Bayesian approach to inverse problems for functions with a variable-index Besov prior

The Bayesian approach has been adopted to solve inverse problems that reconstruct a function from noisy observations. Prior measures play a key role in the Bayesian method. Hence, many probability measures have been proposed, among which total variation (TV) is a well-known prior measure that can preserve sharp edges. However, it has two drawbacks, the staircasing effect and a lack of the discretization-invariant property. The variable-index TV prior has been proposed and analyzed in the area of image analysis for the former, and the Besov prior has been employed recently for the latter. To overcome both issues together, in this paper, we present a variable-index Besov prior measure, which is a non-Gaussian measure. Some useful properties of this new prior measure have been proven for functions defined on a torus. We have also generalized Bayesian inverse theory in infinite dimensions for our new setting. Finally, this theory has been applied to integer- and fractional-order backward diffusion problems. To the best of our knowledge, this is the first time that the Bayesian approach has been used for the fractional-order backward diffusion problem, which provides an opportunity to quantify its uncertainties.

[1]  S. Roudenko,et al.  Function spaces of variable smoothness and integrability , 2007, 0711.2354.

[2]  Yunmei Chen,et al.  Variable Exponent, Linear Growth Functionals in Image Restoration , 2006, SIAM J. Appl. Math..

[3]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[4]  Ting Wei,et al.  Two regularization methods for solving a Riesz–Feller space-fractional backward diffusion problem , 2010 .

[5]  H. Triebel.  Function Spaces and Wavelets on Domains , 2008 .

[6]  A. Stuart,et al.  MAP estimators and their consistency in Bayesian nonparametric inverse problems , 2013, 1303.4795.

[7]  A. N. Tikhonov,et al.  REGULARIZATION OF INCORRECTLY POSED PROBLEMS , 1963 .

[8]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[9]  Jigen Peng,et al.  Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives ✩ , 2012 .

[10]  A. Stuart,et al.  Besov priors for Bayesian inverse problems , 2011, 1105.0889.

[11]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[12]  Joel Franklin,et al.  Well-posed stochastic extensions of ill-posed linear problems☆ , 1970 .

[13]  Martin Burger,et al.  Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems , 2014, 1412.5816.

[14]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[15]  P. Hästö,et al.  Besov spaces with variable smoothness and integrability , 2010 .

[16]  D. Stroock,et al.  Probability Theory: An Analytic View , 1995, The Mathematical Gazette.

[17]  Jigen Peng,et al.  A novel characteristic of solution operator for the fractional abstract Cauchy problem , 2012 .

[18]  James C. Robinson,et al.  Bayesian inverse problems for functions and applications to fluid mechanics , 2009 .

[19]  J. Diestel,et al.  On vector measures , 1974 .

[20]  Sari Lasanen,et al.  Non-Gaussian statistical inverse problems. Part I: Posterior distributions , 2012 .

[21]  Juha Tiirola,et al.  Image Decompositions Using Spaces of Variable Smoothness and Integrability , 2014, SIAM J. Imaging Sci..

[22]  Henning Kempka.  2-Microlocal Besov and Triebel-Lizorkin Spaces of Variable Integrability , 2009 .

[23]  Henning Kempka.  Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability , 2010 .

[24]  R. Danchin,et al.  Fourier Analysis and Nonlinear Partial Differential Equations , 2011 .

[25]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[26]  K. Knopp.  Theory and Application of Infinite Series , 1990 .

[27]  I. Sokolov,et al.  Anomalous transport : foundations and applications , 2008 .

[28]  William Rundell,et al.  A tutorial on inverse problems for anomalous diffusion processes , 2015, 1501.00251.

[29]  G. Roberts,et al.  MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster , 2012, 1202.0709.

[30]  A. Stuart,et al.  The Bayesian Approach to Inverse Problems , 2013, 1302.6989.

[31]  S. Lasanen,et al.  Measurements and infinite‐dimensional statistical inverse theory , 2007 .

[32]  Jijun Liu,et al.  Total variation regularization for a backward time-fractional diffusion problem , 2013 .

[33]  G. Weiss,et al.  A First Course on Wavelets , 1996 .

[34]  Jigen Peng,et al.  WELL-POSEDNESS OF ABSTRACT DISTRIBUTED-ORDER FRACTIONAL DIFFUSION EQUATIONS , 2013 .

[35]  A new characteristic property of Mittag-Leffler functions and fractional cosine functions , 2014 .

[36]  Matti Lassas. Eero Saksman,et al.  Discretization-invariant Bayesian inversion and Besov space priors , 2009, 0901.4220.

[37]  Sari Lasanen,et al.  Posterior convergence for approximated unknowns in non-Gaussian statistical inverse problems , 2011, 1112.0906.

[38]  Chak-Kuen Wong,et al.  Total variation image restoration: numerical methods and extensions , 1997, Proceedings of International Conference on Image Processing.

[39]  P. Hästö,et al.  Lebesgue and Sobolev Spaces with Variable Exponents , 2011 .

[40]  S. Siltanen,et al.  Can one use total variation prior for edge-preserving Bayesian inversion? , 2004 .

[41]  Junxiong Jia,et al.  Existence and uniqueness of mild solutions for abstract delay fractional differential equations , 2011, Comput. Math. Appl..

[42]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .