A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion

2012 ◽  
Vol 34 (3) ◽  
pp. A1460-A1487 ◽  
Author(s):  
James Martin ◽  
Lucas C. Wilcox ◽  
Carsten Burstedde ◽  
Omar Ghattas
Keyword(s):  
Inverse Problems ◽  
Large Scale ◽  
Seismic Inversion ◽  
Mcmc Method ◽  
Statistical Inverse Problems

An object-oriented optimization framework for large-scale inverse problems

2021 ◽  
pp. 104790
Author(s):  
Ettore Biondi ◽  
Guillaume Barnier ◽  
Robert G. Clapp ◽  
Francesco Picetti ◽  
Stuart Farris
Keyword(s):  
Inverse Problems ◽  
Large Scale ◽  
Object Oriented ◽  
Optimization Framework

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.

Object-Oriented Optimization for Small- and Large-Scale Seismic Inversion Procedures

2021 ◽  
Author(s):  
E. Biondi ◽  
G. Barnier ◽  
R. Clapp ◽  
F. Picetti ◽  
S. Farris
Keyword(s):  
Large Scale ◽  
Object Oriented ◽  
Seismic Inversion

Large-Scale Inverse Problems in Imaging

2011 ◽  
pp. 43-86 ◽  
Author(s):  
Julianne Chung ◽  
Sarah Knepper ◽  
James G. Nagy
Keyword(s):  
Inverse Problems ◽  
Large Scale

New insights into one-norm solvers from the Pareto curve

Geophysics ◽  
2008 ◽  
Vol 73 (4) ◽  
pp. A23-A26 ◽  
Author(s):  
Gilles Hennenfent ◽  
Ewout van den Berg ◽  
Michael P. Friedlander ◽  
Felix J. Herrmann
Keyword(s):  
Inverse Problems ◽  
Large Scale ◽  
Scale Problem ◽  
Objective Criterion ◽  
Prior Model ◽  
Trade Off ◽  
Large Scale Problem ◽  
Pareto Curve

Geophysical inverse problems typically involve a trade-off between data misfit and some prior model. Pareto curves trace the optimal trade-off between these two competing aims. These curves are used commonly in problems with two-norm priors in which they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one-norm prior, Pareto curves remain relatively unexplored. We show how these curves lead to new insights into one-norm regularization. First, we confirm theoretical properties of smoothness and convexity of these curves from a stylized and a geophysical example. Second, we exploit these crucial properties to approximate the Pareto curve for a large-scale problem. Third, we show how Pareto curves provide an objective criterion to gauge how different one-norm solvers advance toward the solution.

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