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A conjugate gradient like method for p-norm minimization in functional spaces

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Abstract

We develop an iterative algorithm to recover the minimum p-norm solution of the functional linear equation \(Ax = b,\) where \(A: \mathcal {X}\longrightarrow \mathcal {Y}\,\) is a continuous linear operator between the two Banach spaces \(\mathcal {X}= L^p\), \(1< p < 2\), and \(\mathcal {Y}= L^r\), \(r > 1\), with \(x \in \mathcal {X}\) and \(b \in \mathcal {Y}\). The algorithm is conceived within the same framework of the Landweber method for functional linear equations in Banach spaces proposed by Schöpfer et al. (Inverse Probl 22:311–329, 2006). Indeed, the algorithm is based on using, at the n-th iteration, a linear combination of the steepest current “descent functional” \(A^* J \left( b - A x_n \right) \) and the previous descent functional, where J denotes a duality map of the Banach space \(\mathcal {Y}\). In this regard, the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in Hilbert spaces. We demonstrate that the proposed iterative algorithm converges strongly to the minimum p-norm solution of the functional linear equation \(Ax = b\) and that it is also a regularization method, by applying the discrepancy principle as stopping rule. According to the geometrical properties of \(L^p\) spaces, numerical experiments show that the method is fast, robust in terms of both restoration accuracy and stability, promotes sparsity and reduces the over-smoothness in reconstructing edges and abrupt intensity changes.

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References

  1. Asplund, E.: Positivity of duality mappings. Bull. Am. Math. Soc. 73, 200–203 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging, pp. 98–127. IOP Publishing, London (1998)

    Book  MATH  Google Scholar 

  3. Bonesky, K., Kazimiersky, K., Maass, P., Schöpfer, F., Schuster, T.: Minimization of Tikhonov functional in Banach spaces. Abstr. Appl. Anal. 2008, 1–20 (2008)

  4. Bregman, L.M.: The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  5. Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Probl. 20, 1411–1421 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Butnariu, D., Resmerita, E.: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. 2006, 1–39 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems. In: Hazewind, M. (ed.) Mathematics and its Applications, vol. 62, pp. 1–87. Kluwer Academic Publishers, Dordrecht (1990)

  8. Daubachies, L., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with sparsity constraint. Commun. Pure Appl. Math. 57, 1413–1457 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brianzi, P., Di Benedetto, F., Estatico, C.: Preconditioned iterative regularization in Banach spaces. Comput. Optim. Appl. 54, 263–282 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Grasmair, M.: Generalized Bregman distances and convergences rate for non-convex regularization methods. Inverse Probl. 26, 1–20 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gratton, S., Toint, P.L., Tröltzsch, A.: An active-set trust-region method for derivative-free nonlinear bound-constrained optimization. Optim. Methods Softw. 26, 873–894 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hansen, P.C.: Rank-deficient and discrete ill-posed problem: numerical aspects of linear inversions. SIAM, Philadelphia (1998)

  13. Herzog, R., Wollner, W.: A conjugate direction method for linear systems in Banach spaces. J. Inverse Ill Posed Probl. (2016). doi:10.1515/jiip-2016-0027

  14. Kien, B.T.: The normalized duality mapping and two related characteristic properties of a uniformly convex Banach space. Acta Math. Vietnam. 27, 53–67 (2002)

    MATH  MathSciNet  Google Scholar 

  15. Lenti, F., Nunziata, F., Estatico, C., Migliaccio, M.: Analysis of reconstructions obtained solving \(l^p\)-penalized minimization problems. IEEE Trans. Geosci. Remote Sens. 53, 4876–4886 (2015)

    Article  Google Scholar 

  16. Lenti, F., Nunziata, F., Estatico, C., Migliaccio, M.: Conjugate gradient method in Hilbert and Banach spaces to enhance the spatial resolution of radiometer data. IEEE Trans. Geosci. Remote Sens. 54, 397–406 (2016)

    Article  Google Scholar 

  17. Lindestrauss, J., Tzafriri, L.: Classical Banach Spaces, pp. 53–104. Springer, Berlin (1996)

    Google Scholar 

  18. Milicic, P.M.: On moduli of expansion of the duality mapping of smooth Banach spaces. J. Inequal. Pure Appl. Math. 3, 1–7 (2002)

    MATH  MathSciNet  Google Scholar 

  19. Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21, 1303–1314 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Schöpfer, F., Louis, A.K., Schuster, T.: Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Probl. 22, 311–329 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Schöpfer, F., Schuster, T.: Fast regularizing sequential subspace optimization in Banach spaces. Inverse Probl. 25, 1–22 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Schöpfer, F., Schuster, T., Louis, A.K.: Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods. J. Inverse Ill Posed Probl. 15, 479–506 (2007)

    MATH  MathSciNet  Google Scholar 

  23. Schöpfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Probl. 24, 1–20 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Scheinberg, K., Toint, Ph.L.: Self-correcting geometry in model based algorithms for derivative-free unconstrained optimization. Technical report, TR09/06, Department of Mathematics University of Namur, Belgium (2009)

  25. Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.: Regularization methods in Banach spaces. In: de Gruyter, W. (ed.) Radon Series on Computational and Applied Mathematics, vol. 10. de Gruyter, Berlin (2012)

  26. Xu, Z.B., Roach, G.F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157, 189–210 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The work of C. Estatico is partly supported by PRIN 2012 2012MTE38N and GNCS-INdAM. The work of F. Lenti is supported by the AVENUE Project of the RTRA STAE foundation.

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Estatico, C., Gratton, S., Lenti, F. et al. A conjugate gradient like method for p-norm minimization in functional spaces. Numer. Math. 137, 895–922 (2017). https://doi.org/10.1007/s00211-017-0893-7

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  • DOI: https://doi.org/10.1007/s00211-017-0893-7

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