scholarly journals Nonlocal TV-Gaussian prior for Bayesian inverse problems with applications to limited CT reconstruction

2020 ◽  
Vol 14 (1) ◽  
pp. 117-132 ◽  
Author(s):  
Didi Lv ◽  
◽  
Qingping Zhou ◽  
Jae Kyu Choi ◽  
Jinglai Li ◽  
...  
Keyword(s):  
Inverse Problems ◽  
Ct Reconstruction ◽  
Bayesian Inverse Problems ◽  
Gaussian Prior

scholarly journals Sequential ensemble transform for Bayesian inverse problems

2021 ◽  
Vol 427 ◽  
pp. 110055
Author(s):  
Aaron Myers ◽  
Alexandre H. Thiéry ◽  
Kainan Wang ◽  
Tan Bui-Thanh
Keyword(s):  
Inverse Problems ◽  
Bayesian Inverse Problems ◽  
Ensemble Transform

hIPPYlib

2021 ◽  
Vol 47 (2) ◽  
pp. 1-34
Author(s):  
Umberto Villa ◽  
Noemi Petra ◽  
Omar Ghattas
Keyword(s):  
Inverse Problems ◽  
Large Scale ◽  
Low Rank ◽  
Scalable Algorithms ◽  
Low Rank Approximation ◽  
Dimensional Parameter ◽  
Infinite Dimensional ◽  
Bayesian Inverse Problems ◽  
Rank Approximation ◽  
Extensible Software

We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with (possibly) infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitively expensive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYlib is that the solution of the inverse problem is computed at a cost, measured in linearized forward PDE solves, that is independent of the parameter dimension. The mean of the posterior is approximated by the MAP point, which is found by minimizing the negative log-posterior with an inexact matrix-free Newton-CG method. The posterior covariance is approximated by the inverse of the Hessian of the negative log posterior evaluated at the MAP point. The construction of the posterior covariance is made tractable by invoking a low-rank approximation of the Hessian of the log-likelihood. Scalable tools for sample generation are also discussed. hIPPYlib makes all of these advanced algorithms easily accessible to domain scientists and provides an environment that expedites the development of new algorithms.

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