Methods of solving ill-posed inverse problems

Abstract

A number of methods of solving inverse heat-conduction problems are analyzed from the point of view of their practical use. Problems of determining discrepancy gradients and obtaining smooth solutions are considered as applied to the method of iteration regularization.i. At the present time there exist two directions in the study of effects of heat and mass transfer, viz., the active development of methods of numerical modeling on the basis of equations expressing conservation laws, and continual improvement of experimental methods. If, in this connection, one considers the role of inverse heat-conduction problems (IHCP), they are important for both the first and second direction, and inverse problems particularly make it possible to extract quantitative information on the sources of characteristics appearing in the mathematical model of the effect (coefficients of equations, boundary conditions, etc.), and recover necessary data from experimental studies.Application areas of these methods are thermophysical and heat-technology studies in various fields of science, technology, industry, energetics, machine construction, aviation technology, metallurgy, etc.Obviously, it can be stated that IHCP methods underwent through their development stages of formulation of primary concepts, statement of problems and their solutions, as well as formulation of basic methods and examples of solving problems.The state-of-the-art in this area is characterized by a set of concepts and methods, which can be uncoupled as initial form of the theory and methodology of inverse problems.Further accumulation of abstract and concrete facts must lead to their union in a general theoretical system, based on uniqueness principles, rigorous mathematical studies of correctly stated problems, and further development of the IHCP algorithm apparatus.At the same time, one must carry out an allrounded practical investigation of inverse problem methods, and clarification of the most rational applications.It must be stressed that contemporary IHCP methods are oriented toward using computational technology, and only in exceptional cases are "manual" treatments allowed.2. The concepts of direct and inverse problems always refer to mathematical model effects, separated by causal and consequential characteristics of the process investigated. In the direct problem the "consequence" results from the "cause," while in inverse problems this happens in the opposite direction.The abstract form of an inverse problem is represented by an operator equation of the first kind Au=f, uCU, fCF (1) with a given operator A: U + F and an element f, from which one seeks a solution u.A characteristic feature of inverse problems is that the operator A has no bounded inverse, i.e., problem (i) is incorrect.One possibility is the parametrized statement of the inverse problem, for example, when the quantity u(x) is sought in the form

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