Fast iterative solvers for an optimal transport problem

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.