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Inverse problems in spaces of measures

Published online by Cambridge University Press:  27 March 2012

Kristian Bredies  and
Hanna Katriina Pikkarainen
Kristian Bredies
Affiliation:
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria. kristian.bredies@uni-graz.at
Hanna Katriina Pikkarainen
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria; hanna.pikkarainen@ricam.oeaw.ac.at
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Abstract

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 problemsvector-valued finite Radon measuresTikhonov regularizationdelta-peak solutionsgeneralized conditional gradient methoditerative soft-thresholdingsparse deconvolution
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 190 - 218
Copyright
© EDP Sciences, SMAI, 2012

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References

R.A. Adams and J.J.F. Fournier, Sobolev spaces. Academic Press (2003).
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).
Beck, A. and Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2 (2009) 183–202. Google Scholar
Bonesky, T., Kazimierski, K.S., Maass, P., Schöpfer, F. and Schuster, T., Minimization of Tikhonov functionals in Banach spaces. Abstr. Appl. Anal. (2008) 192679. Google Scholar
Bredies, K. and Lorenz, D.A., Iterated hard shrinkage for minimization problems with sparsity constraints. SIAM J. Sci. Comput. 30 (2008) 657–683. Google Scholar
Bredies, K. and Lorenz, D.A., Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14 (2008) 813–837. Google Scholar
Bredies, K., Lorenz, D.A. and Maass, P., A generalized conditional gradient method and its connection to an iterative shrinkage method. Comput. Optim. Appl. 42 (2009) 173–193. Google Scholar
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.
Burger, M. and Osher, S., Convergence rates of convex variational regularization. Inverse Prob. 20 (2004) 1411–1421. Google Scholar
Candès, E.J., Romberg, J.K. and Tao, T., Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 (2006) 1207–1223. Google Scholar
Clason, C. and Kunisch, K., A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM : COCV 17 (2011) 243–266. Google Scholar
Combettes, P.L. and Wajs, V.R., Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4 (2005) 1168–1200. Google Scholar
J.B. Conway, A course in functional analysis. Springer (1990).
Daubechies, I., Defrise, M. and De Mol, C., An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Comm. Pure Appl. Math. 57 (2004) 1413–1457. Google Scholar
Donoho, D.L., Compressed sensing. IEEE Trans. Inf. Theory 52 (2006) 1289–1306. Google Scholar
Donoho, D.L., Elad, M. and Temlyakov, V.N., Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52 (2006) 6–18. Google Scholar
C. Dossal and S. Mallat, Sparse spike deconvolution with minimum scale, in Proc. of SPARS’05 (2005).
N. Dunford and J.T. Schwartz, Linear Operators. I. General Theory. Interscience Publishers (1958).
Efron, B., Hastie, T., Johnstone, I. and Tibshirani, R., Least angle regression. Ann. Statist. 32 (2004) 407–499. Google Scholar
I. Ekeland and R. Temam, Convex analysis and variational problems. North-Holland (1976).
Engl, H.W. and Landl, G., Convergence rates for maximum entropy regularization. SIAM J. Numer. Anal. 30 (1993) 1509–1536. Google Scholar
H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic Publishers (1996).
Figueiredo, M.A.T., Nowak, R.D. and Wright, S.J., Gradient projection for sparse reconstruction : Application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1 (2007) 586–597. Google Scholar
I. Fonseca and G. Leoni, Modern methods in the calculus of variations : L p spaces. Springer (2007).
Fornasier, M. and Rauhut, H., Recovery algorithms for vector valued data with joint sparsity constraints. SIAM J. Numer. Anal. 46 (2008) 577–613. Google Scholar
Fuchs, J.-J., On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory. 50 (2004) 1341–1344. Google Scholar
Gibbs, A.L. and Su, F.E., On choosing and bounding probability metrics. Int. Stat. Rev. 70 (2002) 419–435. Google Scholar
Grasmair, M., Haltmeier, M. and Scherzer, O., Sparse regularization with ℓ q penalty term. Inverse Prob. 24 (2008) 055020. Google Scholar
Griesse, R. and Lorenz, D.A., A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inverse Prob. 24 (2008) 035007. Google Scholar
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Publishing Limited (1985).
Hein, T., Tikhonov regularization in Banach spaces – improved convergence rates results. Inverse Prob. 25 (2009) 035002. Google Scholar
Hofmann, B., Kaltenbacher, B., Pöschl, C. and Scherzer, O., A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Prob. 23 (2007) 987–1010. Google Scholar
L. Hörmander, The Analysis of Linear Partial Differential Operators I. Springer-Verlag (1990).
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).
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.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function Spaces. Springer (1979).
Lorenz, D.A., Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl. 16 (2008) 463–478. Google Scholar
Lorenz, D.A. and Trede, D., Optimal convergence rates for Tikhonov regularization in Besov scales. Inverse Prob. 24 (2008) 055010. Google Scholar
D.A. Lorenz and D. Trede, Greedy deconvolution of point-like objects, in Proc. of SPARS’09 (2009).
Mao, Y., Dong, B. and Osher, S., A nonlinear PDE-based method for sparse deconvolution. Multiscale Model. Simul. 8 (2010) 965–976. Google Scholar
Mugnier, L.M., Fusco, T. and Conan, J.-M., 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. Google ScholarPubMed
Nesterov, Y.E., A method of solving a convex programming problem with convergence rate O(1/k 2). Soviet Math. Dokl. 27 (1983) 372–376. Google Scholar
Neubauer, A., On enhanced convergence rates for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. Inverse Prob. 25 (2009) 065009. Google Scholar
Resmerita, E. and Scherzer, O., Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Prob. 22 (2006) 801–814. Google Scholar
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.
Stadler, G., Elliptic optimal control problems with L 1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159–181. Google Scholar
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. Google Scholar
Stern, A.S., Donoho, D.L. and Hoch, J.C., NMR data processing using iterative thresholding and minimum l 1-norm reconstruction. J. Magn. Reson. 188 (2007) 295–300. Google Scholar
A.N. Tikhonov, A.S. Leonov and A.G. Yagola, Nonlinear ill-posed problems 1. Chapman & Hall (1998).
Xu, Z.B. and Roach, G.F., Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157 (1991) 189–210. Google Scholar
C. Zălinescu, Convex analysis in general vector spaces. World Scientific (2002).
E. Zeidler, Nonlinear Functional Analysis and its Applications III. Springer-Verlag (1985).