Inverse Modeling Via Linearized Functional Minimization

Abstract:

We present a novel parameter estimation methodology for transient models of geophysical systems with uncertain, spatially distributed, heterogeneous and piece-wise continuous parameters.
The methodology employs a bayesian approach to propose an inverse modeling problem for the spatial configuration of the model parameters.
The likelihood of the configuration is formulated using sparse measurements of both model parameters and transient states.
We propose using total variation regularization (TV) as the prior reflecting the heterogeneous, piece-wise continuity assumption on the parameter distribution.
The maximum a posteriori (MAP) estimator of the parameter configuration is then computed by minimizing the negative bayesian log-posterior using a linearized functional minimization approach.

The computation of the MAP estimator is a large-dimensional nonlinear minimization problem with two sources of nonlinearity: (1) the TV operator, and (2) the nonlinear relation between states and parameters provided by the model's governing equations.
We propose a a hybrid linearized functional minimization (LFM) algorithm in two stages to efficiently treat both sources of nonlinearity.
The relation between states and parameters is linearized, resulting in a linear minimization sub-problem equipped with the TV operator; this sub-problem is then minimized using the Alternating Direction Method of Multipliers (ADMM).

The methodology is illustrated with a transient saturated groundwater flow application in a synthetic domain, stimulated by external point-wise loadings representing aquifer pumping, together with an array of discrete measurements of hydraulic conductivity and transient measurements of hydraulic head.
We show that our inversion strategy is able to recover the overall large-scale features of the parameter configuration, and that the reconstruction is improved by the addition of transient information of the state variable.