.lanl.gov/Research/Publications/Docs/…. This should explain my interest in $\det(D^2u)=ρ$ (but not the use of $Δu=ρ$ by others). Sections 3 and 4 apparently imply a straightforward numerical solution technique for Monge-Ampere --- consider $f_2=const$ in Formulas 14 and 24. How do I reconcile this with "much more difficult to solve numerically" ? Thx for your help. –  Igor Markov Feb 6 '12 at 17:32
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Igor, I confess to being a pure mathematician who knows very little about numerical solutions of PDE's. My answer was based on the general belief that linear PDE's are usually much easier to solve numerically than nonlinear ones. But the paper you found does appear to find a very nice unconstrained variational approach to solving certain types of elliptic Monge-Ampere equation. If that's exactly what you need, then that's great! –  Deane Yang Feb 6 '12 at 17:36

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