Inverse problems in spaces of measures
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 190-218.

The ill-posed problem of solving linear equations in the space of vector-valued finite Radon measures with Hilbert space data is considered. Approximate solutions are obtained by minimizing the Tikhonov functional with a total variation penalty. The well-posedness of this regularization method and further regularization properties are mentioned. Furthermore, a flexible numerical minimization algorithm is proposed which converges subsequentially in the weak* sense and with rate &#x1d4aa;(n-1) in terms of the functional values. Finally, numerical results for sparse deconvolution demonstrate the applicability for a finite-dimensional discrete data space and infinite-dimensional solution space.

DOI : https://doi.org/10.1051/cocv/2011205
Classification : 65J20,  46E27,  49M05
Mots clés : inverse problems, vector-valued finite Radon measures, Tikhonov regularization, delta-peak solutions, generalized conditional gradient method, iterative soft-thresholding, sparse deconvolution
@article{COCV_2013__19_1_190_0,
author = {Bredies, Kristian and Pikkarainen, Hanna Katriina},
title = {Inverse problems in spaces of measures},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {190--218},
publisher = {EDP-Sciences},
volume = {19},
number = {1},
year = {2013},
doi = {10.1051/cocv/2011205},
zbl = {1266.65083},
mrnumber = {3023066},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2011205/}
}
TY  - JOUR
AU  - Bredies, Kristian
AU  - Pikkarainen, Hanna Katriina
TI  - Inverse problems in spaces of measures
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
DA  - 2013///
SP  - 190
EP  - 218
VL  - 19
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2011205/
UR  - https://zbmath.org/?q=an%3A1266.65083
UR  - https://www.ams.org/mathscinet-getitem?mr=3023066
UR  - https://doi.org/10.1051/cocv/2011205
DO  - 10.1051/cocv/2011205
LA  - en
ID  - COCV_2013__19_1_190_0
ER  - 
Bredies, Kristian; Pikkarainen, Hanna Katriina. Inverse problems in spaces of measures. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 190-218. doi : 10.1051/cocv/2011205. http://www.numdam.org/articles/10.1051/cocv/2011205/

[1] R.A. Adams and J.J.F. Fournier, Sobolev spaces. Academic Press (2003). | MR 2424078 | Zbl 1098.46001

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | MR 1857292 | Zbl 0957.49001

[3] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2 (2009) 183-202. | MR 2486527 | Zbl 1175.94009

[4] T. Bonesky, K.S. Kazimierski, P. Maass, F. Schöpfer and T. Schuster, Minimization of Tikhonov functionals in Banach spaces. Abstr. Appl. Anal. (2008) 192679. | MR 2393115

[5] K. Bredies and D.A. Lorenz, Iterated hard shrinkage for minimization problems with sparsity constraints. SIAM J. Sci. Comput. 30 (2008) 657-683. | MR 2385880 | Zbl 1170.46067

[6] K. Bredies and D.A. Lorenz, Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14 (2008) 813-837. | MR 2461608 | Zbl 1175.65061

[7] 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. | MR 2471395 | Zbl 1179.90326

[8] K. Bredies, T. Alexandrov, J. Decker, D.A. Lorenz and H. Thiele, Sparse deconvolution for peak picking and ion charge estimation in mass spectrometry, in Progress in Industrial Mathematics at ECMI 2008, edited by H.-G. Bock et al., Springer (2010) 287-292. | Zbl 1220.78067

[9] M. Burger and S. Osher, Convergence rates of convex variational regularization. Inverse Prob. 20 (2004) 1411-1421. | MR 2109126 | Zbl 1068.65085

[10] E.J. Candès, J.K. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 (2006) 1207-1223. | MR 2230846 | Zbl 1098.94009

[11] C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM : COCV 17 (2011) 243-266. | Numdam | MR 2775195 | Zbl 1213.49041

[12] P.L. Combettes and V.R. Wajs, Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4 (2005) 1168-1200. | MR 2203849 | Zbl 1179.94031

[13] J.B. Conway, A course in functional analysis. Springer (1990). | MR 1070713 | Zbl 0706.46003

[14] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Comm. Pure Appl. Math. 57 (2004) 1413-1457. | MR 2077704 | Zbl 1077.65055

[15] D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52 (2006) 1289-1306. | MR 2241189 | Zbl 1288.94016

[16] D.L. Donoho, M. Elad and V.N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52 (2006) 6-18. | MR 2237332 | Zbl 1288.94017

[17] C. Dossal and S. Mallat, Sparse spike deconvolution with minimum scale, in Proc. of SPARS'05 (2005).

[18] N. Dunford and J.T. Schwartz, Linear Operators. I. General Theory. Interscience Publishers (1958). | MR 117523 | Zbl 0084.10402

[19] B. Efron, T. Hastie, I. Johnstone and R. Tibshirani, Least angle regression. Ann. Statist. 32 (2004) 407-499. | MR 2060166 | Zbl 1091.62054

[20] I. Ekeland and R. Temam, Convex analysis and variational problems. North-Holland (1976). | MR 463994 | Zbl 0322.90046

[21] H.W. Engl and G. Landl, Convergence rates for maximum entropy regularization. SIAM J. Numer. Anal. 30 (1993) 1509-1536. | MR 1239834 | Zbl 0790.65110

[22] H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic Publishers (1996). | MR 1408680 | Zbl 0859.65054

[23] 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.

[24] I. Fonseca and G. Leoni, Modern methods in the calculus of variations : Lp spaces. Springer (2007). | MR 2341508 | Zbl 1153.49001

[25] M. Fornasier and H. Rauhut, Recovery algorithms for vector valued data with joint sparsity constraints. SIAM J. Numer. Anal. 46 (2008) 577-613. | MR 2383204 | Zbl 1211.65066

[26] J.-J. Fuchs, On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory. 50 (2004) 1341-1344. | MR 2094894 | Zbl 1284.94018

[27] A.L. Gibbs and F.E. Su, On choosing and bounding probability metrics. Int. Stat. Rev. 70 (2002) 419-435. | Zbl 1217.62014

[28] M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with ℓq penalty term. Inverse Prob. 24 (2008) 055020. | MR 2438955 | Zbl 1157.65033

[29] R. Griesse and D.A. Lorenz, A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inverse Prob. 24 (2008) 035007. | MR 2421961 | Zbl 1152.49030

[30] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Publishing Limited (1985). | MR 775683 | Zbl 0695.35060

[31] T. Hein, Tikhonov regularization in Banach spaces - improved convergence rates results. Inverse Prob. 25 (2009) 035002. | MR 2480172 | Zbl 1170.65033

[32] B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Prob. 23 (2007) 987-1010. | MR 2329928 | Zbl 1131.65046

[33] L. Hörmander, The Analysis of Linear Partial Differential Operators I. Springer-Verlag (1990). | MR 1065993 | Zbl 0712.35001

[34] V.K. Ivanov, V.V. Vasin and V.P. Tanana, Theory of linear ill-posed problems and its applications, 2nd edition. Inverse and Ill-posed Problems Series, VSP, Utrecht (2002). | MR 2010817 | Zbl 1037.65056

[35] H. Lee, A. Battle, R. Raina and A.Y. Ng, Efficient sparse coding algorithms, in Advances in Neural Information Processing Systems, edited by B. Schölkopf, J. Platt and T. Hoffman. MIT Press 19 (2007) 801-808.

[36] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function Spaces. Springer (1979). | MR 540367 | Zbl 0852.46015

[37] D.A. Lorenz, Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl. 16 (2008) 463-478. | MR 2442066 | Zbl 1161.65041

[38] D.A. Lorenz and D. Trede, Optimal convergence rates for Tikhonov regularization in Besov scales. Inverse Prob. 24 (2008) 055010. | MR 2438945 | Zbl 1147.49030

[39] D.A. Lorenz and D. Trede, Greedy deconvolution of point-like objects, in Proc. of SPARS'09 (2009).

[40] Y. Mao, B. Dong and S. Osher, A nonlinear PDE-based method for sparse deconvolution. Multiscale Model. Simul. 8 (2010) 965-976. | MR 2644319 | Zbl 1201.35025

[41] L.M. Mugnier, T. Fusco and J.-M. Conan, MISTRAL : a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images. J. Opt. Soc. Am. A 21 (2004) 1841-1854. | MR 2164634

[42] Y.E. Nesterov, A method of solving a convex programming problem with convergence rate O(1/k2). Soviet Math. Dokl. 27 (1983) 372-376. | Zbl 0535.90071

[43] A. Neubauer, On enhanced convergence rates for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. Inverse Prob. 25 (2009) 065009. | MR 2506854 | Zbl 1176.65071

[44] E. Resmerita and O. Scherzer, Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Prob. 22 (2006) 801-814. | MR 2235638 | Zbl 1103.65062

[45] O. Scherzer and B. Walch, Sparsity regularization for Radon measures, in Scale Space and Variational Methods in Computer Vision, edited by X.-C. Tai, K. Morken, M. Lysaker and K.-A. Lie. Springer-Verlag (2009) 452-463.

[46] G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159-181. | MR 2556849 | Zbl 1185.49031

[47] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR 192177 | Zbl 0151.15401

[48] A.S. Stern, D.L. Donoho and J.C. Hoch, NMR data processing using iterative thresholding and minimum l1-norm reconstruction. J. Magn. Reson. 188 (2007) 295-300.

[49] A.N. Tikhonov, A.S. Leonov and A.G. Yagola, Nonlinear ill-posed problems 1. Chapman & Hall (1998). | MR 1630660 | Zbl 0920.65038

[50] Z.B. Xu and G.F. Roach, Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157 (1991) 189-210. | MR 1109451 | Zbl 0757.46034

[51] C. Zălinescu, Convex analysis in general vector spaces. World Scientific (2002). | Zbl 1023.46003

[52] E. Zeidler, Nonlinear Functional Analysis and its Applications III. Springer-Verlag (1985). | MR 768749 | Zbl 0583.47051

Cité par Sources :