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

• URL: http://arxiv.org/abs/2112.01878v1
• Date: Fri, 3 Dec 2021 12:30:46 GMT
• Title: Fast $L^2$ optimal mass transport via reduced basis methods for the Monge-Amp$\grave{\rm e}$re equation
• Authors: Shijin Hou, Yanlai Chen, Yinhua Xia
• Abstract summary: We propose a machine learning-like method for solving the parameterized Monge-Amp$graverm e$re equation. Several challenging numerical tests demonstrate the accuracy and high efficiency of our method for solving the Monge-Amp$graverm e$re equation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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