An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions.

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Two Numerical Methods for the elliptic Monge-Ampère equation (2010, methodology): “They point out in [7] “if, despite the smoothness of its data, the above problem has no solution in H(2)(Ω), the above augmented Lagrangian method solves it in a least-squares sense,” and in [8], “in cases where the Monge-Ampère equation under consideration has no smooth solution, [the methods] prov”
A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two (2013, background): “Note that, when using the stopping criterion (ii), numerical results show that the residual ∣∣∣∣∣∣D2h(ψn h ) − pnh∣∣∣∣∣∣0h varies like h(2) approximately, which agrees with the results in [19, 21].”