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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.
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