Discretization-invariant Bayesian inversion and Besov space priors

Previous Article
A greedy method for reconstructing polycrystals from three-dimensional X-ray diffraction data
IPI Home
This Issue
Next Article
The factorization method is independent of transmission eigenvalues

February 2009, 3(1): 87-122. doi: 10.3934/ipi.2009.3.87

Discretization-invariant Bayesian inversion and Besov space priors

Matti Lassas ^1, , Eero Saksman ^1, and Samuli Siltanen ^2,

1.	Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hallstromin katu 2b) FI-00014, Finland, Finland
2.	Tampere University of Technology,Institute of Mathematics,, P.O. Box 553, 33101 Tampere

Received February 2008 Revised November 2008 Published February 2009

Abstract
Related Papers

Bayesian solution of an inverse problem for indirect measurement $M = AU + $ε is considered, where $U$ is a function on a domain of $\R^d$. Here $A$ is a smoothing linear operator and ε is Gaussian white noise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k$ε , where $P_k$ is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to the computational measurement model $M_{kn}=P_k A U_n + P_k$ε . Bayes formula gives then the posterior distribution

$\pi_{kn}(u_n\|\m_{kn})$~ Π _n $(u_n)\exp(-\frac{1}{2}$||$\m_{kn} - P_kA u_n$||$\_2^2)$

in $\R^d$, and the mean $\u_{kn}$:$=\int u_n \ \pi_{kn}(u_n\|\m_k)\ du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing prior distributions Π _n for all $n\geq n_0>0$ by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Π _n represent the same a priori information for all $n$ and that the mean $\u_{kn}$ converges to a limit estimate as $k,n$→$\infty$. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B^1_11$ prior is related to penalizing the $\l^1$ norm of the wavelet coefficients of $U$.

Keywords: wavelet, discretization invariance, Inverse problem, statistical inversion, Besov space., Bayesian inversion, reconstruction.

Mathematics Subject Classification: 60F17, 65C20, 42C4.

Citation: Matti Lassas, Eero Saksman, Samuli Siltanen. Discretization-invariant Bayesian inversion and Besov space priors. Inverse Problems & Imaging, 2009, 3 (1) : 87-122. doi: 10.3934/ipi.2009.3.87

[1]	Lassi Roininen, Janne M. J. Huttunen, Sari Lasanen. Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (2) : 561-586. doi: 10.3934/ipi.2014.8.561
[2]	Lassi Roininen, Mark Girolami, Sari Lasanen, Markku Markkanen. Hyperpriors for Matérn fields with applications in Bayesian inversion. Inverse Problems & Imaging, 2019, 13 (1) : 1-29. doi: 10.3934/ipi.2019001
[3]	Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
[4]	Tan Bui-Thanh, Omar Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. Inverse Problems & Imaging, 2015, 9 (1) : 27-53. doi: 10.3934/ipi.2015.9.27
[5]	Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069
[6]	Tan Bui-Thanh, Quoc P. Nguyen. FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems. Inverse Problems & Imaging, 2016, 10 (4) : 943-975. doi: 10.3934/ipi.2016028
[7]	Linh V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Problems & Imaging, 2009, 3 (4) : 649-675. doi: 10.3934/ipi.2009.3.649
[8]	Igor E. Shparlinski. Close values of shifted modular inversions and the decisional modular inversion hidden number problem. Advances in Mathematics of Communications, 2015, 9 (2) : 169-176. doi: 10.3934/amc.2015.9.169
[9]	Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399
[10]	Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[11]	Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17
[12]	Frank Natterer. Photo-acoustic inversion in convex domains. Inverse Problems & Imaging, 2012, 6 (2) : 315-320. doi: 10.3934/ipi.2012.6.315
[13]	Piotr Fijałkowski. A global inversion theorem for functions with singular points. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 173-180. doi: 10.3934/dcdsb.2018011
[14]	Liying Wang, Weiguo Zhao, Dan Zhang, Linming Zhao. A geometric inversion algorithm for parameters calculation in Francis turbine. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1373-1384. doi: 10.3934/dcdss.2015.8.1373
[15]	Peter Kuchment, Fatma Terzioglu. Inversion of weighted divergent beam and cone transforms. Inverse Problems & Imaging, 2017, 11 (6) : 1071-1090. doi: 10.3934/ipi.2017049
[16]	Laura Poggiolini, Marco Spadini. Local inversion of a class of piecewise regular maps. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2207-2224. doi: 10.3934/cpaa.2018105
[17]	T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems & Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040
[18]	Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
[19]	Josep M. Olm, Xavier Ros-Oton. Approximate tracking of periodic references in a class of bilinear systems via stable inversion. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 197-215. doi: 10.3934/dcdsb.2011.15.197
[20]	Eric Bedford, Kyounghee Kim. Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 977-1013. doi: 10.3934/dcds.2008.21.977

2018 Impact Factor: 1.469

American Institute of Mathematical Sciences