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 unkn own coefficient functions in differential equations from data containing ra ndom errors. The set of PDEs constituting the current forward model describ es a special case of two-phase porous-media flow. We are concerned mainly w ith 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 an alysis (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 t o 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 I by comparing LCA and MCA results, and issue 2 by including also CMN results in the analysis. It is found that CMN and MCA r esults lead to identical and negative conclusions concerning the validity o f LCA. However, if the real concern is parameter subsets where LCA predicts high a ccuracy, these conclusions, based on calculations involving all of the para meters, were often misleading. Use of specially designed subset CMNs is ess ential to avoid this. A potential explanation, which may have implications also for other parameter estimation problems, is presented.