# A computationally efficient parallel Levenberg-Marquardt 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, computationally-efficient parallel Levenberg-Marquardt method for solving inverse modeling problems with a highly parameterized model space. Levenberg-Marquardt methods require the solution of a linear system of equations which can be prohibitively expensive to compute for moderate to large-scale 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 Levenberg-Marquardtmore »

- 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):
- LA-UR-16-22377

Journal ID: ISSN 0043-1397

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Water Resources Research

- Additional Journal Information:
- Journal Name: Water Resources Research; Journal ID: ISSN 0043-1397

- 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 Levenberg-Marquardt 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 Levenberg-Marquardt algorithm for highly parameterized inverse model analyses*. United States. doi:10.1002/2016WR019028.

```
Lin, Youzuo, O'Malley, Daniel, and Vesselinov, Velimir V. Thu .
"A computationally efficient parallel Levenberg-Marquardt 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 Levenberg-Marquardt 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, computationally-efficient parallel Levenberg-Marquardt method for solving inverse modeling problems with a highly parameterized model space. Levenberg-Marquardt methods require the solution of a linear system of equations which can be prohibitively expensive to compute for moderate to large-scale 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 Levenberg-Marquardt methods using standard linear inversion techniques such as QR or SVD methods, our Levenberg-Marquardt method yields a speed-up ratio on the order of ~101 to ~102 in a multi-core computational environment. Furthermore, our new inverse modeling method is a powerful tool for characterizing subsurface heterogeneity for moderate- to large-scale problems.},

doi = {10.1002/2016WR019028},

journal = {Water Resources Research},

number = ,

volume = ,

place = {United States},

year = {Thu Sep 01 00:00:00 EDT 2016},

month = {Thu Sep 01 00:00:00 EDT 2016}

}

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