A computationally efficient parallel LevenbergMarquardt algorithm for highly parameterized inverse model analyses
Abstract
Inverse modeling seeks model parameters given a set of observations. However, for practical problems because the number of measurements is often large and the model parameters are also numerous, conventional methods for inverse modeling can be computationally expensive. We have developed a new, computationallyefficient parallel LevenbergMarquardt method for solving inverse modeling problems with a highly parameterized model space. LevenbergMarquardt methods require the solution of a linear system of equations which can be prohibitively expensive to compute for moderate to largescale problems. Our novel method projects the original linear problem down to a Krylov subspace, such that the dimensionality of the problem can be significantly reduced. Furthermore, we store the Krylov subspace computed when using the first damping parameter and recycle the subspace for the subsequent damping parameters. The efficiency of our new inverse modeling algorithm is significantly improved using these computational techniques. We apply this new inverse modeling method to invert for random transmissivity fields in 2D and a random hydraulic conductivity field in 3D. Our algorithm is fast enough to solve for the distributed model parameters (transmissivity) in the model domain. The algorithm is coded in Julia and implemented in the MADS computational framework (http://mads.lanl.gov). By comparing with LevenbergMarquardtmore »
 Authors:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 LANL EP Program; USDOE
 OSTI Identifier:
 1312574
 Report Number(s):
 LAUR1622377
Journal ID: ISSN 00431397
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Water Resources Research
 Additional Journal Information:
 Journal Name: Water Resources Research; Journal ID: ISSN 00431397
 Publisher:
 American Geophysical Union (AGU)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Earth Sciences
Citation Formats
Lin, Youzuo, O'Malley, Daniel, and Vesselinov, Velimir V. A computationally efficient parallel LevenbergMarquardt algorithm for highly parameterized inverse model analyses. United States: N. p., 2016.
Web. doi:10.1002/2016WR019028.
Lin, Youzuo, O'Malley, Daniel, & Vesselinov, Velimir V. A computationally efficient parallel LevenbergMarquardt algorithm for highly parameterized inverse model analyses. United States. doi:10.1002/2016WR019028.
Lin, Youzuo, O'Malley, Daniel, and Vesselinov, Velimir V. 2016.
"A computationally efficient parallel LevenbergMarquardt algorithm for highly parameterized inverse model analyses". United States.
doi:10.1002/2016WR019028. https://www.osti.gov/servlets/purl/1312574.
@article{osti_1312574,
title = {A computationally efficient parallel LevenbergMarquardt algorithm for highly parameterized inverse model analyses},
author = {Lin, Youzuo and O'Malley, Daniel and Vesselinov, Velimir V.},
abstractNote = {Inverse modeling seeks model parameters given a set of observations. However, for practical problems because the number of measurements is often large and the model parameters are also numerous, conventional methods for inverse modeling can be computationally expensive. We have developed a new, computationallyefficient parallel LevenbergMarquardt method for solving inverse modeling problems with a highly parameterized model space. LevenbergMarquardt methods require the solution of a linear system of equations which can be prohibitively expensive to compute for moderate to largescale problems. Our novel method projects the original linear problem down to a Krylov subspace, such that the dimensionality of the problem can be significantly reduced. Furthermore, we store the Krylov subspace computed when using the first damping parameter and recycle the subspace for the subsequent damping parameters. The efficiency of our new inverse modeling algorithm is significantly improved using these computational techniques. We apply this new inverse modeling method to invert for random transmissivity fields in 2D and a random hydraulic conductivity field in 3D. Our algorithm is fast enough to solve for the distributed model parameters (transmissivity) in the model domain. The algorithm is coded in Julia and implemented in the MADS computational framework (http://mads.lanl.gov). By comparing with LevenbergMarquardt methods using standard linear inversion techniques such as QR or SVD methods, our LevenbergMarquardt method yields a speedup ratio on the order of ~101 to ~102 in a multicore computational environment. Furthermore, our new inverse modeling method is a powerful tool for characterizing subsurface heterogeneity for moderate to largescale problems.},
doi = {10.1002/2016WR019028},
journal = {Water Resources Research},
number = ,
volume = ,
place = {United States},
year = 2016,
month = 9
}
Web of Science

A truncated LevenbergMarquardt algorithm for the calibration of highly parameterized nonlinear models
We propose a modification to the LevenbergMarquardt minimization algorithm for a more robust and more efficient calibration of highly parameterized, strongly nonlinear models of multiphase flow through porous media. The new method combines the advantages of truncated singular value decomposition with those of the classical LevenbergMarquardt algorithm, thus enabling a more robust solution of underdetermined inverse problems with complex relations between the parameters to be estimated and the observable state variables used for calibration. The truncation limit separating the solution space from the calibration null space is reevaluated during the iterative calibration process. In between these reevaluations, fewer forward simulationsmore » 
USING THE LEVENBERGMARQUARDT METHOD FOR SOLUTIONS OF INVERSE TRANSPORT PROBLEMS IN ONE AND TWODIMENSIONAL GEOMETRIES
Determining the components of a radioactive source/shield system using the system's radiation signature, a type of inverse transport problem, is one of great importance in homeland security, material safeguards, and waste management. Here, the LevenbergMarquardt (or simply 'Marquardt') method, a standard gradientbased optimization technique, is applied to the inverse transport problems of interface location identification, shield material identification, source composition identification, and material mass density identification (both separately and combined) in multilayered radioactive source/shield systems. Onedimensional spherical problems using leakage measurements of neutroninduced gammaray lines and twodimensional cylindrical problems using flux measurements of uncollided passive gammaray lines are considered. Gradientsmore »