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.
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 ?
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.
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 ?
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).