Optimal Design Techniques for Distributed Parameter Systems

Abstract

A wide number of inverse problems consist in selecting best parameter values of a given mathematical model based fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data. The problem of collecting data in the "best way" in order to assure a statistically efficient estimate of the parameter is known as Optimal Design. In this work we consider the problem of finding optimal locations for source identification in the 3D unit sphere from data on its boundary. We apply three different optimal design criteria to this 3D problem: the Incremental Generalized Sensitivity Function (IGSF), the classical D-optimal criterion and the SE-criterion recently introduced in [3]. The estimation of the parameters is then obtained by means of the Ordinary Least Square procedure on the resulting optimal observation points and compared to that for a uniform observation mesh.. In order to analyze the performance of each strategy, the data are numerically simulated and the estimated values are compared with the values used for simulation.

Optimal Estimation of Thermal Diffusivity in an Energy Transfer Problem

“…where Θ is the set of admissible values for the parameter to be estimated. We look for the parameter value α 2 that minimizes the functional (5), that is,…”

Parameter Estimation in Distributed Systems: Optimal Design

“…Since the sensitivity equations cannot be easily solved for in the models chosen here and in [15,16,17,18] to illustrate this method, one can use a modified version of tssolve.m [5], which implements the myAD package developed in [23]. Solving (19) requires using a nonlinear constrained optimization algorithm.…”

“…Such models where there may be a wide range of variables to possibly observe are not only ideal on which to test the proposed methodology, but also are widely encountered in applications. For example these methods have been recently used in [17,18,19] to design optimal data collection in terms of the location of sensors and the number needed for optimal design in electroencephalography (EEG) in the recording of electrical activity along the scalp. The underlying models in these applications are nonhomogeneous second order elliptic partial differential equations.…”

Parameter Estimation in Distributed Systems: Optimal Design

“…Since the sensitivity equations cannot be easily solved for in the models chosen here and in [15,16,17,18] to illustrate this method, one can use a modified version of tssolve.m [5], which implements the myAD package developed in [23]. Solving (19) requires using a nonlinear constrained optimization algorithm.…”