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Handbook of Geomathematics pp 1–33Cite as

Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery

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Abstract

Inverse problems in seismic tomography are often cast in the form of an optimization problem involving a cost function composed of a data misfit term and regularizing constraint or penalty. Depending on the noise model that is assumed to underlie the data acquisition, these optimization problems may be non-smooth. Another source of lack of smoothness (differentiability) of the cost function may arise from the regularization method chosen to handle the ill-posed nature of the inverse problem. A numerical algorithm that is well suited to handle minimization problems involving two non-smooth convex functions and two linear operators is studied. The emphasis lies on the use of some simple proximity operators that allow for the iterative solution of non-smooth convex optimization problems. Explicit formulas for several of these proximity operators are given and their application to seismic tomography is demonstrated.

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References

  • Afonso MV, Bioucas-Dias JM, Figueiredo MAT (2010) Fast image recovery using variable splitting and constrained optimization. IEEE Trans Image Process 19(9):2345–2356

    Article  MathSciNet  Google Scholar 

  • Beck A, Teboulle M (2009) Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans Image Process 18:2419–2434

    Article  MathSciNet  Google Scholar 

  • Beck A, Teboulle M (2009) A fast iterative shrinkage-threshold algorithm for linear inverse problems. SIAM J Imaging Sci 2:183–202

    Article  MathSciNet  MATH  Google Scholar 

  • Bonesky T (2009) Morozov’s discrepancy principle and Tikhonov-type functionals. Inverse Probl 25(1):015015 (11pp)

    Google Scholar 

  • Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press

    Book  MATH  Google Scholar 

  • Bredies K (2009) A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space. Inverse Probl 25:015005

    Article  MathSciNet  Google Scholar 

  • Bruckstein AM, Donoho DL, Elad M (2009) From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev 51(1):34–81

    Article  MathSciNet  MATH  Google Scholar 

  • Candès EJ, Romberg J (2004) Practical signal recovery from random projections. In: Wavelet applications in signal and image processing XI, Proceedings of the SPIE Conference, vol 5914, San Diego CA, USA

    Google Scholar 

  • Chambolle A, Pock T (2011) A first-order primal-dual algorithm for convex problems with applications to imaging. J Math Imaging Vis 40:120–145 hal-00490826

    Google Scholar 

  • Chen P, Huang J, Zhang X (2013) A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration. Inverse Probl 29(2):025011 (33pp)

    Google Scholar 

  • Claerbout JF, Muir F (1973) Robust modelling with erratic data. Geophysics 38(5):826–844

    Article  Google Scholar 

  • Combettes PL, Pesquet J-C (2011) Proximal splitting methods in signal processing, chapter 1. In: Bauschke HH, Burachik RS, Combettes PL, Elser V, Luke DR, Wolkowicz H (eds) Fixed-point algorithms for inverse problems in science and engineering. Springer-Verlag, pp 185–212

    Google Scholar 

  • Combettes PL, Wajs VR (2005) Signal recovery by proximal forward-backward splitting. Multiscale Model Simul 4(4):1168–1200

    Article  MathSciNet  MATH  Google Scholar 

  • Daubechies I, Defrise M, De Mol C (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun Pure Appl Math 57(11):1413–1457

    Article  MATH  Google Scholar 

  • Daubechies I, Teschke G, Vese L (2007) Iteratively solving linear inverse problems under general convex constraints. Inverse Probl Imaging 1:29–46

    Article  MathSciNet  MATH  Google Scholar 

  • Daubechies I, Fornasier M, Loris I (2008) Accelerated projected gradient method for linear inverse problems with sparsity constraints. J Fourier Anal Appl 14(5–6):764–792

    Article  MathSciNet  MATH  Google Scholar 

  • Defrise M, Vanhove C, Liu X (2011) An algorithm for total variation regularization in high-dimensional linear problems. Inverse Probl 27(6):065002

    Article  MathSciNet  Google Scholar 

  • Engl HW, Hanke M, Neubauer A (2000) Regularization of inverse problems. Kluwer Academic Publishers, Dordrecht (The Netherlands)

    Google Scholar 

  • Esser JE (2010) Primal dual algorithms for convex models and applications to image restoration, registration and nonlocal inpainting. PhD thesis, University of California

    Google Scholar 

  • Esser E, Zhang X, Chan TF (2010) A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J Imaging Sci 3(4):1015–1046

    Article  MathSciNet  MATH  Google Scholar 

  • Figueiredo MAT, Nowak RD, Wright SJ (2007) Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J Sel Top Signal Process (Special Issue Convex Optim Methods Signal Process) 1:586–598

    Article  Google Scholar 

  • Fornasier M, Rauhut H (2008) Recovery algorithms for vector-valued data with joint sparsity constraints. SIAM J Numer Anal 46:577–613

    Article  MathSciNet  MATH  Google Scholar 

  • Gholami A, Siahkoohi HR (2010) Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints. Geophys J Int 180(2):871–882

    Article  Google Scholar 

  • Goldstein AA (1964) Convex programming in Hilbert space. Bull Am Math Soc 70:709–710

    Article  MATH  Google Scholar 

  • Grasmair M, Haltmeier M, Scherzer O (2011) The residual method for regularizing ill-posed problems. Appl Math Comput 218(6):2693–2710

    Article  MathSciNet  MATH  Google Scholar 

  • Hale E, Yin W, Zhang Y (2008) Fixed-point continuation for ℓ 1-minimization: Methodology and convergence. SIAM J Optim 19(3):1107–1130

    Article  MathSciNet  MATH  Google Scholar 

  • Hansen C (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34(4):561–580

    Article  MathSciNet  MATH  Google Scholar 

  • Hennenfent G, van den Berg E, Friedlander MP, Herrmann FJ (2008) New insights into one-norm solvers from the Pareto curve. Geophysics 73(4):A23–A26

    Article  Google Scholar 

  • Herrmann FJ, Hennenfent G (2008) Non-parametric seismic data recovery with curvelet frames. Geophys J Int 173(1):233–248

    Article  Google Scholar 

  • Herrmann FJ, Moghaddam PP, Stolk C (2008) Sparsity- and continuity-promoting seismic image recovery with curvelet frames. Appl Comput Harmonic Anal 24(2):150–173

    Article  MathSciNet  MATH  Google Scholar 

  • Ivanov VV (1976) The theory of approximate methods and their application to the numerical solution of singular integral equations. Nordhoff International, Leyden

    MATH  Google Scholar 

  • Kim S-J, Koh K, Lustig M, Boyd S, Gorinevsky D (2007) An interior-point method for large-scale ℓ 1-regularized least squares. IEEE J Sel Top Signal Process 1(4):606–617

    Article  Google Scholar 

  • Levitin ES, Polyak BT (1966) Constrained minimization problems. USSR Comput Math Math Phys 6:1–50

    Article  Google Scholar 

  • Loris I, Verhoeven C (2011) On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty. Inverse Probl 27(12):125007

    Article  MathSciNet  Google Scholar 

  • Loris I, Verhoeven C (2012) Iterative algorithms for total variation-like reconstructions in seismic tomography. Int J Geomath 3(2):179–208

    Article  MathSciNet  MATH  Google Scholar 

  • Loris I, Nolet G, Daubechies I, Dahlen FA (2007) Tomographic inversion using ℓ 1-norm regularization of wavelet coefficients. Geophys J Int 170(1):359–370

    Article  Google Scholar 

  • Loris I, Bertero M, De Mol C, Zanella R, Zanni L (2009) Accelerating gradient projection methods for ℓ 1-constrained signal recovery by steplength selection rules. Appl Comput Harmon Anal 27(2):247–254

    Article  MathSciNet  MATH  Google Scholar 

  • Loris I, Douma H, Nolet G, Daubechies I, Regone C (2010) Nonlinear regularization techniques for seismic tomography. J Comput Phys 229(3):890–905

    Article  MathSciNet  MATH  Google Scholar 

  • Mi T, Li S, Liu Y (2012) Fast thresholding algorithms with feedbacks for sparse signal recovery. ArXiv: abs/1204.3700

    Google Scholar 

  • Moreau JJ (1965) Proximité et dualité dans un espace hilbertien. Bull Soc Math France 93:273–299

    MathSciNet  MATH  Google Scholar 

  • Morozov VA (1966) On the solution of functional equations by the method of regularization. Soviet Math Dokl 7:414–417

    MathSciNet  MATH  Google Scholar 

  • Nikazad T, Davidi R, Herman GT (2012) Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction. Inverse Probl 28(3):035005 (19pp)

    Google Scholar 

  • Nolet G (2008) A breviary of seismic tomography. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Polyak BT (1987) Introduction to optimization. Optimization Software, New York

    Google Scholar 

  • Rezghi M, Hosseini SM (2009) A new variant of L-curve for Tikhonov regularization. J Comput Appl Math 231(2):914–924

    Article  MathSciNet  MATH  Google Scholar 

  • Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Phys D Nonlin Phenom 60(1–4):259–268

    Article  MATH  Google Scholar 

  • Santosa F, Symes WW (1986) Linear inversion of band-limited reflection seismograms. SIAM J Sci Stat Comput 7:1307–1330

    Article  MathSciNet  MATH  Google Scholar 

  • Strong D, Chan T (2003) Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl 19(6):S165

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor HL, Banks SC, McCoy JF (1979) Deconvolution with the ℓ 1 norm. Geophysics 44:39–52

    Article  Google Scholar 

  • Tikhonov AN (1963) Solution of incorrectly formulated problems and the regularization method. Soviet Math Dokl 4:1035–1038

    Google Scholar 

  • Trampert J, Woodhouse JH (1995) Global phase-velocity maps of Love and Rayleigh-waves between 40 and 150 seconds. Geophys J Int 122(2):675–690

    Article  Google Scholar 

  • Trampert J, Woodhouse JH (1996) High resolution global phase velocity distributions. Geophys Res Lett 23(1):21–24

    Article  Google Scholar 

  • Trampert J, Woodhouse JH (2001) Assessment of global phase velocity models. Geophys J Int 144(1):165–174. doi: 10.1046/j.1365-246x.2001.00307.x

    Article  Google Scholar 

  • van den Berg E, Friedlander MP (2007) SPGL1: a solver for large-scale sparse reconstructionhttp://www.cs.ubc.ca/labs/scl/spgl1

  • van den Berg E, Friedlander MP (2008) Probing the Pareto frontier for basis pursuit solutions. SIAM J Sci Comput 31(2):890–912

    Article  MathSciNet  MATH  Google Scholar 

  • van den Berg E, Friedlander MP (2011) Sparse optimization with least-squares constraints. SIAM J Optim 21(4):1201–1229

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang X, Burger M, Osher S (2011) A unified primal-dual algorithm framework based on Bregman iteration. J Sci Comput 46:20–46

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu M, Chan T (2008) An efficient primal-dual hybrid gradient algorithm for total variation image restoration. Technical report, UCLA

    Google Scholar 

Download references

Acknowledgements

I.L. is a research associate of the Fonds de la Recherche Scientifique (FNRS).

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Authors and Affiliations

  1. Université libre de Bruxelles, Bruxelles, Belgium

    Ignace Loris

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  1. Ignace Loris
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Correspondence to Ignace Loris .

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Editors and Affiliations

  1. Geomathematische Gruppe, TU Kaiserslautern FB Mathematik, Kaiserslautern, Germany

    Willi Freeden

  2. University of Central Florida Dept. Mathematics, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Inst. Analysis, TU Braunschweig FB 1 Mathematik und Informatik, Braunschweig, Germany

    Thomas Sonar

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Loris, I. (2014). Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27793-1_65-3

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  • DOI: https://doi.org/10.1007/978-3-642-27793-1_65-3

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  • Online ISBN: 978-3-642-27793-1

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Chapter history

  1. Latest

    Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery
    Published:
    17 September 2014

    DOI: https://doi.org/10.1007/978-3-642-27793-1_65-3

  2. Original

    Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery
    Published:
    21 August 2014

    DOI: https://doi.org/10.1007/978-3-642-27793-1_65-2

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