A Method for Model Identification and Parameter Estimation

M. Bambach
Institute of Metal Forming
RWTH Aachen University

M. Heinkenschloss
Department of Computational and Applied Mathematics
Rice University

M. Herty
Department of Mathematics
RWTH Aachen University



Inverse Problems, 2013, Vol. 29, No. 2, pages 025009

Abstract

We propose and analyze a new method for the identification of a parameter-dependent model that best describes a given system. This problem arises, for example, in the mathematical modeling of material behavior where several competing constitutive models are available to describe a given material, and one has to determine the best-suited constitutive model for a given material and application from experiments. We assume that the true model is one of $N$ possible parameter-dependent models. To identify the correct model and the corresponding parameters we can perform experiments, where for each experiment we prescribe an input to the system and observe a part of the system state. Our approach consists of two stages. In the first stage, for each pair of models we determine the experiment, i.e., system input and observation, that best differentiates between the two models, and measure the distance between the two models. Then we conduct $N(N-1)$ or, depending on the approach taken, $N(N-1)/2$ experiments and use the result of the experiments as well as the previously computed model distances to determine the true model. We provide sufficient conditions on the model distances and measurement errors which guarantee that our approach identifies the correct model. Given the model, we identify the corresponding model parameters in the second stage. The problem in the second stage is a standard parameter estimation problem and we use a method suitable for the given application. We illustrate our approach on three examples, including one where the models are elliptic partial differential equations with different parameterized right hand sides and an example where we identify the constitutive equation in a problem from computational viscoplasticity.

Keywords

Model identification, parameter estimation.