A truncated Levenberg–Marquardt algorithm for the calibration of highly parameterized nonlinear models

Abstract

We propose a modification to the Levenberg-Marquardt 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 Levenberg-Marquardt 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 re-evaluated during the iterative calibration process. In between these re-evaluations, fewer forward simulations are required, compared to the standard approach, to calculate the approximate sensitivity matrix. Truncated singular values are used to calculate the Levenberg-Marquardt parameter updates, ensuring that safe small steps along the steepest-descent direction are taken for highly correlated parameters of low sensitivity, whereas efficient quasi-GaussNewton steps are taken for independent parameters with high impact. The performance of 1 Corresponding author; SAFinsterle@lbl.gov, Earth Sciences Division, 1 Cyclotron Road, MS 90-1116, Berkeley, CA 94720; phone: (510) 486-5205; fax: (510) the proposed scheme is demonstrated for a synthetic data set representing infiltration into a partially saturated, heterogeneous soil, where hydrogeological, petrophysical, and geostatistical parameters are estimated based on the joint inversion of hydrological and geophysical data.

A computationally efficient parallel Levenberg‐Marquardt algorithm for highly parameterized inverse model analyses

“…The classical Levenberg-Marquardt algorithm has been widely used in inverse modeling, because it is more robust than the Gauss-Newton method and yields a faster rate of convergence than the steepest descent method [Finsterle and Kowalsky, 2011;Fienen et al, 2009;Nowak and Cirpka, 2004;Simunek et al, 1998]. However, the efficiency and robustness of Levenberg-Marquardt algorithm is dependent on the correct selection of the damping parameter [Nielsen, 1999].…”

“…In this version, a parallel run manager, YAMR (Yet Another run ManageR), has been integrated. Another example of the coarse-grained parallelism is the ''Parallel PEST'' option used in the software packages of PEST [Doherty and Hunt, 2010].The Levenberg-Marquardt (LM) algorithm has been used extensively in solving nonlinear inverse problems in groundwater modeling because of its robustness [Finsterle and Kowalsky, 2011;Tonkin and Doherty, 2005;Nowak and Cirpka, 2004;Cooley, 1985]. In this work, we also use it to solve our minimization problem.…”

Estimation of plume distribution for carbon sequestration using parameter estimation with limited monitoring data

“…Main methods for gradient-based optimization are the Levenberg-Marquardt formulation and the conjugate gradient or quasi-Newton approaches. For the Levenberg-Marquardt formulation, studies and algorithm development are included by Zhang et al [2003], Tonkin and Doherty [2005], Finsterle and Kowalsky [2011], and Finsterle and Zhang [2011]. For the conjugate gradient or quasi-Newton approach, Cheng et al [2003] and Oyerinde et al [2009] are examples.…”

Solving iTOUGH2 simulation and optimization problems using the PEST protocol

“…The concept of estimating superparameters (Tonkin 290 and Doherty, 2005), implemented in PEST, is a powerful method to address highly 291 parameterized inverse problems. The regularization approach employed by iTOUGH2 is 292 described in Finsterle and Kowalsky (2011). In addition to the algorithm, iTOUGH2 provides the local and global minimization methods summarized in 294 Table 1.…”