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 2‐D and a random hydraulic conductivity field in 3‐D. 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 multicore computational environment. Therefore, our new inverse modeling method is a powerful tool for characterizing subsurface heterogeneity for moderate to large‐scale problems.

A Novel Hybrid Approach for Numerical Modeling of the Nucleating Flow in Laval Nozzle and Transonic Steam Turbine Blades

“…To prevent high iterations for the Levenberg-Matquardt's method, some criteria have been proposed by Denis which are shown in Equation (27) [37,38]:…”

“…where the terms ε 1 , ε 2 and ε 3 are small numbers for convergence [35]. The conditions given by Equation (27) are tests for the minimum sum of squared errors which is considerably small, and it is expected that it finds the closest answer to the real solution [39], where by establishing convergence in Equations (25)- (27), the final index is selected. In this research, first, the hybrid (scalar (Jameson) + CUSP) modeling approach will be followed to model two-phase steam flow for five nozzles with different geometries (Moore nozzle types A and B, Young nozzle type C, Barschdorff nozzle) and also the dry flow between the blades of a turbine with moderate slope.…”

“…Based on the authors' experience, the dry steam flow is run for a limited number of times, e.g., 10 runs for the scalar (Jameson) and then the CUSP is run for the same number of times. Then the results are passed onto the inverse method for analysis and finally every combination will be checked for the final convergence of Equations (25)- (27), and these procedures are conducted in order ( Figure 2). In the second step of the flow solution, the equation for real gases alongside dry flow conditions are considered and procedures similar to step one are continued.…”

“…The Levenberg-Marquardt algorithm is used to find the minimum of a multivariate nonlinear function; it is designed as a standard method for solving least squares problems for nonlinear functions, and exhibits high efficiency in solving inverse engineering problems. As of present, this algorithm can be used to solve a variety of nonlinear equations [27].…”

Quantifying model structural error: Efficient Bayesian calibration of a regional groundwater flow model using surrogates and a data‐driven error model

“…As numerical groundwater models are increasingly used to inform water resources management decisions and policies under future scenarios of climate and land use change, there is a need to ensure accuracy and quantify the intrinsic uncertainties of these models. In the last two decades, there have been significant advances in least square regression‐based calibration and uncertainty quantification techniques [e.g., Doherty et al ., ; Hill and Tiedeman , ; Lin et al ., ; Tonkin et al ., ]. However, common practices often focus on parameter uncertainty while neglecting model structural error, which is ubiquitous in groundwater models due to simplified or even incorrect conceptualiztion of the real hydrogeologic system [ Cooley , ; Gupta et al ., ; Refsgaard et al ., ].…”