An Efficient Iterative Approach for Large-Scale Separable Nonlinear Inverse Problems

2010 ◽  
Vol 31 (6) ◽  
pp. 4654-4674 ◽  
Author(s):  
Julianne Chung ◽  
James G. Nagy
Keyword(s):  
Inverse Problems ◽  
Large Scale ◽  
Iterative Approach ◽  
Nonlinear 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.

Projection methods and applications for seismic nonlinear inverse problems with multiple constraints

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. R251-R269 ◽  
Author(s):  
Bas Peters ◽  
Brendan R. Smithyman ◽  
Felix J. Herrmann
Keyword(s):  
Inverse Problems ◽  
Prior Information ◽  
Waveform Inversion ◽  
Projection Methods ◽  
Multiple Constraints ◽  
Local Minima ◽  
Low Frequencies ◽  
Trade Off ◽  
Nonlinear Inverse Problems ◽  
Easy Integration

Nonlinear inverse problems are often hampered by local minima because of missing low frequencies and far offsets in the data, lack of access to good starting models, noise, and modeling errors. A well-known approach to counter these deficiencies is to include prior information on the unknown model, which regularizes the inverse problem. Although conventional regularization methods have resulted in enormous progress in ill-posed (geophysical) inverse problems, challenges remain when the prior information consists of multiple pieces. To handle this situation, we have developed an optimization framework that allows us to add multiple pieces of prior information in the form of constraints. The proposed framework is more suitable for full-waveform inversion (FWI) because it offers assurances that multiple constraints are imposed uniquely at each iteration, irrespective of the order in which they are invoked. To project onto the intersection of multiple sets uniquely, we use Dykstra’s algorithm that does not rely on trade-off parameters. In that sense, our approach differs substantially from approaches, such as Tikhonov/penalty regularization and gradient filtering. None of these offer assurances, which makes them less suitable to FWI, where unrealistic intermediate results effectively derail the inversion. By working with intersections of sets, we avoid trade-off parameters and keep objective calculations separate from projections that are often much faster to compute than objectives/gradients in 3D. These features allow for easy integration into existing code bases. Working with constraints also allows for heuristics, where we built up the complexity of the model by a gradual relaxation of the constraints. This strategy helps to avoid convergence to local minima that represent unrealistic models. Using multiple constraints, we obtain better FWI results compared with a quadratic penalty method, whereas all definitions of the constraints are in terms of physical units and follow from the prior knowledge directly.

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