Fast $L^2$ optimal mass transport via reduced basis methods for the Monge-Amp$\grave{\rm e}$re equation

TL;DR

This paper introduces a fast and accurate reduced basis method for solving the parameterized Monge-Ampère equation, significantly improving efficiency in applications like image registration.

Contribution

It develops a novel reduced residual reduced over-collocation (R2-ROC) approach tailored for the nonlinear Monge-Ampère equation, enabling efficient online solutions without approximating nonlinearity.

Findings

Achieves high accuracy in numerical tests

Demonstrates significant computational efficiency

Handles various parametric boundary conditions

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.