# American Institute of Mathematical Sciences

May  2012, 6(2): 183-200. doi: 10.3934/ipi.2012.6.183

## Besov priors for Bayesian inverse problems

 1 Department of Mathematics, University of Sussex, Brighton BN1 5DJ, United Kingdom 2 School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom 3 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received  May 2011 Revised  March 2012 Published  May 2012

We consider the inverse problem of estimating a function $u$ from noisy, possibly nonlinear, observations. We adopt a Bayesian approach to the problem. This approach has a long history for inversion, dating back to 1970, and has, over the last decade, gained importance as a practical tool. However most of the existing theory has been developed for Gaussian prior measures. Recently Lassas, Saksman and Siltanen (Inv. Prob. Imag. 2009) showed how to construct Besov prior measures, based on wavelet expansions with random coefficients, and used these prior measures to study linear inverse problems. In this paper we build on this development of Besov priors to include the case of nonlinear measurements. In doing so a key technical tool, established here, is a Fernique-like theorem for Besov measures. This theorem enables us to identify appropriate conditions on the forward solution operator which, when matched to properties of the prior Besov measure, imply the well-definedness and well-posedness of the posterior measure. We then consider the application of these results to the inverse problem of finding the diffusion coefficient of an elliptic partial differential equation, given noisy measurements of its solution.
Citation: Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems and Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
##### References:
 [1] F. Abramovich and B. W. Silverman, Wavelet decomposition approaches to statistical inverse problems, Biometrika, 85 (1998), 115-129. doi: 10.1093/biomet/85.1.115. [2] F. Abramovich, T. Sapatinas and B. W. Silverman, Wavelet thresholding via a Bayesian approach, J. R. Stat. Soc. Ser. B Stat. Methodol., 60 (1998), 725-749. doi: 10.1111/1467-9868.00151. [3] A. Beskos, G. O. Roberts, A. M. Stuart and J. Voss, MCMC methods for diffusion bridges, Stochastic Dynamics, 8 (2008), 319-350. doi: 10.1142/S0219493708002378. [4] A. Beskos, G. O. Roberts and A. M. Stuart, Optimal scalings for local Metropolis-Hastings chains on nonproduct targets in high dimensions, Ann. Appl. Prob., 19 (2009), 863-898. doi: 10.1214/08-AAP563. [5] R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, in "Acta Numerica," Acta Numer., 7, Cambridge Univ. Press, Cambridge, (1998), 1-49. [6] A. Chambolle, R. A. DeVore, N. Lee and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), 319-335. doi: 10.1109/83.661182. [7] S. L. Cotter, M. Dashti, J. C. Robinson and A. M. Stuart, Bayesian inverse problems for functions and applications to fluid mechanics, Inverse Problems, 25 (2009), 115008, 43 pp. [8] S. L. Cotter, M. Dashti and A. M. Stuart, Approximation of Bayesian inverse problems for PDEs, SIAM J. Num. Anal., 48 (2010), 322-345. doi: 10.1137/090770734. [9] S. L.Cotter, M. Dashti and A. M.Stuart, Variational data assimilation using targetted random walks, Int. J. Num. Meth. Fluids, To appear, 2011. [10] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. [11] M. Dashti and A. Stuart, Uncertainty quantificationand weak approximation of an elliptic inverse problem, SIAM J. Num.Anal., to appear, 2011, arXiv:1102.0143. [12] I. Daubechies, "Ten Lectures on Wavelets," CBMS-NSF Regional Conference Series in Applied Mathematics, 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. [13] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042. [14] D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425-455. doi: 10.1093/biomet/81.3.425. [15] 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. [16] J. N. Franklin, Well-posed stochastic extensions of ill-posed linear problems, J. Math. Anal. Appl., 31 (1970), 682-716. doi: 10.1016/0022-247X(70)90017-X. [17] T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the Mumuford-Shah functional, Inverse Problems, 27 (2011), 015008, 32 pp. doi: 10.1088/0266-5611/27/1/015008. [18] M. Hairer, A. M. Stuart and J. Voss, Analysis of SPDEs arising in path sampling. II. The nonlinear case, Annals of Applied Probability, 17 (2007), 1657-1706. doi: 10.1214/07-AAP441. [19] J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Applied Mathematical Sciences, 160, Springer-Verlag, New York, 2005. [20] J.-P. Kahane, "Some Random Series of Functions," Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985. [21] S. Lasanen, Discretizations of generalized random variables with applications to inverse problems, Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 130, University of Oulu, Oulu, 2002. [22] S. Lasanen, Measurements and infinite-dimensional statistical inverse theory, PAMM, 7 (2007), 1080101-1080102. doi: 10.1002/pamm.200700068. [23] 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. [24] M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalized random variables, Inverse Problems, 5 (1989), 599-612. doi: 10.1088/0266-5611/5/4/011. [25] A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Z. Wahrsch. Verw. Gebiete, 65 (1984), 385-397. [26] Y. Meyer, "Wavelets and Operators," Translated from the 1990 French original by D. H. Salinger, Cambridge Studies in Advanced Mathematics, 37, Cambridge University Press, Cambridge, 1992. [27] P. Piiroinen, "Statistical Measruements, Experiments and Applications," Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 143, University of Helsinki, Helsinki, 2005. [28] Ch. Schwab and A. M. Stuart, Sparse deterministic approximation of Bayesian inverse problems, submitted, arXiv:1103.4522, 2011. [29] P. D. Spanos and R. Ghanem, Stochastic finite element expansion for random media, J. Eng. Mech., 115 (1989), 1035-1053. doi: 10.1061/(ASCE)0733-9399(1989)115:5(1035). [30] A. M. Stuart, Inverse problems: A Bayesian approach, Acta Numerica, 2010. doi: 10.1017/S0962492910000061. [31] H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. [32] P. Wojtaszczyk, "A Mathematical Introduction to Wavelets," London Mathematical Society Student Texts, 37, Cambridge University Press, Cambridge, 1997.

show all references

##### References:
 [1] F. Abramovich and B. W. Silverman, Wavelet decomposition approaches to statistical inverse problems, Biometrika, 85 (1998), 115-129. doi: 10.1093/biomet/85.1.115. [2] F. Abramovich, T. Sapatinas and B. W. Silverman, Wavelet thresholding via a Bayesian approach, J. R. Stat. Soc. Ser. B Stat. Methodol., 60 (1998), 725-749. doi: 10.1111/1467-9868.00151. [3] A. Beskos, G. O. Roberts, A. M. Stuart and J. Voss, MCMC methods for diffusion bridges, Stochastic Dynamics, 8 (2008), 319-350. doi: 10.1142/S0219493708002378. [4] A. Beskos, G. O. Roberts and A. M. Stuart, Optimal scalings for local Metropolis-Hastings chains on nonproduct targets in high dimensions, Ann. Appl. Prob., 19 (2009), 863-898. doi: 10.1214/08-AAP563. [5] R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, in "Acta Numerica," Acta Numer., 7, Cambridge Univ. Press, Cambridge, (1998), 1-49. [6] A. Chambolle, R. A. DeVore, N. Lee and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), 319-335. doi: 10.1109/83.661182. [7] S. L. Cotter, M. Dashti, J. C. Robinson and A. M. Stuart, Bayesian inverse problems for functions and applications to fluid mechanics, Inverse Problems, 25 (2009), 115008, 43 pp. [8] S. L. Cotter, M. Dashti and A. M. Stuart, Approximation of Bayesian inverse problems for PDEs, SIAM J. Num. Anal., 48 (2010), 322-345. doi: 10.1137/090770734. [9] S. L.Cotter, M. Dashti and A. M.Stuart, Variational data assimilation using targetted random walks, Int. J. Num. Meth. Fluids, To appear, 2011. [10] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. [11] M. Dashti and A. Stuart, Uncertainty quantificationand weak approximation of an elliptic inverse problem, SIAM J. Num.Anal., to appear, 2011, arXiv:1102.0143. [12] I. Daubechies, "Ten Lectures on Wavelets," CBMS-NSF Regional Conference Series in Applied Mathematics, 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. [13] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042. [14] D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425-455. doi: 10.1093/biomet/81.3.425. [15] 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. [16] J. N. Franklin, Well-posed stochastic extensions of ill-posed linear problems, J. Math. Anal. Appl., 31 (1970), 682-716. doi: 10.1016/0022-247X(70)90017-X. [17] T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the Mumuford-Shah functional, Inverse Problems, 27 (2011), 015008, 32 pp. doi: 10.1088/0266-5611/27/1/015008. [18] M. Hairer, A. M. Stuart and J. Voss, Analysis of SPDEs arising in path sampling. II. The nonlinear case, Annals of Applied Probability, 17 (2007), 1657-1706. doi: 10.1214/07-AAP441. [19] J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Applied Mathematical Sciences, 160, Springer-Verlag, New York, 2005. [20] J.-P. Kahane, "Some Random Series of Functions," Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985. [21] S. Lasanen, Discretizations of generalized random variables with applications to inverse problems, Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 130, University of Oulu, Oulu, 2002. [22] S. Lasanen, Measurements and infinite-dimensional statistical inverse theory, PAMM, 7 (2007), 1080101-1080102. doi: 10.1002/pamm.200700068. [23] 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. [24] M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalized random variables, Inverse Problems, 5 (1989), 599-612. doi: 10.1088/0266-5611/5/4/011. [25] A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Z. Wahrsch. Verw. Gebiete, 65 (1984), 385-397. [26] Y. Meyer, "Wavelets and Operators," Translated from the 1990 French original by D. H. Salinger, Cambridge Studies in Advanced Mathematics, 37, Cambridge University Press, Cambridge, 1992. [27] P. Piiroinen, "Statistical Measruements, Experiments and Applications," Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 143, University of Helsinki, Helsinki, 2005. [28] Ch. Schwab and A. M. Stuart, Sparse deterministic approximation of Bayesian inverse problems, submitted, arXiv:1103.4522, 2011. [29] P. D. Spanos and R. Ghanem, Stochastic finite element expansion for random media, J. Eng. Mech., 115 (1989), 1035-1053. doi: 10.1061/(ASCE)0733-9399(1989)115:5(1035). [30] A. M. Stuart, Inverse problems: A Bayesian approach, Acta Numerica, 2010. doi: 10.1017/S0962492910000061. [31] H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. [32] P. Wojtaszczyk, "A Mathematical Introduction to Wavelets," London Mathematical Society Student Texts, 37, Cambridge University Press, Cambridge, 1997.
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