Analysis of parameter sensitivity and experimental design for a class of nonlinear partial differential equations

Abstract

The purpose of this work is to analyse the parameter sensitivity problem for a class of nonlinear elliptic partial differential equations, and to show how numerical simulations can help to optimize experiments for the estimation of parameters in such equations. As a representative example we consider the Laplace–Young problem describing the free surface between two fluids in contact with the walls of a bounded domain, with the parameters being those associated with surface tension and contact. We investigate the sensitivity of the solution and associated functionals to the parameters, examining in particular under what conditions the solution is sensitive to parameter choice. From this, the important practical question of how to optimally design experiments is discussed; i.e. how to choose the shape of the domain and the type of measurements to be performed, such that a subsequent inversion of the measured data for the model parameters yields maximal accuracy in the parameters. We investigate this through numerical studies of the behaviour of the eigenvalues of the sensitivity matrix and their relation to experimental design. These studies show that the accuracy with which parameters can be identified from given measurements can be improved significantly by numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.