Abstract
Repeatedly solving the parameterized optimal mass transport (pOMT) problem is
a frequent task in applications such as image registration and adaptive grid
generation. It is thus critical to develop a highly efficient reduced solver
that is equally accurate as the full order model. In this paper, we propose
such a machine learning-like method for pOMT by adapting a new reduced basis
(RB) technique specifically designed for nonlinear equations, the reduced
residual reduced over-collocation (R2-ROC) approach, to the parameterized
Monge-Amp$\grave{\rm e}$re equation. It builds on top of a narrow-stencil
finite different method (FDM), a so-called truth solver, which we propose in
this paper for the Monge-Amp$\grave{\rm e}$re equation with a transport
boundary. Together with the R2-ROC approach, it allows us to handle the strong
and unique nonlinearity pertaining to the Monge-Amp$\grave{\rm e}$re equation
achieving online efficiency without resorting to any direct approximation of
the nonlinearity. Several challenging numerical tests demonstrate the accuracy
and high efficiency of our method for solving the Monge-Amp$\grave{\rm e}$re
equation with various parametric boundary conditions.