A computationally efficient parallel Levenberg‐Marquardt algorithm for highly parameterized inverse model analyses. Issue 9 (15th September 2016) - Lin, Youzuo; O'Malley, Daniel; Vesselinov, Velimir V.
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 ∼ 10 1 to ∼ 10 2 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. Key Points: We generate a Krylov subspace and obtain a much smaller approximated problem by projecting the original problem down to the subspace We employ both coarse and fine‐grained parallelism to our LM method to parallelize the implementation of the LM algorithm in two levels We employ a subspace recycling technique to take the advantage of the solution space previously generated