Preconditioned ADMM with Nonlinear Operator Constraint

  • Martin Benning
  • Florian Knoll
  • Carola-Bibiane Schönlieb
  • Tuomo Valkonen
Conference paper
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 494)


We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).


ADMM Primal-dual Nonlinear inverse problems Parallel MRI Proximal point method Operator splitting Iterative Bregman method 



MB, CS and TV acknowledge EPSRC grant EP/M00483X/1. FK ackowledges National Institutes of Health grant NIH P41 EB017183.

EPSRC Data Statement: the corresponding code and data are available for download at


  1. 1.
    Aubert-Broche, B., Evans, A.C., Collins, L.: A new improved version of the realistic digital brain phantom. NeuroImage 32(1), 138–145 (2006)CrossRefGoogle Scholar
  2. 2.
    Bachmayr, M., Burger, M.: Iterative total variation schemes for nonlinear inverse problems. Inverse Prob. 25(10), 105004 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benning, M., Gladden, L., Holland, D., Schönlieb, C.-B., Valkonen, T.: Phase reconstruction from velocity-encoded MRI measurements-a survey of sparsity-promoting variational approaches. J. Magn. Reson. 238, 26–43 (2014)CrossRefGoogle Scholar
  5. 5.
    Bernstein, M.A., King, K.F., Zhou, X.J.: Handbook of MRI Pulse Sequences. Elsevier, Amsterdam (2004)Google Scholar
  6. 6.
    Block, K.T., Uecker, M., Frahm, J.: Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint. Magn. Reson. Med. 57(6), 1086–1098 (2007)CrossRefGoogle Scholar
  7. 7.
    Bonettini, S., Loris, I., Porta, F., Prato, M.: Variable metric inexact line-search based methods for nonsmooth optimization. Siam J. Optim. 26, 891–921 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Candes, E.J., et al.: Compressive sampling. In: Proceedings of the International Congress of Mathematicians, vol. 3, Madrid, Spain, pp. 1433–1452 (2006)Google Scholar
  9. 9.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gabay, D.: Applications of the method of multipliers to variational inequalities. Stud. Math. Appl. 15, 299–331 (1983)Google Scholar
  12. 12.
    Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Knoll, F., Clason, C., Bredies, K., Uecker, M., Stollberger, R.: Parallel imaging with nonlinear reconstruction using variational penalties. Magn. Reson. Med. 67(1), 34–41 (2012)CrossRefGoogle Scholar
  14. 14.
    Möllenhoff, T., Strekalovskiy, E., Möller, M., Cremers, D.: The primal-dual hybrid gradient method for semiconvex splittings. SIAM J. Imaging Sci. 8(2), 827–857 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Möller, M., Benning, M., Schönlieb, C., Cremers, D.: Variational depth from focus reconstruction. IEEE Trans. Image Process. 24(12), 5369–5378 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: inertial proximal algorithm for nonconvex optimization. SIAM J. Imaging Sci. 7(2), 1388–1419 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ramani, S., Fessler, J., et al.: Parallel MR image reconstruction using augmented Lagrangian methods. IEEE Trans. Med. Imaging 30(3), 694–706 (2011)CrossRefGoogle Scholar
  18. 18.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, 46:49. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  19. 19.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom. 60(1), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sbrizzi, A., Hoogduin, H., Lagendijk, J.J., Luijten, P., den Berg, C.A.T.: Robust reconstruction of B1+ maps by projection into a spherical functions space. Magn. Reson. Med. 71(1), 394–401 (2014)CrossRefGoogle Scholar
  21. 21.
    Uecker, M., Hohage, T., Block, K.T., Frahm, J.: Image reconstruction by regularized nonlinear inversion joint estimation of coil sensitivities and image content. Magn. Reson. Med. 60(3), 674–682 (2008)CrossRefGoogle Scholar
  22. 22.
    Valkonen, T.: A primal-dual hybrid gradient method for nonlinear operators with applications to MRI. Inverse Prob. 30(5), 055012 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, X., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46(1), 20–46 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  • Martin Benning
    • 1
  • Florian Knoll
    • 2
  • Carola-Bibiane Schönlieb
    • 1
  • Tuomo Valkonen
    • 3
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  2. 2.Center for Advanced Imaging, Innovation and ResearchNew York UniversityNew YorkUSA
  3. 3.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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