Functional Identification of the Nonlinear Thermal-Conductivity Coefficient by Gradient Methods. I. Conjugate Operators

Abstract

UDC 536.2Consideration is given to the gradient methods of solution of the inverse heat-conduction problem on determination of the nonlinear coefficient λ(T) without its preliminary finite-dimensional approximation.Introduction. Gradient methods of numerical solution of inverse heat-conduction problems have been developed in many works, mainly in [1][2][3]. In particular, the problem of identification of the nonlinear thermal-conductivity coefficient λ(T) has been considered in [3][4][5][6]. In [1-3, 7, 8], gradient methods have been used for restoration and evaluation of the power of heat sources.One problem frequently arising when gradient methods are used is numerical realization of the values of conjugate (adjoint) operators. For example, in the case of identification of λ(T), the operator conjugate to the internal-superposition operator (other names [9]: the substitution operator, the weighted-shift operator, the operator of replacement of a variable, and the composite operator) is present in the scheme of the method of conjugate gradients. The wellknown approach presented in [3] leads to a complex and difficult-to-control procedure of computation of the values of the operator conjugate to the internal-superposition operator. Therefore, a finite-dimensional approximation of the sought nonlinear coefficients by any system of basis functions has been used in [3] and in subsequent works, thus reducing inverse heat-conduction problems to a problem of restoration of a finite number of parameters. In this connection, such approaches to solution of inverse heat-conduction problems are frequently called parametric ones.In the present work, we consider heat-conduction problems without preliminary approximation of the functions sought. Such an approach is conventionally called functional (or finite-dimensional) identification. Functional identification of the nonlinear thermal-conductivity coefficient by gradient methods is based on new representations of the operator conjugate to the internal-superposition operator; these representations enable one to obtain formulas of the values of a conjugate operator, which are convenient for numerical calculations. We note that similar representations were used earlier in the theory of controlled integro-differential and functional-differential systems [10][11][12].The results of the work are presented in two papers. In the first paper, we describe the algorithm of functional identification of the coefficient λ(T); the emphasis is on finding the gradient of the square of the residual functional for λ(T) in the space L 2 [T (1) , T (2) ] of functions summable with the square and in the Sobolev space W 2 [T

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