Inverse Problems and Imaging (IPI)

Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors

Pages: 857 - 874, Volume 11, Issue 5, October 2017      doi:10.3934/ipi.2017040

       Abstract        References        Full Text (990.3K)       Related Articles       

T. J. Sullivan - Free University of Berlin and Zuse Institute Berlin, Takustraße 7, 14195 Berlin, Germany (email)

Abstract: This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen--Loève expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.

Keywords:  Bayesian inverse problems, heavy-tailed distribution, Karhunen--Loève expansion, quasi-Banach spaces, stable distribution, uncertainty quantification, well-posedness.
Mathematics Subject Classification:  Primary: 65J22; Secondary: 35R30, 60E07, 62F15, 62G35, 60B11, 28C20.

Received: May 2016;      Revised: November 2016;      Available Online: July 2017.