# Computation of Parameters in some Mathematical Models

In computational science it’s quite common to explain dynamic systems by mathematical models in forms of differential or integral equations. These types of designs could have variables that have to be computed for the model to be complete. For the unique kind of ordinary differential equations analyzed within this dissertation, the resulting parameter estimation problem is a separable nonlinear least squares problem with equality restrictions. This challenge could be resolved by iteration, but because of complex calculations of derivatives and the existence of numerous local minima, so called short-cut techniques might be an alternative solution. These techniques are dependant on simple versions of the primary problem. An algorithm, referred to as modified Kaufman algorithm, is suggested and it takes the separability into consideration. Furthermore, different types of discretizations and formulations of the optimisation problem are reviewed plus the effect of ill-conditioning.Calculation of parameters frequently consists of as a part solution of linear system of equations Ax = b. The related pseudoinverse solution depends upon the attributes of the matrix A and vector b. The singular value decomposition of A may then be utilized to construct error propagation matrices and by use of these it’s possible to analyze how alterations in the input data impact the solution x….

Contents

1 Why this kind of Research?
2 The Area of Research – Scientiﬁc Computing
2.1 Computational Science and Engineering
2.1.1 Computation of Model Parameters
2.2 Examples of Applications
2.3 Some Important Concepts
2.3.1 Dynamic Models and Differential Equations
2.3.2 Direct and Inverse Problems
2.3.3 Optimization and Linear Algebra
2.3.4 Computerizing and Approximations
2.3.5 Validation and Error Reduction
2.4 Historical Notes and Visions for the Future
2.5 Bibliography
3
4 Summary of the Papers
4.1 Paper I & II
4.1.1 Contributions
4.1.2 Future Work
4.2 Paper III & IV
4.2.1 Contributions
4.2.2 Future Work…

Source: Umea University