Fast iterative solvers for an optimal transport problem

2018/01/12 by Herzog, Roland, Pearson, John W., Stoll, Martin
#65F10 #65F50 #65N22 #76D07 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1801.04172

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.

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