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 𝒪(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.
Keywords: 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},
year = {2013},
publisher = {EDP Sciences},
volume = {19},
number = {1},
doi = {10.1051/cocv/2011205},
mrnumber = {3023066},
zbl = {1266.65083},
language = {en},
url = {https://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 SP - 190 EP - 218 VL - 19 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2011205/ DO - 10.1051/cocv/2011205 LA - en ID - COCV_2013__19_1_190_0 ER -
%0 Journal Article %A Bredies, Kristian %A Pikkarainen, Hanna Katriina %T Inverse problems in spaces of measures %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 190-218 %V 19 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2011205/ %R 10.1051/cocv/2011205 %G en %F COCV_2013__19_1_190_0
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
[1] and , Sobolev spaces. Academic Press (2003). | Zbl | MR
[2] , and , Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | Zbl | MR
[3] and , A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2 (2009) 183-202. | Zbl | MR
[4] , , , and , Minimization of Tikhonov functionals in Banach spaces. Abstr. Appl. Anal. (2008) 192679. | MR
[5] and , Iterated hard shrinkage for minimization problems with sparsity constraints. SIAM J. Sci. Comput. 30 (2008) 657-683. | Zbl | MR
[6] and , Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14 (2008) 813-837. | Zbl | MR
[7] , and , A generalized conditional gradient method and its connection to an iterative shrinkage method. Comput. Optim. Appl. 42 (2009) 173-193. | Zbl | MR
[8] , , , and , 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
[9] and , Convergence rates of convex variational regularization. Inverse Prob. 20 (2004) 1411-1421. | Zbl | MR
[10] , and , Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 (2006) 1207-1223. | Zbl | MR
[11] and , A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM : COCV 17 (2011) 243-266. | Zbl | MR | Numdam
[12] and , Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4 (2005) 1168-1200. | Zbl | MR
[13] , A course in functional analysis. Springer (1990). | Zbl | MR
[14] , and , An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Comm. Pure Appl. Math. 57 (2004) 1413-1457. | Zbl | MR
[15] , Compressed sensing. IEEE Trans. Inf. Theory 52 (2006) 1289-1306. | Zbl | MR
[16] , and , Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52 (2006) 6-18. | Zbl | MR
[17] and , Sparse spike deconvolution with minimum scale, in Proc. of SPARS'05 (2005).
[18] and , Linear Operators. I. General Theory. Interscience Publishers (1958). | Zbl | MR
[19] , , and , Least angle regression. Ann. Statist. 32 (2004) 407-499. | Zbl | MR
[20] and , Convex analysis and variational problems. North-Holland (1976). | Zbl | MR
[21] and , Convergence rates for maximum entropy regularization. SIAM J. Numer. Anal. 30 (1993) 1509-1536. | Zbl | MR
[22] , and , Regularization of Inverse Problems. Kluwer Academic Publishers (1996). | Zbl | MR
[23] , and , Gradient projection for sparse reconstruction : Application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1 (2007) 586-597.
[24] and , Modern methods in the calculus of variations : Lp spaces. Springer (2007). | Zbl | MR
[25] and , Recovery algorithms for vector valued data with joint sparsity constraints. SIAM J. Numer. Anal. 46 (2008) 577-613. | Zbl | MR
[26] , On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory. 50 (2004) 1341-1344. | Zbl | MR
[27] and , On choosing and bounding probability metrics. Int. Stat. Rev. 70 (2002) 419-435. | Zbl
[28] , and , Sparse regularization with ℓq penalty term. Inverse Prob. 24 (2008) 055020. | Zbl | MR
[29] and , A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inverse Prob. 24 (2008) 035007. | Zbl | MR
[30] , Elliptic Problems in Nonsmooth Domains. Pitman Publishing Limited (1985). | Zbl | MR
[31] , Tikhonov regularization in Banach spaces - improved convergence rates results. Inverse Prob. 25 (2009) 035002. | Zbl | MR
[32] , , and , A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Prob. 23 (2007) 987-1010. | Zbl | MR
[33] , The Analysis of Linear Partial Differential Operators I. Springer-Verlag (1990). | Zbl | MR
[34] , and , Theory of linear ill-posed problems and its applications, 2nd edition. Inverse and Ill-posed Problems Series, VSP, Utrecht (2002). | Zbl | MR
[35] , , and , 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] and , Classical Banach Spaces II. Function Spaces. Springer (1979). | Zbl | MR
[37] , Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl. 16 (2008) 463-478. | Zbl | MR
[38] and , Optimal convergence rates for Tikhonov regularization in Besov scales. Inverse Prob. 24 (2008) 055010. | Zbl | MR
[39] and , Greedy deconvolution of point-like objects, in Proc. of SPARS'09 (2009).
[40] , and , A nonlinear PDE-based method for sparse deconvolution. Multiscale Model. Simul. 8 (2010) 965-976. | Zbl | MR
[41] , and , 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
[42] , A method of solving a convex programming problem with convergence rate O(1/k2). Soviet Math. Dokl. 27 (1983) 372-376. | Zbl
[43] , On enhanced convergence rates for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. Inverse Prob. 25 (2009) 065009. | Zbl | MR
[44] and , Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Prob. 22 (2006) 801-814. | Zbl | MR
[45] and , 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] , Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159-181. | Zbl | MR
[47] , Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Zbl | MR | Numdam
[48] , and , NMR data processing using iterative thresholding and minimum l1-norm reconstruction. J. Magn. Reson. 188 (2007) 295-300.
[49] , and , Nonlinear ill-posed problems 1. Chapman & Hall (1998). | Zbl | MR
[50] and , Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157 (1991) 189-210. | Zbl | MR
[51] , Convex analysis in general vector spaces. World Scientific (2002). | Zbl
[52] , Nonlinear Functional Analysis and its Applications III. Springer-Verlag (1985). | Zbl | MR
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