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1

Conditional stability estimate for an ill-posed elliptic equation by using nonlocal conditions

Benrabah Abderafik

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 3

5--17

EN

We consider an ill-posed linear homogeneous fourth-order elliptic equation. We show that the problem is ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the given data. We propose a regularization method via nonlocal conditions and under some a priori bound assumptions different estimates for the regularized solution are obtained. Numerical examples for a rectangle domain show the effectiveness of the new method in providing highly accurate numerical solutions as the noise level tends to zero.

2

An effective estimate for selecting the regularization parameter in the 3D inversion of magnetotelluric data

Ghaedrahmati Reza, Moradzadeh Ali, Moradpouri Farzad

Acta Geophysica

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2022

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Vol. 70, no 2

609--621

EN

The magnetotelluric (MT) inverse problem is a nonlinear and strongly ill-posed problem. Therefore, to avoid the problem of non-uniqueness of response, this problem is mainly solved by Tikhonov regularization method. The purpose of this study is to present a suitable method for selecting the regularization parameters in the 3D MT inverse problem, with regard to the accuracy and speed of the inversion. In this research, the regularization parameter is simply estimated in each iteration of inversion as the ratio of the data misfit to sum of the data misfit and model norm in the pre-iteration. This scheme is applied in the well-known 3D inversion algorithm, WSInv3DMT, instead of the discrepancy principle method. The accuracy of this scheme is assessed by performing the inversion on synthetic models and real data. Results from the inversion for the synthetic and real data indicate that the data misfit and the model norm are reduced with an acceptable rate during the inversion operation. The inverse model has been smoothly converged to an appropriate model and that unrealistic structures have not been included in the model. The results also show that estimation of the regularization parameter by the discrepancy principle method and continuing the inversion to achieve the target data misfit may lead to the production of a model with non-realistic structures, while in the proposed scheme the inversion has not encountered this problem and it converges to an appropriate model after fewer iterations of inversion. In addition, the results show that the time consumed for the inversion of a set of real data with 41 stations and 16 measurement frequencies would decrease up to 27 percent compared to the time devoted for inverting the same set of data by the discrepancy principle method. Also the inversion does not deviate toward unrealistic models and it closely converges to the model of real geological structures.

3

Rekonstrukcja obrazów w elektrycznej tomografii pojemnościowej

Smolik W. T.

Prace Naukowe Politechniki Warszawskiej. Elektronika

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2013

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z. 188

3--226

PL

W rozprawie zaprezentowano technikę obrazowania metodą elektrycznej tomografii pojemnościowej. Opisano podstawy fizyczne, wybrane metody rozwiązywania problemu prostego i algorytmy rekonstrukcji obrazów oraz konstrukcję aparatury pomiarowej. Omówiono metody wyznaczania rozkładu pola elektrycznego i macierzy wrażliwości w kontekście modelowania tomograficznych sond pojemnościowych. Numeryczne algorytmy wyznaczania rozkładu potencjału, optymalizowane przez autora pod względem czasu obliczeń, mają duże znaczenie w nieliniowej, trójwymiarowej rekonstrukcji obrazów. Zaprezentowano występujące w tomografii elektrycznej nieliniowe zagadnienie odwrotne wiąz metodami regularyzacji problemu źle uwarunkowanego i technikami doboru parametru regularyzacji. Opisano algebraiczne, liniowe i nieliniowe metody rekonstrukcji obrazów, a w szczególności zaproponowane przez autora algorytmy przedziałami liniowe. W rozprawie przedstawiono, opracowane iv ramach p tac badawczych, oprogramowanie do numerycznego modelowania i rekonstrukcji obrazów w elektrycznej tomografii pojemnościowej. Opisano konstrukcję pojemnościowych sond tomograficznych oraz metody pomiaru bardzo małych pojemności. Zaprezentowano opracowany w ramach prac doświadczalnych tomograf pojemnościowy. Rozprawa jest podsumowaniem badań prowadzonych przez autora w dziedzinie tomografu elektrycznej.

EN

This dissertation presents an imaging technique by means of electrical capacitance tomography. The physical basis, selected methods of forward problem solution and image reconstruction algorithms, as well the design of the measurement system were described. The methods of electric field distribution and sensitivity matrix calculation were discussed m the context of the design of capacitance tomographic sensors. Numerical algorithms for potential distribution computation, optimized by die author regarding calculation speed, are of great importance in nonlinear three-dimensional image reconstruction. Nonlinear inverse problems occurring in electrical tomography together with the regularization methods for ill-conditioned problems and selection techniques of a regularization parameter were presented. Algebraic linear and nonlinear image reconstruction methods, particularly range linear algorithms proposed by the author, were described. The dissertation also presents software for numerical modelling and image reconstruction in electrical capacitance tomography elaborated within the framework of the author's research and describes the design of tomographic capacitance sensor and the methods for very small capacitance measurement. The capacitance tomography scanner elaborated within the framework of experimental work was presented. This dissertation is a summary of the research carried out by the author in the field of electrical tomography.