Numerical Analysis/Partial Differential Equations Numerical solution of the Monge-Ampère equation by a Newton's algorithm

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Optimal Mass Transport: Signal processing and machine-learning applications (2017, methodology): “Several authors have proposed numerical methods to obtain the optimal transport map through solving the Monge–Ampère PDE in (11) [7], [33].”
Two Numerical Methods for the elliptic Monge-Ampère equation (2010, methodology): “Loeper and Rapetti [18] solve the equation (with periodic boundary conditions) by linearization and iterate using Newton’s method.”
An introduction to sampling via measure transport (2016, background): “ computation of general non-triangular transports see [61]; for a generalization of the framework to compositions of maps see [61,66]; and for the computation of optimal transports see, for instance, [88,10,4,50]. Now let ˇdenote any unnormalized version of the target density. For any map Tin the fe”
A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two (2013, background): “, [2, 5, 6, 14, 24, 37, 40, 41, 43, 44, 50], with applications in geometry, mechanics and physics.”
Unidimensional and Evolution Methods for Optimal Transportation (2013, methodology): “Or, using the peculiar form the optimal map should have, T (x ) = x − ∇S (x ), one can start from a non-optimal potential S0, and then apply Newton’s method to catch the optimal S [40].”