Convergence rates for linear inverse problems in the presence of an additive normal soise

Stochastic Analysis and Applications

Volumn 27, Issue 2, 2009, Pages 240-257

  Hofinger, Andreas ^a   Pikkarainen, Hanna K ^a  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

Author keywords

Convergence rates; Linear inverse problems; Parameter choice rules; Statistical inversion theories

Indexed keywords

EID: 61349093573     PISSN: 07362994     EISSN: 15329356     Source Type: Journal    
DOI: 10.1080/07362990802558295     Document Type: Article

References (20)

1. Billingsley, P. 1968. Convergence of Probability Measures. John Wiley & Sons, New York.
2. Dudley, R.M. 2003. Real Analysis and Probability. Wadsworth, Belmont, CA.
3. Engl, H.W., Hanke, M., and Neubauer, A. 1996. Regularization of Inverse Problems. Kluwer, New York.
4. Engl, H.W., Hofinger, A., and Kindermann, S. 2005. Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems. Inverse Problems 21:399-412.
5. Engl, H.W., and Wakolbinger, A. 1985. Continuity properties of the extension of a locally Lipschitz continuous map to the space of probability measures. Monatshefte fur Mathematik 100:85-103.
6. Fan, K. 1944. Entfernung zweier zufälliger Größen und die Konvergenz nach Wahrscheinlichkeit. Mathematische Zeitschrift 49:681-683.
7. Groetsch, C.W., and Vogel, C.R. 1987. Asymptotic theory of filtering for linear operator equations with discrete noisy data. Mathematical Computations 49:499-506.
8. Hofinger, A. 2006. Ill-Posed Problems: Extending the Deterministic Theory to a Stochastic Setup. Trauner Verlag. (Doctoral thesis).
9. Hofinger, A. 2006. The metrics of Prokhorov and Ky Fan for assessing uncertainty in inverse problems. Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 215:107-125.
10. Hofinger, A., and Pikkarainen, H.K. 2007. Convergence rate for the Bayesian approach to linear inverse problems. Inverse Problems 23:2469-2484.
11. Huber, P. 1981. Robust Statistics. John Wiley & Sons, New York.
12. Kaipio, J.P., and Somersalo, E. 2004. Statistical and Computational Inverse Problems. Springer-Verlag, New York.
13. Kuo, H.-H. 1975. Gaussian Measures in Banach Spaces. Springer-Verlag, New York.
14. Lawson, C., and Hanson, R. 1974. Solving Least Squares Problems. Prentice-Hall, Upper Saddle River, NJ.
15. Lehtinen, M.S., Päivärinta, L., and Somersalo, E. 1989. Linear inverse problems for generalised random variables. Inverse Problems 5:599-612.
16. Lukas, M.A. 1988. Convergence rates for regularized solutions. Mathematical Computations 51:107-131.
17. Luschgy, H. 1995. Linear estimators and radonifying operators. Theory Probability and Applications 40:167-175.
18. Mair, B.A., and Ruymgaart, F.H. 1996. Statistical inverse estimation in Hilbert scales. SIAM Journal of Applied Mathematics56:1424-1444.
19. Parthasarathy, K.R. 1967. Probability Measures on Metric Spaces. Academic Press, New York.
20. Wahba, G. 1977. Practical approximate solutions to linear operator equations when the data are noisy. SIAM Journal of Numerical Analysis 14:651-667.