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Oracle-type posterior contraction rates in Bayesian inverse problems
1. | School of Mathematical Sciences, Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University, Shanghai, 200433, China, China |
2. | Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany |
References:
[1] |
S. Agapiou, Aspects of Bayesian Inverse Problems, Ph.D. thesis, University of Warwick, 2013. |
[2] |
S. Agapiou, S. Larsson and A. M. Stuart, Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems, Stochastic Process. Appl., 123 (2013), 3828-3860.
doi: 10.1016/j.spa.2013.05.001. |
[3] |
S. Agapiou and P. Mathé, Preconditioning the prior to avoid saturation in Bayesian inverse problems, submitted, arXiv:1409.6496, 2014. |
[4] |
F. Bauer and M. A. Lukas, Comparing parameter choice methods for regularization of ill-posed problems, Math. Comput. Simulation, 81 (2011), 1795-1841.
doi: 10.1016/j.matcom.2011.01.016. |
[5] |
N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise, Inverse Problems, 20 (2004), 1773-1789.
doi: 10.1088/0266-5611/20/6/005. |
[6] |
G. Blanchard and P. Mathé, Discrepancy principle for statistical inverse problems with application to conjugate gradient regularization, Inverse Problems, 28 (2012), 115011, 23pp.
doi: 10.1088/0266-5611/28/11/115011. |
[7] |
A. Caponnetto and E. De Vito, Optimal rates for the regularized least-squares algorithm, Found. Comput. Math., 7 (2007), 331-368.
doi: 10.1007/s10208-006-0196-8. |
[8] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
doi: 10.1007/978-94-009-1740-8. |
[9] |
B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse Problems, 7 (1991), 675-702.
doi: 10.1088/0266-5611/7/5/003. |
[10] |
J.-P. Florens and A. Simoni, Regularizing priors for linear inverse problems, Scand. J. Stat., 39 (2012), 214-235.
doi: 10.1111/j.1467-9469.2011.00784.x. |
[11] |
P. C. Hansen, Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), 1-35.
doi: 10.1007/BF02149761. |
[12] |
J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer-Verlag, New York, 2005. |
[13] |
B. T. Knapik, A. W. van der Vaart and H. van Zanten, Bayesian inverse problems with Gaussian priors, Ann. Statist., 39 (2011), 2626-2657.
doi: 10.1214/11-AOS920. |
[14] |
S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Probl. Imaging, 6 (2012), 215-266.
doi: 10.3934/ipi.2012.6.215. |
[15] |
S. Lasanen, Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns, Inverse Probl. Imaging, 6 (2012), 267-287.
doi: 10.3934/ipi.2012.6.267. |
[16] |
M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122.
doi: 10.3934/ipi.2009.3.87. |
[17] |
M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563.
doi: 10.1088/0266-5611/20/5/013. |
[18] |
M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-3-642-20212-4. |
[19] |
M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612.
doi: 10.1088/0266-5611/5/4/011. |
[20] |
S. Lu and S. V. Pereverzev, Regularization Theory for Ill-Posed Problems. SElected Topics, Inverse and Ill-posed Problems Series, 58, De Gruyter, Berlin, 2013.
doi: 10.1515/9783110286496. |
[21] |
S. Lu and P. Mathé, Discrepancy based model selection in statistical inverse problems, J. Complexity, 30 (2014), 290-308.
doi: 10.1016/j.jco.2014.02.002. |
[22] |
A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Z. Wahrsch. Verw. Gebiete, 65 (1984), 385-397.
doi: 10.1007/BF00533743. |
[23] |
C. Marteau and P. Mathé, General regularization schemes for signal detection in inverse problems, Math. Meth. Statist., 23 (2014), 176-200.
doi: 10.3103/S1066530714030028. |
[24] |
P. Mathé and B. Hofmann, How general are general source conditions?, Inverse Problems, 24 (2008), 015009, 5pp.
doi: 10.1088/0266-5611/24/1/015009. |
[25] |
P. Mathé and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 789-803.
doi: 10.1088/0266-5611/19/3/319. |
[26] |
P. Mathé and S. V. Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data, Math. Comp., 75 (2006), 1913-1929.
doi: 10.1090/S0025-5718-06-01873-4. |
[27] |
P. Mathé and U. Tautenhahn, Enhancing linear regularization to treat large noise, J. Inverse Ill-Posed Probl., 19 (2011), 859-879.
doi: 10.1515/JIIP.2011.052. |
[28] |
A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.
doi: 10.1017/S0962492910000061. |
show all references
References:
[1] |
S. Agapiou, Aspects of Bayesian Inverse Problems, Ph.D. thesis, University of Warwick, 2013. |
[2] |
S. Agapiou, S. Larsson and A. M. Stuart, Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems, Stochastic Process. Appl., 123 (2013), 3828-3860.
doi: 10.1016/j.spa.2013.05.001. |
[3] |
S. Agapiou and P. Mathé, Preconditioning the prior to avoid saturation in Bayesian inverse problems, submitted, arXiv:1409.6496, 2014. |
[4] |
F. Bauer and M. A. Lukas, Comparing parameter choice methods for regularization of ill-posed problems, Math. Comput. Simulation, 81 (2011), 1795-1841.
doi: 10.1016/j.matcom.2011.01.016. |
[5] |
N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise, Inverse Problems, 20 (2004), 1773-1789.
doi: 10.1088/0266-5611/20/6/005. |
[6] |
G. Blanchard and P. Mathé, Discrepancy principle for statistical inverse problems with application to conjugate gradient regularization, Inverse Problems, 28 (2012), 115011, 23pp.
doi: 10.1088/0266-5611/28/11/115011. |
[7] |
A. Caponnetto and E. De Vito, Optimal rates for the regularized least-squares algorithm, Found. Comput. Math., 7 (2007), 331-368.
doi: 10.1007/s10208-006-0196-8. |
[8] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
doi: 10.1007/978-94-009-1740-8. |
[9] |
B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse Problems, 7 (1991), 675-702.
doi: 10.1088/0266-5611/7/5/003. |
[10] |
J.-P. Florens and A. Simoni, Regularizing priors for linear inverse problems, Scand. J. Stat., 39 (2012), 214-235.
doi: 10.1111/j.1467-9469.2011.00784.x. |
[11] |
P. C. Hansen, Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), 1-35.
doi: 10.1007/BF02149761. |
[12] |
J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer-Verlag, New York, 2005. |
[13] |
B. T. Knapik, A. W. van der Vaart and H. van Zanten, Bayesian inverse problems with Gaussian priors, Ann. Statist., 39 (2011), 2626-2657.
doi: 10.1214/11-AOS920. |
[14] |
S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Probl. Imaging, 6 (2012), 215-266.
doi: 10.3934/ipi.2012.6.215. |
[15] |
S. Lasanen, Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns, Inverse Probl. Imaging, 6 (2012), 267-287.
doi: 10.3934/ipi.2012.6.267. |
[16] |
M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122.
doi: 10.3934/ipi.2009.3.87. |
[17] |
M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563.
doi: 10.1088/0266-5611/20/5/013. |
[18] |
M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-3-642-20212-4. |
[19] |
M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612.
doi: 10.1088/0266-5611/5/4/011. |
[20] |
S. Lu and S. V. Pereverzev, Regularization Theory for Ill-Posed Problems. SElected Topics, Inverse and Ill-posed Problems Series, 58, De Gruyter, Berlin, 2013.
doi: 10.1515/9783110286496. |
[21] |
S. Lu and P. Mathé, Discrepancy based model selection in statistical inverse problems, J. Complexity, 30 (2014), 290-308.
doi: 10.1016/j.jco.2014.02.002. |
[22] |
A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Z. Wahrsch. Verw. Gebiete, 65 (1984), 385-397.
doi: 10.1007/BF00533743. |
[23] |
C. Marteau and P. Mathé, General regularization schemes for signal detection in inverse problems, Math. Meth. Statist., 23 (2014), 176-200.
doi: 10.3103/S1066530714030028. |
[24] |
P. Mathé and B. Hofmann, How general are general source conditions?, Inverse Problems, 24 (2008), 015009, 5pp.
doi: 10.1088/0266-5611/24/1/015009. |
[25] |
P. Mathé and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 789-803.
doi: 10.1088/0266-5611/19/3/319. |
[26] |
P. Mathé and S. V. Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data, Math. Comp., 75 (2006), 1913-1929.
doi: 10.1090/S0025-5718-06-01873-4. |
[27] |
P. Mathé and U. Tautenhahn, Enhancing linear regularization to treat large noise, J. Inverse Ill-Posed Probl., 19 (2011), 859-879.
doi: 10.1515/JIIP.2011.052. |
[28] |
A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.
doi: 10.1017/S0962492910000061. |
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