Assessing the validity of a linearized accuracy measure for a nonlinear parameter estimation problem

We consider accuracy assessment for the inverse problem of recovery of unknown coefficient functions in differential equations from data containing random errors. The set of PDEs constituting the current forward model describes a special case of two-phase porous-media flow. We are concerned mainly with two issues. (1) When is it valid to calculate parameter accuracies for the current nonlinear estimation problem by a linearized method, linearized covariance analysis (LCA)? (2) Can the validity of LCA be assessed without performing an accurate, but computationally very expensive, Monte Carlo analysis (MCA)? For both issues, special emphasis is put on parameter subsets for which LCA predicts high accuracy. The curvature measures of nonlinearity (CMNs) are a potential alternative to MCA. CMNs are approximate, but considerably less expensive to compute. In this paper, we apply LCA, CMNs and MCA to several instances of the current model. We address issue 1 by comparing LCA and MCA results, and issue 2 by including also CMN results in the analysis. It is found that CMN and MCA results lead to identical and negative conclusions concerning the validity of LCA. However, if the real concern is parameter subsets where LCA predicts high accuracy, these conclusions, based on calculations involving all of the parameters, were often misleading. Use of specially designed subset CMNs is essential to avoid this. A potential explanation, which may have implications also for other parameter estimation problems, is presented.