Advances in Computational Mathematics

, Volume 45, Issue 2, pp 495–517 | Cite as

Fast iterative solvers for an optimal transport problem

  • Roland Herzog
  • John W. Pearson
  • Martin StollEmail author


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.


PDE-constrained optimisation Saddle point systems Time-dependent PDE-constrained optimisation Preconditioning Krylov subspace solver Optical flow Optimal transport 

Mathematics Subject Classification (2010)

65F08 68U10 49J20 35L45 65M06 


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The authors would like to thank two anonymous referees for their helpful comments and suggestions. JWP gratefully acknowledges support from the Engineering and Physical Sciences (EPSRC) Fellowship EP/M018857/2.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of MathematicsTechnische Universität ChemnitzChemnitzGermany
  2. 2.School of MathematicsThe University of EdinburghEdinburghUK

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