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Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements

Mroczka J., Szczuczyński D.

Metrology and Measurement Systems

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2009

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Vol. 16, nr 3

333-357

EN

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.

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Wpływ modelu rozproszenia światła na jakość rozwiązań zagadnienia odwrotnego w pomiarach nefelometrycznych

Mroczka J., Szczuczyński D.

Pomiary Automatyka Kontrola

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2007

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R. 53, nr 9 bis

242--245

PL

W pracy przedstawiono rezultaty badań symulacyjnych wpływu stosowanego modelu matematycznego pomiarów nefelometrycznych na jakość rozwiązań zagadnienia odwrotnego polegającego na wyznaczaniu funkcji rozkładu wielkości częstek fazy zdyspergowanej układu dyspersyjnego na podstawie wyników pomiarów nefelometrycznych. Analizie poddane zostały dwa modele matematyczne: model oparty na ogólnej teorii Mie rozpraszania światła na cząstce sferycznej oraz znacznie prostszy model bazujący na teorii dyfakcji Fraunhofera stanowiącej przybliżenie teorii Mie dla pewnych szczególnych warunków. Uzyskane wyniki wykazały, że rozwiązania rozważanego zagadnienia odwrotnego otrzymywane z zastosowaniem modelu Fraunhofera charakteruzują się ogólnie mniejszym błędem oraz znacznie mniejszą podatnością na niekorzystne efekty złego postawienia problemu w porównaniu z rozwiązaniami uzyskanymi w oparciu o bardziej skomplikowany model Mie, o ile spełnione są warunki stosowalności teorii Fraunhofera.

EN

The work presents results of the simulation research on the influence of applied mathematical model of nephelometric measurements on the quality of solutions of the inverse problem consisting in determination of the particle size distribution of the dispersed phase of the dispersed system basig on results of nephelometric measurements. Two mathematical models were analyzed: the model based on the general Mie theory of light scattering by a spherical particle and the considerably simpler modelbased on the Fraunhofer diffraction theory which is an approximation of the Mie theory for certain particular conditions. Obtained results demonstarted that the solutions of considered inverse problem gained by application of the Fraunhofer model are characterized by generally smaller error and significantly smaller susceptibility to harmful effects of illposedness of the problem comparing to solutins gained basing on more complicated Mie model as long as the conditions of applicability of the Fraunhofer theory are fulfilled.

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Problem odwrotny - jakość rekonstrukcji funkcji rozkładu wielkości cząstek w pomiarach nefelometrycznych i turbidymetrycznych

Mroczka J., Szczuczyński D.

Pomiary Automatyka Kontrola

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2007

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R. 53, nr 9 bis

246--249

PL

W pracy przedstawiono wyniki badań symulacyjnych mających na celu porównanie jakości rekonstrukcji funkcji rozkładu wielkości cząstek fazy zdyspergowanej układu dyspersyjnego realizowanej poprzez rozwiązywanie problemu odwrotnego dla wyników pomiarów nefelometrycznych oraz dla wyników pomiarów turbidymetrycznych o róznym stopniu zakłócenia przez błędy losowe. W przypadku obu technik pomiarowych zastosowano modele matematyczne oparte na teorii rozpraszania światła Mie. Uzyskane rezultaty wykazały, że rekonstrukcja runkcji rozkładu na podstawie wyników pomiarów turbidymetrycznych charakteryzuje się ogólnie większą dokładności od rekonstrukcji na podstawie wyników pomiarów neflometrycznych. Przewaga rekonstrukcji realizowanej w oparciu o wyniki pomiarów turbidymetrycznych wyraźnie zwiększa się w przypadku, gdy dane pomiarowe obu rodzajów obciążone są błędami losowymi.

EN

This work presents results of the simulation research aiming comparison of the quality of teh reconstruction of the particle size distribution of the dispersed phase of the dispersed system performed by solution of the inverse problem for results of nephelometric and turbidimetric measurements interfered to a various extent by random errors. In case of both measurement techniques mathematical models based on Mie light scattering theory were applied. Obtained results demonstrated that the reconstruction of the particle size distribution on the basis of the results of turbidimetric measurements is characterized by generally bigger accuracy than the reconstruction on the basis of the results of nephelometric measurements. The advantage of the reconstruction performed basing on the results of turbidimetric measurements increases considerably in case when the measruement data of both kinds are affected by random errors.