## 1. Introduction

[2] Nonlinear models are frequently used to model physical phenomena and engineering applications. In this paper, we refer to a nonlinear model very broadly: the output of the model is a nonlinear function of the parameters [*Draper and Smith*, 1998]. Thus nonlinear models can include systems of partial differential equations (PDEs). Some examples include CFD (computational fluid dynamics) [*Oberkampf and Barone*, 2005], groundwater flow [*Beauheim and Roberts*, 2002], etc. Nonlinear models also include functional approximations of uncertain data via regression or response surface models. Many nonlinear models require the solution of some type of optimization problem to determine the optimal parameter settings for the model. Statisticians have long worried about how to determine the optimal parameters for nonlinear regression models [*Seber and Wild*, 2003]. In the case of nonlinear regression, optimization methods have been used to determine the parameters which “best fit” the data, according to minimizing a least squares expression. The optimization methods may not always converge and find the true solution, although advances in the optimization methods have improved nonlinear least squares solvers.

[3] In this paper, we are concerned about determining confidence intervals around the parameter values in a nonlinear model. The parameters may be parameters in an approximation model such as a regression model, or physics modeling parameters which are used in physical simulation models such as PDEs. We refer to data as separate from parameters: data are physical data which are input either to a regression or physical simulation, and parameters are variables which are used in the representation and solution of the nonlinear model. For example, in groundwater flow modeling, parameters include hydraulic conductivity, specific storage, etc. Data may include measured flow rates from well tests.

[4] The focus of this paper is on calculating and evaluating joint confidence intervals. A joint confidence interval is one that simultaneously bounds the parameters; it is also called a simultaneous confidence interval. It is not a set of individual confidence intervals for each parameter in the problem. Individual confidence intervals on parameters usually assume independence of the parameters, which may lead to large errors if one is trying to infer a region where the parameters may jointly exist, for example, with 95% confidence. A joint confidence interval for a problem with two parameters may be an ellipse, where all points within the ellipse represent potential combinations of the parameters that fall within the confidence region.

[5] We are interested in calculating joint confidence intervals to understand the range or potential spread in the parameter values. Given various sources of uncertainty, it is unlikely that the optimal parameters which minimize some least squares formulation are the only reasonable parameters. There are several sources of uncertainty that can contribute to difficulty in identifying optimal parameter values in nonlinear problems. The data itself may have significant uncertainties: there may be missing values, measurement error, systematic biases, etc. Nonlinear inverse problems may involve discontinuities which result in multiple values for the optimal parameters, due to complexities in the underlying physics (e.g., significant heterogeneities in material models). The parameters themselves may have significant variability as part of their inherent randomness. Finally, model form can also influence the parameter settings [*Sun*, 1999]. For these reasons, one should not always trust the optimal parameter values obtained by a nonlinear least squares solutions. Looking at the joint confidence intervals on parameter values will give a more complete picture about the optimal values for the parameters, and their correlation.

[6] This paper examines three methods for determining joint confidence intervals in nonlinear models and compares them in case studies. In two of our cases, the nonlinear model is a groundwater flow code; it is a set of PDEs. The first section of this paper provides background to nonlinear regression and determination of parameters in nonlinear models. The second section describes three methods which are used to determine joint confidence intervals for parameters in nonlinear models. The third section outlines the example problems used in the case studies, and the fourth section discusses the results, with a fifth section providing conclusions.