The Numerical Simulations of Inverse Problems on the Parameter Estimation

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The problem of determination an estimator through sample data is a part of the inverse problems. Generally, the inverse problem has no solution in ordinary sense since the observation data have been contaminated by noises. The minimizer of the least square functional is usually taken as the solution of the inverse problem. The optimal design on parameter estimation uses the Fisher information matrix (FIM) as a tool for optimal criteria that minimizes some cost functional over set of FIM's. The consideration is based on the Cramer-Rao lower bound inequality can only be attained by the inverse of FIM. This article demonstrates how to implement the inverse problems in connection with parameter estimation numerically where the set of noises is generated independently from one trial to another. The numerical simulation is applied to a distributed parameter system of parabolic equation to find the optimal sensor locations for the parameter. The simulation is also carried out to a model of dynamical system to obtain the optimal time for measurements. Correspondingly, two algorithms are composed to do numerical realization by computer. The results from numerical experiments are confirmed to theoretical background. In particular, the accuracy of estimators are compared to the prior supposed nominal parameters and the variance of estimators are contrasted with the lower bound of Cramer-Rao inequality through the functional value acting on FIM's. The numerical results are also confirmed to the premise used in the parameter estimation that the information content on the parameter may vary considerably from one time measurement to another.