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 proble...

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Bibliographic Details
Published inAdvances in computational mathematics Vol. 45; no. 2; pp. 495 - 517
Main Authors Herzog, Roland, Pearson, John W., Stoll, Martin
Format Journal Article
LanguageEnglish
Published New York Springer US 02.04.2019
Springer Nature B.V
Subjects
Computational mathematics
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Iterative methods
Linear systems
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematical models
Mathematics
Mathematics and Statistics
Newton methods
Nonlinearity
Optimization
Solvers
Transport equations
Visualization
68U10
Time-dependent PDE-constrained optimisation
Krylov subspace solver
49J20
65M06
PDE-constrained optimisation
Saddle point systems
35L45
Preconditioning
Optimal transport
65F08
Optical flow
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ISSN1019-7168
1572-9044
DOI10.1007/s10444-018-9625-5

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Summary: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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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-018-9625-5