Comparison of optimal design methods in inverse problems

Abstract

Typical optimal design methods for inverse or parameter estimation problems are designed to choose optimal sampling distributions through minimization of a specific cost function related to the resulting error in parameter estimates. It is hoped that the inverse problem will produce parameter estimates with increased accuracy using data collected according to the optimal sampling distribution. Here we formulate the classical optimal design problem in the context of general optimization problems over distributions of sampling times. We present a new Prohorov metric based theoretical framework that permits one to treat succinctly and rigorously any optimal design criteria based on the Fisher Information Matrix (FIM). A fundamental approximation theory is also included in this framework. A new optimal design, SE-optimal design (standard error optimal design), is then introduced in the context of this framework. We compare this new design criteria with the more traditional D-optimal and E-optimal designs. The optimal sampling distributions from each design are used to compute and compare standard errors; the standard errors for parameters are computed using asymptotic theory or bootstrapping and the optimal mesh. We use three examples to illustrate ideas: the Verhulst-Pearl logistic population model [13], the standard harmonic oscillator model [13] and a popular glucose regulation model [16, 19, 29].

D-Optimal Design Applied to Calibration of Strapdown Three-axis Magnetometer

“…There are several optimality criteria proposed to get an optimal design, for example, A-optimal design, D-optimal design, E-optimal design and SE-optimal design [3]. Among them, the Doptimal is the most popular one.…”

“…Since the quality of data sets may dramatically influence the accuracy and robust of parameter estimation [2], a major question we face is how to best collect measurements to enable the calibration method to efficiently and accurately estimate model parameters. This is the well-known optimal design problem [3]. An optimal design of maneuver should be carefully considered before the procedure of magnetometer calibration.…”

Mathematical Models of Dividing Cell Populations: Application to CFSE Data

“…Significantly, the models (3.16) and (3.17) are not limited to a specific cell type or stimulation condition. A validated mathematical model might also be used to aid [15] in the design of experiments (e.g., to determine when to take measurements in a manner that maximizes parameter information while minimizing the amount of blood needed for the experiment).…”

Modeling the Dynamics of the Cardiovascular-respiratory System (CVRS) in Humans, a Survey

“…In [7] the design criterion was introduced in order to define SE-optimal designs in the context of very general sampling strategies characterized by probability measures on the sampling interval. See [8] for a comparison of D-optimal, E-optimal and SE-optimal designs and [9] for a Monte Carlo based analysis. The problems discussed in Subsection 5.1 require to select those parameters of a model for which the parameter estimation problem is well-posed.…”