Model Calibration Under Uncertainty: Matching Distribution Information

Abstract

*† ‡ We develop an approach for estimating model parameters which result in the “best distribution fit” between experimental and simulation data. Best distribution fit means matching moments of experimental data to those of a simulation (and possibly matching a full probability distribution). This approach extends typical nonlinear least squares methods which identify parameters maximizing agreement between experimental points and computational simulation results. Several analytic formulations for the distribution matching problem are provided, along with results for solving test problems and comparisons of this parameter estimation technique with a deterministic least squares approach. I. Introduction Nonlinear models are frequently used to model physical phenomena, including 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 [Draper98]. Thus, nonlinear models can include systems of partial differential equations (PDEs). Some examples include CFD (computational fluid dynamics), groundwater flow, heat transport, etc. Nonlinear models also include functional approximations of uncertain data via regression or response surface models. In most cases, we have some type of simulation model which is a nonlinear model, so we use the term nonlinear model and simulation model interchangeably in this paper. In addition to a simulation model, we assume that we have experimental data which may be used to calibrate the model. The calibration often requires the solution of an optimization problem to determine the optimal parameter settings for the simulation model. In this paper, we are concerned with identifying model parameters which result in a “best fit” between experimental data and simulation results in a nondeterministic context. That is, instead of matching point estimates we are concerned with matching moments (e.g., mean or variance) between experimental and simulation data where the variability in the model output is due to parametric uncertainty. We develop an approach extending the typical nonlinear least squares formulation to allow for this distribution matching. Note that “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 distinguish data from parameters: data are physical data which are input either to a regression or physical simulation. For example, in groundwater flow modeling, hydraulic conductivity is a parameter and data may include measured flow rates from well tests. In this paper, we denote parameters that will be calibrated as θ, and the independent input data (e.g., state variables, configuration data, boundary conditions) as x. We also assume that there are uncertain variables, denoted by u, that represent inherent variability or lack of exact knowledge influencing the simulation, but which we cannot observe. The effect of the uncertain variables is reflected in both the output variability of the nonlinear model and the experimental data, but we can only explicitly account for these uncertain variables in the simulation model. Thus, our simulation model, f,