Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Book cover Splitting Methods in Communication, Imaging, Science, and Engineering

Part of the book series: Scientific Computation ((SCIENTCOMP))

  • 3965 Accesses

Abstract

The main goal of this chapter is to present a brief overview of operator splitting methods and algorithms when applied to the solution of initial value problems and optimization problems, topics to be addressed with many more details in the following chapters of this book. The various splitting algorithms, methods, and schemes to be considered and discussed include: the Lie scheme, the Strang symmetrized scheme, the Douglas-Rachford and Peaceman-Rachford alternating direction methods, the alternating direction method of multipliers (ADMM), and the split Bregman method. This chapter also contains a brief description of (most of) the following chapters of this book.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends®; in Machine Learning 3 (1), 1–122 (2011)

    Google Scholar 

  2. Burger, M., Gilboa, G., Osher, S., Xu, J.: Nonlinear inverse scale space methods. Communications in Mathematical Sciences 4 (1), 179–212 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Burger, M., Möller, M., Benning, M., Osher, S.: An adaptive inverse scale space method for compressed sensing. Mathematics of Computation 82 (281), 269–299 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40 (1), 120–145 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chorin, A.J., Hughes, T.J., McCracken, M.F., Marsden, J.E.: Product formulas and numerical algorithms. Communications on Pure and Applied Mathematics 31 (2), 205–256 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  6. Descombes, S.: Convergence of a splitting method of high order for reaction-diffusion systems. Mathematics of Computation 70 (236), 1481–1501 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Descombes, S., Schatzman, M.: Directions alternées d’ordre élevé en réaction-diffusion. C. R. Acad. Sci. Paris Sér. I Math. 321 (11), 1521–1524 (1995)

    MATH  Google Scholar 

  8. Descombes, S., Schatzman, M.: On Richardson extrapolation of Strang formula for reaction-diffusion equations. diffusion equations. In: Equations aux Dérivées Partielles et Applications: Articles dédiés á J.L. Lions pp. 429–452 (1998)

    Google Scholar 

  9. Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Transactions of the American Mathematical Society pp. 421–439 (1956)

    Google Scholar 

  10. Gabay, D.: Application de la méthode des multiplicateurs aux inéquations variationnelles. In: M. Fortin, R. Glowinski (eds.) Lagrangiens Augmentés: Application à la Résolution Numérique des Problèmes aux Limites pp. 279–307. Dunod, Paris (1982)

    Google Scholar 

  11. Gabay, D.: Applications of the method of multipliers to variational inequalities. In: M. Fortin, R. Glowinski (eds.) Augmented Lagrangians: Application to the Numerical Solution of Boundary Value Problems pp. 299–331. North–Holland, Amsterdam (1983)

    Google Scholar 

  12. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Computers & Mathematics with Applications 2 (1), 17–40 (1976)

    Article  MATH  Google Scholar 

  13. Glowinski, R.: Splitting methods for the numerical solution of the incompressible Navier-Stokes equations. In: J. A. V. Balakrishnan A. A. Dorodtnitsyn, L. Lions (eds.) Vistas in Applied Mathematics, pp. 57–95. Optimization Software (1986)

    Google Scholar 

  14. Glowinski, R.: Finite element methods for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis Vol. IX, pp. 3–1176. North–Holland, Amsterdam (2003)

    Google Scholar 

  15. Glowinski, R.: On alternating direction methods of multipliers: A historical perspective. In: Fitzgibbon, W., Kuznetsov, Y.A., Neittaanmäki, P., Pironneau, O. (eds.) Modeling, Simulation and Optimization for Science and Technology Vol. 34, pp. 59–82. Springer, Dordrecht (2014)

    Google Scholar 

  16. Glowinski, R.: Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems SIAM, Philadelphia, PA (2015)

    Book  MATH  Google Scholar 

  17. Glowinski, R., Marroco, A.: On the approximation by finite elements of order one, and solution by penalisation-duality of a class of nonlinear Dirichlet problems. ESAIM: Mathematical Modelling and Numerical Analysis - Mathematical Modelling and Numerical Analysis 9 (R2), 41–76 (1975)

    MATH  Google Scholar 

  18. Godlewsky, E.: Méthodes à pas Multiples et de Directions Alternées pour la Discrétisation d’Equations d’Evolution. Doctoral Dissertation, Université P. & M. Curie, Paris (1980)

    Google Scholar 

  19. Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM Journal on Imaging Sciences 2 (2), 323–343 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Layton, W.J., Maubach, J.M., Rabier, P.J.: Parallel algorithms for maximal monotone operators of local type. Numerische Mathematik 71 (1), 29–58 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lie, S., Engel, F.: Theorie der transformationsgruppen (Vol. 1). American Soc., Providence, RI (1970)

    Google Scholar 

  22. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM Journal on Numerical Analysis 16 (6), 964–979 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  23. Marchuk, G.I.: Splitting and alternating direction methods. In: Ciarlet, P.G., Lions, J.L.(eds.) Handbook of Numerical Analysis Vol. I, pp. 197–462. North–Holland, Amsterdam (1990)

    Chapter  Google Scholar 

  24. McLachlan, R.I., Reinout, G., Quispel, W.: Splitting methods. Acta Numerica 11, 341–434 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Modeling & Simulation 4 (2), 460–489 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Osher, S., Ruan, F., Yao, Y., Yin, W., Xiong, J.: Sparse recovery via differential inclusions. UCLA CAM Report 14–16 (2014)

    Google Scholar 

  27. Peaceman, D.W., Rachford Jr, H.H.: The numerical solution of parabolic and elliptic differential equations. Journal of the Society for Industrial and Applied Mathematics 3 (1), 28–41 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sheng, Q.: Solving linear partial differential equations by exponential splitting. IMA Journal of Numerical Analysis 9 (2), 199–212 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Sheng, Q.: Global error estimates for exponential splitting. IMA Journal of Numerical Analysis 14 (1), 27–56 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  30. Strang, G.: On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis 5 (3), 506–517 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  31. Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM Journal on Numerical Analysis 46 (4), 2022–2038 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM Journal on Control and Optimization 29 (1), 119–138 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  33. Usadi, A., Dawson, C.: 50 Years of ADI Methods: Celebrating the Contributions of Jim Douglas, Don Peaceman and Henry Rachford. SIAM News 39 (2) (2006)

    Google Scholar 

  34. Wang, Y., Yang, J., Yin, W., Zhang, Y.: A New Alternating Minimization Algorithm for Total Variation Image Reconstruction. SIAM Journal on Imaging Sciences 1 (3), 248–272 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yin, W., Osher, S.: Error forgetting of Bregman iteration. Journal of Scientific Computing 54 (2–3), 684–695 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  36. Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for ℓ 1-minimization with applications to compressed sensing. SIAM Journal on Imaging Sciences 1 (1), 143–168 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM Journal on Imaging Sciences 3 (3), 253–276 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. UCLA CAM Report pp. 08–34 (2008)

    Google Scholar 

Download references

Acknowledgements

All the chapters in this book have been peer-reviewed. We greatly appreciate the voluntary work and experted reviews by the anonymous reviewers. We want to express our deep and sincere gratitude to all the authors, who have made tremendous contributions and offered generous support to this book.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Roland Glowinski

  2. Department of Mathematics, UCLA, Los Angeles, CA, 90095, USA

    Stanley J. Osher & Wotao Yin

Authors
  1. Roland Glowinski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stanley J. Osher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wotao Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roland Glowinski .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, USA

    Roland Glowinski

  2. Department of Mathematics, UCLA, Los Angeles, California, USA

    Stanley J. Osher

  3. Department of Mathematics, UCLA, Los Angeles, California, USA

    Wotao Yin

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Glowinski, R., Osher, S.J., Yin, W. (2016). Introduction. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-41589-5_1

Download citation

Publish with us

Policies and ethics

Search

Navigation