METHOD OF MINIMAX OPTIMIZATION IN THE COEFFICIENT INVERSE HEAT-CONDUCTION PROBLEM

Consideration has been given to the inverse problem on identifi cation of a temperature-dependent thermal-conductivity coeffi cient. The problem was formulated in an extremum statement as a problem of search for a quantity considered as the optimum control of an object with distributed parameters, which is described by a nonlinear homogeneous spatially one-dimensional Fourier partial equation with boundary conditions of the second kind. As the optimality criterion, the authors used the error (minimized on the time interval of observation) of uniform approximation of the temperature computed on the object′s model at an assigned point of the segment of variation in the spatial variable to its directly measured value. Pre-parametrization of the sought control action, which a priori records its description accurate to assigning parameters of representation in the class of polynomial temperature functions, ensured the reduction of the problem under study to a problem of parametric optimization. To solve the formulated problem, the authors used an analytical minimax-optimization method taking account of the alternance properties of the sought optimum solutions based on which the algorithm of computation of the optimum values of the sought parameters is reduced to a system (closed for these unknowns) of equations fixing minimax deviations of the calculated values of temperature from those observed on the time interval of identifi cation. The obtained results confi rm the effi ciency of the proposed method for solution of a certain range of applied problems. The authors have studied the infl uence of the coordinate of a point of temperature measurement on the exactness of solution of the inverse problem.
Consideration has been given to the inverse problem on identifi cation of a temperature-dependent thermal-conductivity coeffi cient. The problem was formulated in an extremum statement as a problem of search for a quantity considered as the optimum control of an object with distributed parameters, which is described by a nonlinear homogeneous spatially one-dimensional Fourier partial equation with boundary conditions of the second kind. As the optimality criterion, the authors used the error (minimized on the time interval of observation) of uniform approximation of the temperature computed on the object′s model at an assigned point of the segment of variation in the spatial variable to its directly measured value. Pre-parametrization of the sought control action, which a priori records its description accurate to assigning parameters of representation in the class of polynomial temperature functions, ensured the reduction of the problem under study to a problem of parametric optimization. To solve the formulated problem, the authors used an analytical minimax-optimization method taking account of the alternance properties of the sought optimum solutions based on which the algorithm of computation of the optimum values of the sought parameters is reduced to a system (closed for these unknowns) of equations fixing minimax deviations of the calculated values of temperature from those observed on the time interval of identifi cation. The obtained results confi rm the effi ciency of the proposed method for solution of a certain range of applied problems. The authors have studied the infl uence of the coordinate of a point of temperature measurement on the exactness of solution of the inverse problem.
Author:  A. N. Diligenskaya and É. Ya. Rapoport
Keywords:  coefficient inverse heat-conduction problem, parametric optimization, minimax-optimization method
Page:  1008