### A Computational and Statistical Comparison of Approximation Methods for the Estimation of Probability Distributions on Parameters

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Jimena Davis, North Carolina State University
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We will present and compare computational and statistical results of some approximation methods for a general class of inverse problems in which the underlying dynamics are described by partial differential equations and the unknown quantity of interest is a probability distribution. We are interested in determining the distribution P* from a given family P(Q) that gives the best fit of the underlying model to some given data. In general, this optimization problem involves both an infinite dimensional state space and an infinite dimensional parameter space. Therefore, computationally efficient approximation methods are desired. In choosing these methods, we want the finite dimensional sets P^M(Q) to converge to P(Q) in some sense, and in our efforts, we use the Prohorov metric of weak star convergence of measures for these results. The approximation methods that we will present are applicable to a variety of inverse problems, including Type I problems in which individual longitudinal data is available for members in the population and Type II problems in which only aggregate or population level longitudinal data is available. The results that we will present are for problems of Type II, where the population of interest is a size-structured mosquitofish population.

Mosquitofish are being used in the place of chemicals by biologists to control mosquito populations in rice fields in an effort to protect the environment. While they have used mosquitofish in the place of pesticides for some time, they have not completely understood the control of the growth of mosquitofish populations. In order to determine the optimal amount of mosquitofish to use for control purposes, biologists would like a mathematical model that is capable of accurately predicting the growth and decline of the mosquitofish populations. We will present the Sinko-Streifer population model modified as in the Growth Rate Distribution model of Banks-Botsford-Kappel-Wang. We will also present and compare computational and statistical results of a delta function approximation method, a spline based approximation method, and a parameterized ordinary least squares (OLS) formulation. The latter uses an a priori probability distribution in the inverse problem for estimation of distributions of growth rates in size-structured mosquitofish populations. The approximation methods are tested with experimental data collected from rice fields.

**Abstract Author(s): **Jimena L. Davis and H.T. Banks