Bregman distances in inverse problems and partial differential equations

Springer Optimization and Its Applications

Volumn 109, Issue , 2016, Pages 3-33

  Burger, Martin ^a  

^a UNIVERSITY OF MÜNSTER   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84975886694     PISSN: 19316828     EISSN: 19316836     Source Type: Book Series    
DOI: 10.1007/978-3-319-30785-5_2     Document Type: Chapter

References (63)

1. Ambrosio, L., Gigli, N., Savare, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhaeuser, Basel/Boston/Berlin (2005)
2. Arnold, A., Unterreiter, A.: Entropy decay of discretized Fokker-Planck equations I: temporal semidiscretization. Comput. Math. Appl. 46, 1683–1690 (2003)
3. Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Equ. 26, 43–100 (2001)
4. Arnold, A., Carrillo, J.A., Desvillettes, L., Dolbeault, J., Jüngel, A., Lederman, C., Markowich, P., Toscani, G., Villani, C.: Entropies and equilibria of many-particle systems: an essay on recent research. Mon. Hefte Math. 142, 35–43 (2004)
5. Arnold, A., Carlen, E., Ju, Q.: Large-time behavior of non-symmetric Fokker-Planck type equations. Commun. Stoch. Anal. 2, 153–175 (2008)
6. Bartels, S.: Error control and adaptivity for a variational model problem defined on functions of bounded variation. Math. Comput. 84, 1217–1240 (2015)
7. Benamou, J.-D., Carlier, G., Cuturi, M., Nenna, L. Peyre, G., Iterative Bregman projections for regularized transportation problems. SIAM J. Sci. Comput. 37, A1111–A1138 (2015)
8. Benning, M., Burger, M.: Error estimates for general fidelities. Electron. Trans. Numer. Anal. 38, 44–68 (2011)
9. Benning, M., Burger, M.: Ground states and singular vectors of convex variational regularization methods. Methods Appl. Anal. 20, 295–334 (2013)
10. Bodineau, T., Lebowitz, J., Mouhot, C., Villani, C.: Lyapunov functionals for boundary-driven nonlinear drift-diffusion equations. Nonlinearity 27, 2111 (2014)
11. Bregman, L.M.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)
12. Brune, C., Sawatzky, A., Burger, M.: Primal and dual Bregman methods with application to optical nanoscopy. Int. J. Comput. Vis. 92, 211–229 (2011)
13. Burger, M., Lucka, F.: Maximum-a-posteriori estimates in linear inverse problems with logconcave priors are proper Bayes estimators. Inverse Prob. 30, 114004 (2014)
14. Burger,M., Osher, S.: Convergence rates of convex variational regularization. Inverse Prob. 20, 1411–1421 (2004)
15. Burger, M., Osher, S.: A guide to the TV zoo. In: Burger, M., Osher, S. (eds.) Level Set and PDE-Based Reconstruction Methods in Imaging. Springer, Berlin (2013)
16. Burger, M., Gilboa, G., Osher, S., Xu, J.: Nonlinear inverse scale space methods. Commun. Math. Sci. 4, 179–212 (2006)
17. Burger, M., Frick, K., Osher, S., Scherzer, O.: Inverse total variation flow. Multiscale Model. Simul. 6, 366–395 (2007)
18. Burger, M., Resmerita, E., He, L.: Error estimation for Bregman iterations and inverse scale space methods in image restoration. Computing 81, 109–135 (2007)
19. Burger, M., Di Francesco, M., Pietschmann, J.F., Schlake, B.: Nonlinear cross-diffusion with size exclusion. SIAM J. Math. Anal. 42, 2842–2871 (2010)
20. Burger, M., Moeller, M., Benning, M., Osher, S.: An adaptive inverse scale space method for compressed sensing. Math. Comput. 82, 269–299 (2013)
21. Burger, M., Eckardt, L., Gilboa, G., Moeller, M.: Spectral representation of one-homogeneous functionals. In: Aujol, J.F., Niholovq, M., Pzpadikis, N. (eds.) Scale Space and Variation Methods in Computer Vision, Springer, Berlin, Heibelly, 16–27
22. Cai, J., Osher, S., Shen, Z.: Linearized Bregman iterations for compressed sensing. Math. Comput. 78, 1515–1536 (2009)
23. Carlier, G., Jimenez, C.: On Monge’s problem for Bregman-like cost functions. J. Convex Anal. 14, 647–655 (2007)
24. Carrillo, J.A., Jüngel, A., Markowich, P., Toscani, G., Unterreiter, A.: Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Mon. Hefte Math. 133, 1–82 (2001)
25. Censor, Y., Lent, A.: An iterative row-actionmethod for interval convex programming J. Optim. Theory Appl. 34, 321–353 (1981)
26. Censor, Y., Zenios, S.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford University Press, Oxford (1998)
27. Chavent, G., Kunisch, K.: Regularization of linear least squares problems by total bounded variation. ESAIM: Control Optim. Calc. Var. 2, 359–376 (1997)
28. Choi, K., Fahimian, B.P., Li, T., Suh, T.S., Lei, X.: Enhancement of four-dimensional conebeam computed tomography by compressed sensing with Bregman iteration. J. X-Ray Sci. Technol. 21, 177–192 (2013)
29. Di Francesco, M., Fellner, K., Markowich, P.: The entropy dissipation method for inhomogeneous reaction-diffusion systems. Proc. R. Soc. A 464, 3272–3300 (2008)
30. Diening, L., Kreuzer, C.: Linear convergence of an adaptive finite element method for the p-Laplacian equation. SIAM J. Numer. Anal. 46, 614–638 (2008)
31. Droniou, J., Vazquez, J.L.: Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions. Calc. Var. Partial Differ. Equ. 34, 413–434 (2009)
32. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)
33. Engl, H., Hanke-Bourgeois, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)
34. Flemming, J.: Theory and examples of variational regularization with non-metric fitting functionals. J. Inverse III-Posed Probl. 18, 677–699 (2010)
35. Goldstein, T., Osher, S.: The split Bregman method for l1 regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2008)
36. Grasmair, M.: Generalized Bregman distances and convergence rates for non-convex regularization methods. Inverse Problems 11, 115014 (2010)
37. Grasmair, M.: Variational inequalities and higher order convergence rates for Tikhonov regularisation on Banach spaces. J. Inverse III-Posed Probl. 21, 379–394 (2013)
38. Hein, T.: Tikhonov regularization in Banach spaces - improved convergence rates results. Inverse Prob. 25, 035002 (2009)
39. Helin, T., Burger, M.: Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems. Preprint arXiv:1412.5816 (2015)
40. Jüngel, A.: The boundedness-by-entropy method for cross-diffusion systems. Nonlinearity 28, 1963–2001 (2015)
41. Jüngel, A., Matthes, D.: Entropiemethoden für nichtlineare partielle Differentialgleichungen. Int. Math. Nachr. 209, 1–14 (2008)
42. Jüngel, A., Stelzer, I.V.: Entropy structure of a cross-diffusion tumor-growth model. Math. Mod. Meth. Appl. Sci. 22, 1250009, (2012)
43. Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer, New York (2005)
44. Kiwiel, K.C.: Proximal minimization methods with generalized Bregman functions. SIAM J. Control Optim. 35, 1142–1168 (1997)
45. Mielke, A., Haskovec, J., Markowich, P.A.: On uniform decay of the entropy for reactiondiffusion systems. J. Dyn. Differ. Equ. 27, 897–928 (2015)
46. Mielke, A., Rossi, R., Savare, G.: Nonsmooth analysis of doubly nonlinear evolution equations. Calc. Var. Partial Differ. Equ. 46, 253–310 (2013)
47. Moeller, M.: Multiscale methods for polyhedral regularizations and applications in high dimensional imaging. Ph.D. thesis, University of Muenster (2012)
48. Moeller, M., Burger, M.: Multiscale methods for polyhedral regularizations. SIAM J. Optim. 23, 1424–1456 (2013)
49. Müller, J., Brune, C., Sawatzky, A., Kösters, T., Schäfers, K.P., Burger, M.: Reconstruction of short time PET scans using Bregman iterations. In: Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), pp. 2383–2385. IEEE, New York (2011)
50. Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. SIAM Multiscale Model. Simul. 4, 460–489 (2005)
51. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)
52. Pöschl, C.: An overview on convergence rates for Tikhonov regularization methods for nonlinear operators. J. Inverse III-posed Probl. 17, 77–83 (2009)
53. Reid, M.: Meet the Bregman divergences. http://mark.reid.name/blog/meet-the-bregmandivergences.html (2013)
54. Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Problems 21, 1303 (2005)
55. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom. 60, 259–268 (1992)
56. Saint-Raymond, L.: Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. Arch. Ration. Mech. Anal. 166, 47–80 (2003)
57. Scherzer, O., Groetsch, C.: Inverse scale space theory for inverse problems. In: Scale-Space and Morphology in Computer Vision, pp. 317–325. Springer, Berlin/Heidelberg (2001)
58. Werner, F., Hohage, T.: Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data. Inverse Problems 28, 104004 (2012)
59. Witkin, A.P.: Scale-space filtering: a new approach to multi-scale description. In: IEEE International Conference on ICASSP’84 Acoustics, Speech, and Signal Processing, vol. 9, pp. 150–153 (1984)
60. Xu J., Osher, S.: Iterative regularization and nonlinear inverse scale space applied to waveletbased denoising. IEEE Trans. Image Process. 16, 534–544 (2007)
61. Yin, W.: Analysis and generalizations of the linearized Bregman method. SIAM J. Imaging Sci. 3, 856–877 (2010)
62. Zamponi, N., Jüngel, A.: Analysis of degenerate cross-diffusion population models with volume filling, TU Vienna. Preprint (2015)
63. Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3, 253–276 (2010)