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Parameter Range Reduction in Systems of Differential Equations

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Title: Parameter Range Reduction in Systems of Differential Equations
Author: Skelton, Andrew
Department: Department of Mathematics and Statistics
Program: Mathematics and Statistics
Advisor: Willms, Allan
Abstract: This thesis presents an algorithm for parameter range reduction in systems of ordinary differential, differential algebraic and partial differential equations. When parameter values are known only to lie in potentially large regions of parameter space, traditional parameter estimation schemes can fail to converge to useful results in reasonable time. In this algorithm, interval arithmetic and discretization formulae are used to prune regions of parameter space that are inconsistent with given experimental data. For each model parameter, the algorithm outputs both a reduced range and an initial guess to be used as input to a parameter estimation algorithm. The parameter range reduction algorithm in this thesis requires lower and upper bounds of each state variable at any point in the observation window. In the ordinary differential equation case, an algorithm is presented that efficiently converts discrete time series data to a continuous, piecewise linear band that encloses the data and retains the trends of the data to a user-specified tolerance. It is not, however, always possible to obtain experimental data for each state variable. Techniques are developed to allow effective parameter range reduction in the presence of partial data sets. In the partial differential equation case, procedures are developed to allow interpolation on higher dimensional data. The parameter range reduction algorithm is tested on a variety of differential equation models and significant reductions are obtained in a variety of cases, including when experimental data is unavailable for multiple state variables. The reduced ranges and initial guesses output by the algorithm are shown to significantly improve the performance of traditional parameter estimation methods. The parameter range reduction algorithm is also shown to be computationally fast, thus making it an effective aid in the mathematical modelling process.
Date: 2014-04
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