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.
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}, mrnumber = {3023066}, zbl = {1266.65083}, 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 SP - 190 EP - 218 VL - 19 IS - 1 PB - EDP-Sciences UR - http://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 http://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. http://www.numdam.org/articles/10.1051/cocv/2011205/
[1] Sobolev spaces. Academic Press (2003). | MR | Zbl
and ,[2] Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | MR | Zbl
, and ,[3] A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2 (2009) 183-202. | MR | Zbl
and ,[4] Minimization of Tikhonov functionals in Banach spaces. Abstr. Appl. Anal. (2008) 192679. | MR
, , , and ,[5] Iterated hard shrinkage for minimization problems with sparsity constraints. SIAM J. Sci. Comput. 30 (2008) 657-683. | MR | Zbl
and ,[6] Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14 (2008) 813-837. | MR | Zbl
and ,[7] A generalized conditional gradient method and its connection to an iterative shrinkage method. Comput. Optim. Appl. 42 (2009) 173-193. | MR | Zbl
, and ,[8] 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
, , , and ,[9] Convergence rates of convex variational regularization. Inverse Prob. 20 (2004) 1411-1421. | MR | Zbl
and ,[10] Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 (2006) 1207-1223. | MR | Zbl
, and ,[11] A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM : COCV 17 (2011) 243-266. | Numdam | MR | Zbl
and ,[12] Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4 (2005) 1168-1200. | MR | Zbl
and ,[13] A course in functional analysis. Springer (1990). | MR | Zbl
,[14] An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Comm. Pure Appl. Math. 57 (2004) 1413-1457. | MR | Zbl
, and ,[15] Compressed sensing. IEEE Trans. Inf. Theory 52 (2006) 1289-1306. | MR | Zbl
,[16] Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52 (2006) 6-18. | MR | Zbl
, and ,[17] Sparse spike deconvolution with minimum scale, in Proc. of SPARS'05 (2005).
and ,[18] Linear Operators. I. General Theory. Interscience Publishers (1958). | MR | Zbl
and ,[19] Least angle regression. Ann. Statist. 32 (2004) 407-499. | MR | Zbl
, , and ,[20] Convex analysis and variational problems. North-Holland (1976). | MR | Zbl
and ,[21] Convergence rates for maximum entropy regularization. SIAM J. Numer. Anal. 30 (1993) 1509-1536. | MR | Zbl
and ,[22] Regularization of Inverse Problems. Kluwer Academic Publishers (1996). | MR | Zbl
, and ,[23] Gradient projection for sparse reconstruction : Application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1 (2007) 586-597.
, and ,[24] Modern methods in the calculus of variations : Lp spaces. Springer (2007). | MR | Zbl
and ,[25] Recovery algorithms for vector valued data with joint sparsity constraints. SIAM J. Numer. Anal. 46 (2008) 577-613. | MR | Zbl
and ,[26] On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory. 50 (2004) 1341-1344. | MR | Zbl
,[27] On choosing and bounding probability metrics. Int. Stat. Rev. 70 (2002) 419-435. | Zbl
and ,[28] Sparse regularization with ℓq penalty term. Inverse Prob. 24 (2008) 055020. | MR | Zbl
, and ,[29] A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inverse Prob. 24 (2008) 035007. | MR | Zbl
and ,[30] Elliptic Problems in Nonsmooth Domains. Pitman Publishing Limited (1985). | MR | Zbl
,[31] Tikhonov regularization in Banach spaces - improved convergence rates results. Inverse Prob. 25 (2009) 035002. | MR | Zbl
,[32] A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Prob. 23 (2007) 987-1010. | MR | Zbl
, , and ,[33] The Analysis of Linear Partial Differential Operators I. Springer-Verlag (1990). | MR | Zbl
,[34] Theory of linear ill-posed problems and its applications, 2nd edition. Inverse and Ill-posed Problems Series, VSP, Utrecht (2002). | MR | Zbl
, and ,[35] 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.
, , and ,[36] Classical Banach Spaces II. Function Spaces. Springer (1979). | MR | Zbl
and ,[37] Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl. 16 (2008) 463-478. | MR | Zbl
,[38] Optimal convergence rates for Tikhonov regularization in Besov scales. Inverse Prob. 24 (2008) 055010. | MR | Zbl
and ,[39] Greedy deconvolution of point-like objects, in Proc. of SPARS'09 (2009).
and ,[40] A nonlinear PDE-based method for sparse deconvolution. Multiscale Model. Simul. 8 (2010) 965-976. | MR | Zbl
, and ,[41] 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
, and ,[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. | MR | Zbl
,[44] Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Prob. 22 (2006) 801-814. | MR | Zbl
and ,[45] 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.
and ,[46] Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159-181. | MR | Zbl
,[47] Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR | Zbl
,[48] NMR data processing using iterative thresholding and minimum l1-norm reconstruction. J. Magn. Reson. 188 (2007) 295-300.
, and ,[49] Nonlinear ill-posed problems 1. Chapman & Hall (1998). | MR | Zbl
, and ,[50] Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157 (1991) 189-210. | MR | Zbl
and ,[51] Convex analysis in general vector spaces. World Scientific (2002). | Zbl
,[52] Nonlinear Functional Analysis and its Applications III. Springer-Verlag (1985). | MR | Zbl
,Cité par Sources :