Numerical Solution of Inverse Heat Transfer Problems by Parameters Estimation of Ordinary Differential Equations

In this paper, algorithms for solving of two inverse onedimensional heat transfer problems are considered. They are: the coe cient inverse problem and the determination of phase interface location in a melting problem. These inverse problems are reduced to parameters estimation of ordinary di erential equations (ODE) using a nite element method. The algorithm based on the sensitivity analysis is applied to parameters estimation of the resultant ODE systems. Some results of numerical experiments are given. INTRODUCTION It is known that spatial discretization of the initial boundary value problem for a one-dimensional heat equation results in the Cauchy problem for a sti ODE system. If the heat equation contains some unknown parameters, then as a result, we obtain the ODE system containing unknown parameters. Actually, there are many e ective algorithms for the numerical solution of sti ODE systems. The Rosenbrock type methods refer to them. These methods have stability properties inherent to implicit methods and, as opposed to implicit methods, they do not require iterations in performance. In (Gusev, 1983) an e ective method of identi cation of parameters of sti ODE systems was developed. The unknown parameters of ODE systems are estimated by using the least squares and sensitivity analysis. In addition, the identi cation method includes the e ective Rosenbrock type method of the second order with a variable integration step. Application of gradient methods for minimization of an objective function results in the necessity of obtaining derivatives of the solution of the ODE system with respect to the parameters. These derivatives are referred to as sensitivity functions. We name the ODE system, for which it is required to determine parameters, as the basic system. The sensitivity functions can be obtained by a simultaneous solution of the basic system and sensitivity equations. The sensitivity equations arise as a result of di erentiation of the basic system with respect to parameters. It follows from the construction of sensitivity equations that dimension of the ODE system considerably grows in computation of sensitivity functions. It is also known, that solution of ODE's by a Rosenbrock type method demands at each integration step the solution of linear equations systems of the same dimension as that of the ODE system . The proposed approximation of the numerical solution of the basic system, being included in sensitivity equations, enables to solve block diagonal systems of linear equations thus considerably redusing the execution time. THE PARAMETERS ESTIMATION METHOD OF ODE SYSTEM Let us consider a Cauchy problem of an ODE system _ y = f(y; t; p); (1) y(0; p) = y0; where y; y0 2 R; p 2 R is a vector of parameters. 1 Copyright c 1999 by ASME Let us denote Z = (z1; : : : ; zs) T a vector of observables. We assume, the components of Z to be su uciently smooth functions of the solution to system (1). Let us designate Z k values of the vector of observations at the times tk; k = 1; : : : ; s, which are obtained as a result of measurements. The estimated parameters are obtained as a result of minimization of the following function