We consider the problem of estimating amodeling parameter θ using a weighted least squares criterion for given data y by introducing an abstract framework involving generalized measurement procedures characterized by probability measures. We take an optimal design perspective, the general premise (illustrated via examples) being that in any data collected, the information content with respect to estimating θ may vary considerably from one time measurement to another, and in this regard some measurements may be much more informative than others. We propose mathematical tools which can be used to collect data in an almost optimal way, by specifying the duration and distribution of time sampling in the measurements to be taken, consequently improving the accuracy (i.e., reducing the uncertainty in estimates) of the parameters to be estimated. We recall the concepts of traditional and generalized sensitivity functions and use these to develop a strategy to determine the “optimal” final time T for an experiment; this is based on the time evolution of the sensitivity functions and of the condition number of the Fisher information matrix. We illustrate the role of the sensitivity functions as tools in optimal design of experiments, in particular in finding “best” sampling distributions. Numerical examples are presented throughout to motivate and illustrate the ideas.

In this note we present a critical review of the some of the positive features as well as some of the shortcomings of the generalized sensitivity functions (GSF) of Thomaseth-Cobelli in comparison to traditional sensitivity functions (TSF). We do this from a computational perspective of ordinary least squares estimation or inverse problems using two illustrative examples: the Verhulst-Pearl logistic growth model and a recently developed agricultural production network model. Because GSF provide information on the relevance of data measurements for the identification of certain parameters in a typical parameter estimation problems, we argue that they provide the basis for new tools for investigators in design of inverse problem studies.Keywords: Inverse problems, sensitivity and generalized sensitivity functions, Fisher information matrix, logistic growth model, agricultural production networks. In this note we present a critical review of the some of the positive features as well as some of the shortcomings of the generalized sensitivity functions (GSF) of Thomaseth-Cobelli in comparison to traditional sensitivity functions (TSF). We do this from a computational perspective of ordinary least squares estimation or inverse problems using two illustrative examples: the Verhulst-Pearl logistic growth model and a recently developed agricultural production network model. Because GSF provide information on the relevance of data measurements for the identification of certain parameters in a typical parameter estimation problems, we argue that they provide the basis for new tools for investigators in design of inverse problem studies.

In this note we present a critical review of the some of the positive features as well as some of the shortcomings of the generalized sensitivity functions (GSF) of Thomaseth-Cobelli in comparison to traditional sensitivity functions (TSF). We do this from a computational perspective of ordinary least squares estimation or inverse problems using two illustrative examples: the Verhulst-Pearl logistic growth model and a recently developed agricultural production network model. Because GSF provide information on the relevance of data measurements for the identification of certain parameters in a typical parameter estimation problems, we argue that they provide the basis for new tools for investigators in design of inverse problem studies.Keywords: Inverse problems, sensitivity and generalized sensitivity functions, Fisher information matrix, logistic growth model, agricultural production networks. In this note we present a critical review of the some of the positive features as well as some of the shortcomings of the generalized sensitivity functions (GSF) of Thomaseth-Cobelli in comparison to traditional sensitivity functions (TSF). We do this from a computational perspective of ordinary least squares estimation or inverse problems using two illustrative examples: the Verhulst-Pearl logistic growth model and a recently developed agricultural production network model. Because GSF provide information on the relevance of data measurements for the identification of certain parameters in a typical parameter estimation problems, we argue that they provide the basis for new tools for investigators in design of inverse problem studies.

We present an eigensystem decomposition method to recover weak inhomogeneities in a waveguide from knowledge of the far-field scattered acoustic fields. Due to the particular geometry of the waveguide, which supports only a finite number of propagating modes, the problem of recovering inhomogeneities in a waveguide has a different set of challenges than the corresponding problem in free space. Our method takes advantage of the spectral properties of the far-field matrix, and by using its eigenvalues and its eigenvectors we obtain a representation of the linearized solution to the inverse problem in terms of products of fields which are linear combinations of the propagating modes of the waveguide with weights given by the eigenvectors of the far-field matrix. The problem of finding the unknown inhomogeneity reduces to a problem of determining some coefficients in a finite system of linear equations, whose coefficients depend on the background medium and the eigenvalues and the eigenvectors of the far-field matrix. By numerically implementing our inverse algorithm we reconstructed the inhomogeneities present in the waveguide. We show that (1) even with as few as seven propagating modes we obtain a good recovery of the size and shape of the inhomogeneity; (2) multiple inhomogeneities can be well recovered and, as expected, (3) the recovery improves when the number of propagating modes increases.