Fast iterative solvers for an optimal transport problem

TL;DR

This paper develops fast iterative solvers for a PDE-constrained optimal transport problem related to image metamorphosis, introducing discretisation strategies, preconditioners, and dimensionality reduction techniques to improve computational efficiency.

Contribution

It introduces novel preconditioners and a radial basis function discretisation approach for efficiently solving nonlinear optimal transport problems.

Findings

Effective preconditioners for linear systems in Gauss-Newton iterations.

Dimensionality reduction via superpixel centers enhances computational speed.

Method demonstrates improved convergence in PDE-constrained optimal transport.

Abstract

Optimal transport problems pose many challenges when considering their numerical treatment. We investigate the solution of a PDE-constrained optimisation problem subject to a particular transport equation arising from the modelling of image metamorphosis. We present the nonlinear optimisation problem, and discuss the discretisation and treatment of the nonlinearity via a Gauss--Newton scheme. We then derive preconditioners that can be used to solve the linear systems at the heart of the (Gauss--)Newton method. With the optical flow in mind, we further propose the reduction of dimensionality by choosing a radial basis function discretisation that uses the centres of superpixels as the collocation points. Again, we derive suitable preconditioners that can be used for this formulation.