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An efficient projection method for nonlinear inverse problems with sparsity constraints
1. | School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210023, China, China, China |
2. | School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798 Singapore |
References:
[1] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[2] |
K. Bredies, D. A. Lorenz and P. Maass, A generalized conditional gradient method and its connection to an iterative shrinkage method, Comput. Optim. Appl., 42 (2009), 173-193.
doi: 10.1007/s10589-007-9083-3. |
[3] |
X. J. Cai, G. Y. Gu and B. S. He, A proximal point algorithm revisit on the alternating direction method of multipliers, Sci. China Math., 56 (2013), 2179-2186.
doi: 10.1007/s11425-013-4683-0. |
[4] |
I. Daubechies, M. Defrise and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457.
doi: 10.1002/cpa.20042. |
[5] |
I. Daubechies, M. Fornasier and I. Loris, Accelerated projected gradient methods for linear inverse problems with sparsity constraints, J. Fourier Anal. Appl., 14 (2008), 764-792.
doi: 10.1007/s00041-008-9039-8. |
[6] |
I. Daubechies, G. Teschke and L. Vese, Iteratively solving linear inverse problems with general convex constraints, Inverse Problems Imag., 1 (2007), 29-46.
doi: 10.3934/ipi.2007.1.29. |
[7] |
I. Daubechies, G. Teschke and L. Vese, On some iterative concepts for image restoration, Adv. Imag. Electron Phys., 150 (2008), 1-51.
doi: 10.1016/S1076-5670(07)00001-8. |
[8] |
Y. C. Eldar, T. G. Dvorkind and E. Matusiak, Nonlinear and non-ideal sampling: Theory and methods, IEEE Trans. Signal Process., 56 (2008), 5874-5890.
doi: 10.1109/TSP.2008.929872. |
[9] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Springer, Netherlands, 1996.
doi: 10.1007/978-94-009-1740-8. |
[10] |
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volumes I and II, Springer-Verlag, Berlin, 2003. |
[11] |
M. Fornasier, Domain decomposition methods for linear inverse problems with sparsity constraints, Inverse Problems, 23 (2007), 2505-2526.
doi: 10.1088/0266-5611/23/6/014. |
[12] |
M. A. T. Figueiredo, R. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Sel. Top. Signal Process., 1 (2007), 586-597.
doi: 10.1109/JSTSP.2007.910281. |
[13] |
M. Fornasier and H. Rauhut, Recovery algorithms for vector valued data with joint sparsity constraint, SIAM J. Numer. Anal., 46 (2008), 577-613.
doi: 10.1137/0606668909. |
[14] |
Z. L. Ge, G. Qian and D. R. Han, Global convergence of an inexact operator splitting method for monotone variational inequalities, J. Ind. Man. Optim., 7 (2011), 1013-1026.
doi: 10.3934/jimo.2011.7.1013. |
[15] |
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strong monotone variational inequality, J. Optim. Theory Appl., 112 (2002), 129-143.
doi: 10.1023/A:1013048729944. |
[16] |
T. Hein and K. S. Kazimierski, Accelerated Landweber iteration in Banach spaces, Inverse Problems, 26 (2010), 055002 (17pp).
doi: 10.1088/0266-5611/26/5/055002. |
[17] |
B. Kaltenbacher, F. Schöpfer and T. Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems, Inverse Problems, 25 (2009), 065003 (19pp).
doi: 10.1088/0266-5611/25/6/065003. |
[18] |
R. Ramlau and G. Teschke, Tikhonov replacement functionals for iteratively solving nonlinear operator equations, Inverse Problems, 21 (2005), 1571-1592.
doi: 10.1088/0266-5611/21/5/005. |
[19] |
R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints, Numer. Math., 104 (2006), 177-203.
doi: 10.1007/s00211-006-0016-3. |
[20] |
G. Teschke, Multi-frame representations in linear inverse problems with mixed multi-constraints, Appl. Comput. Harmon. Anal., 22 (2007), 43-60.
doi: 10.1016/j.acha.2006.05.003. |
[21] |
G. Teschke and C. Borries, Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints, Inverse Problems, 26 (2010), 025007 (23pp).
doi: 10.1088/0266-5611/26/2/025007. |
[22] |
E. van den Berg and M. P. Friedlander, Probing the Pareto frontier for basic pursuit solutions, SIAM J. Sci. Comput., 31 (2008), 890-912.
doi: 10.1137/080714488. |
[23] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications III, Springer, New York, 1985.
doi: 10.1007/978-1-4612-5020-3. |
[24] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Springer, New York, 1986.
doi: 10.1007/978-1-4612-4838-5. |
[25] |
W. X. Zhang, D. R. Han and Z. B. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Problems, 25 (2009), 115001 (16pp).
doi: 10.1088/0266-5611/25/11/115001. |
[26] |
T. Zhu and Z. Yu, A simple proof for some important properties of the projection mapping, Mathematical Inequalities and Applications, 7 (2004), 453-456.
doi: 10.7153/mia-07-45. |
show all references
References:
[1] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[2] |
K. Bredies, D. A. Lorenz and P. Maass, A generalized conditional gradient method and its connection to an iterative shrinkage method, Comput. Optim. Appl., 42 (2009), 173-193.
doi: 10.1007/s10589-007-9083-3. |
[3] |
X. J. Cai, G. Y. Gu and B. S. He, A proximal point algorithm revisit on the alternating direction method of multipliers, Sci. China Math., 56 (2013), 2179-2186.
doi: 10.1007/s11425-013-4683-0. |
[4] |
I. Daubechies, M. Defrise and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457.
doi: 10.1002/cpa.20042. |
[5] |
I. Daubechies, M. Fornasier and I. Loris, Accelerated projected gradient methods for linear inverse problems with sparsity constraints, J. Fourier Anal. Appl., 14 (2008), 764-792.
doi: 10.1007/s00041-008-9039-8. |
[6] |
I. Daubechies, G. Teschke and L. Vese, Iteratively solving linear inverse problems with general convex constraints, Inverse Problems Imag., 1 (2007), 29-46.
doi: 10.3934/ipi.2007.1.29. |
[7] |
I. Daubechies, G. Teschke and L. Vese, On some iterative concepts for image restoration, Adv. Imag. Electron Phys., 150 (2008), 1-51.
doi: 10.1016/S1076-5670(07)00001-8. |
[8] |
Y. C. Eldar, T. G. Dvorkind and E. Matusiak, Nonlinear and non-ideal sampling: Theory and methods, IEEE Trans. Signal Process., 56 (2008), 5874-5890.
doi: 10.1109/TSP.2008.929872. |
[9] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Springer, Netherlands, 1996.
doi: 10.1007/978-94-009-1740-8. |
[10] |
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volumes I and II, Springer-Verlag, Berlin, 2003. |
[11] |
M. Fornasier, Domain decomposition methods for linear inverse problems with sparsity constraints, Inverse Problems, 23 (2007), 2505-2526.
doi: 10.1088/0266-5611/23/6/014. |
[12] |
M. A. T. Figueiredo, R. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Sel. Top. Signal Process., 1 (2007), 586-597.
doi: 10.1109/JSTSP.2007.910281. |
[13] |
M. Fornasier and H. Rauhut, Recovery algorithms for vector valued data with joint sparsity constraint, SIAM J. Numer. Anal., 46 (2008), 577-613.
doi: 10.1137/0606668909. |
[14] |
Z. L. Ge, G. Qian and D. R. Han, Global convergence of an inexact operator splitting method for monotone variational inequalities, J. Ind. Man. Optim., 7 (2011), 1013-1026.
doi: 10.3934/jimo.2011.7.1013. |
[15] |
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strong monotone variational inequality, J. Optim. Theory Appl., 112 (2002), 129-143.
doi: 10.1023/A:1013048729944. |
[16] |
T. Hein and K. S. Kazimierski, Accelerated Landweber iteration in Banach spaces, Inverse Problems, 26 (2010), 055002 (17pp).
doi: 10.1088/0266-5611/26/5/055002. |
[17] |
B. Kaltenbacher, F. Schöpfer and T. Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems, Inverse Problems, 25 (2009), 065003 (19pp).
doi: 10.1088/0266-5611/25/6/065003. |
[18] |
R. Ramlau and G. Teschke, Tikhonov replacement functionals for iteratively solving nonlinear operator equations, Inverse Problems, 21 (2005), 1571-1592.
doi: 10.1088/0266-5611/21/5/005. |
[19] |
R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints, Numer. Math., 104 (2006), 177-203.
doi: 10.1007/s00211-006-0016-3. |
[20] |
G. Teschke, Multi-frame representations in linear inverse problems with mixed multi-constraints, Appl. Comput. Harmon. Anal., 22 (2007), 43-60.
doi: 10.1016/j.acha.2006.05.003. |
[21] |
G. Teschke and C. Borries, Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints, Inverse Problems, 26 (2010), 025007 (23pp).
doi: 10.1088/0266-5611/26/2/025007. |
[22] |
E. van den Berg and M. P. Friedlander, Probing the Pareto frontier for basic pursuit solutions, SIAM J. Sci. Comput., 31 (2008), 890-912.
doi: 10.1137/080714488. |
[23] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications III, Springer, New York, 1985.
doi: 10.1007/978-1-4612-5020-3. |
[24] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Springer, New York, 1986.
doi: 10.1007/978-1-4612-4838-5. |
[25] |
W. X. Zhang, D. R. Han and Z. B. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Problems, 25 (2009), 115001 (16pp).
doi: 10.1088/0266-5611/25/11/115001. |
[26] |
T. Zhu and Z. Yu, A simple proof for some important properties of the projection mapping, Mathematical Inequalities and Applications, 7 (2004), 453-456.
doi: 10.7153/mia-07-45. |
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