scholarly journals Generalized parallel tempering on Bayesian inverse problems

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Jonas Latz ◽  
Juan P. Madrigal-Cianci ◽  
Fabio Nobile ◽  
Raúl Tempone
Keyword(s):  
Inverse Problems ◽  
Continuous Time ◽  
Chem Phys ◽  
Parallel Tempering ◽  
Sampling Efficiency ◽  
Discrete Time Markov Chain ◽  
Bayesian Inverse Problems ◽  
State Dependent ◽  
Random Walk Metropolis ◽  
Sampling Algorithms

AbstractIn the current work we present two generalizations of the Parallel Tempering algorithm in the context of discrete-time Markov chain Monte Carlo methods for Bayesian inverse problems. These generalizations use state-dependent swapping rates, inspired by the so-called continuous time Infinite Swapping algorithm presented in Plattner et al. (J Chem Phys 135(13):134111, 2011). We analyze the reversibility and ergodicity properties of our generalized PT algorithms. Numerical results on sampling from different target distributions, show that the proposed methods significantly improve sampling efficiency over more traditional sampling algorithms such as Random Walk Metropolis, preconditioned Crank–Nicolson, and (standard) Parallel Tempering.

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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