Thermalization of rate-independent processes by entropic regularization
T. J. Sullivan M. Koslowski F. Theil Michael Ortiz
Discrete & Continuous Dynamical Systems - S 2013, 6(1): 215-233 doi: 10.3934/dcdss.2013.6.215
We consider the effective behaviour of a rate-independent process when it is placed in contact with a heat bath. The method used to ``thermalize'' the process is an interior-point entropic regularization of the Moreau--Yosida incremental formulation of the unperturbed process. It is shown that the heat bath destroys the rate independence in a controlled and deterministic way, and that the effective dynamics are those of a non-linear gradient descent in the original energetic potential with respect to a different and non-trivial effective dissipation potential.
keywords: non-linear evolution equations Gradient descent thermodynamics.
Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors
T. J. Sullivan
Inverse Problems & Imaging 2017, 11(5): 857-874 doi: 10.3934/ipi.2017040

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–Loéve 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

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