INVERSE PROBLEMS FORMULATED IN TERMS OF FIRST-KIND FREDHOLM INTEGRAL EQUATIONS IN INDIRECT MEASUREMENTS

Direct measurements of many properties of real-world systems are not possible. Information on these properties can only be inferred from the result of measurements of other quantities which may be measured directly. The process comprising direct measurements of certain characteristics of the object followed by inference on its sought-for properties from the directly measured characteristics based on a mathematical relation between unknown properties and measured characteristics is called indirect measurement, whereas inference is referred to as an inverse problem in indirect measurement. In general an inverse problem consists either in determining the characteristics of a system under study, driven by controlled or known exciting signals, or in reconstructing exciting signals acting on a system whose internal characteristics are known. In both cases, it is formulated in terms of a mathematical model relating unknown and measured characteristics and signals. One can distinguish continuous and discrete inverse problems, depending on whether the measured and sought-for quantities are represented by functions or by vectors (tuples), respectively. Very many nontrivial inverse problems in indirect measurements are ill-posed which means that they have no solution or the solution exists but is non-unique or unstable, i.e. very small disturbances in the measurement data result in large disturbances in the result of inference. High error amplification is referred to as ill-conditioning. Ill-posedness and ill-conditioning result from the lack of information on sought-for quantities, carried by the measurement data. Therefore, a priori knowledge about the space of admissible solutions has to be employed for solving such inverse problems. The theory of inverse problems and - in particular - effective numerical methods for solving them are of great importance for measurement science and technology; they are crucial for the development of many measurement, imaging and diagnostic techniques. Indirect measurements may be formulated using various mathematical models of the measurement object followed by a measuring system. A broad class of inverse problems, being of importance for indirect measurements, is formulated in terms of Fredholm integral equations of the first kind. These problems are ill-posed and strongly ill-conditioned after discretization. Therefore, sophisticated inverse procedures, utilizing various kinds of a priori knowledge, are applied for solving them. In this paper, theoretical and numerical aspects of inverse problem in indirect measurements are reviewed. In particular the concept of generalized solution (pseudosolution) and the notion of well-posedness is presented and analysed. The review is focused on inverse problems formulated in terms of Fredholm integral equations of the first kind: a general presentation of such problems, at the level of functional analysis, is followed by an overview of numerical aspects of their discretized versions. A concise presentation of selected groups of numerical methods, called inverse methods, for solving inverse problems is also provided.