Ensemble methods for large scale inverse problems

Abstract

Variational data assimilation, also sometimes simply called the ‘adjoint method’, is used very often for large scale model calibration problems. Using the available data, the uncertain parameters in the model are identified by minimizing a certain cost function that measures the difference between the model results and the data. A variational scheme requires the implementation of the adjoint of (the tangent linear approximation of) the original model which is a tremendous programming effort, that hampers new applications of the method. Recently a new ensemble approach to variational inverse modelling using Proper Orthogonal Decomposition (POD) model reduction has been proposed that does not require the implementation of the adjoint model. Using an ensemble of forward model simulations an approximation of the covariance matrix of the model variability is determined. A limited number of leading eigenvectors of this matrix are selected to define a model sub space. By projecting the original model onto this subspace an approximate linear model is obtained. Once this reduced model is available the minimization process can be solved completely in reduced space with negligible computational costs. Schemes based on the well-known Kalman filtering algorithm are also used recently for inverse modeling. The last years a number of ensemble based algorithms have been proposed, e.g., the Ensemble Kalman filter (EnKF), the Reduced Rank Square Root filter (RRSQRT) and the Ensemble Square Root filter (ESRF). Although introduced for linear state estimation, these new algorithms 102 Ensemble methods for large scale inverse problems are able to handle nonlinear models accurately and, therefore, are very attractive for solving combined state and parameter estimation problems. It has been shown recently that the so-called symmetric version of the ESRF introduces the smallest increments and, therefore, is in most applications more accurate then the original version of this algorithm.