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1

Positronium imaging in J-PET with an iterative activity reconstruction and a multistage fitting algorithm

Shopa Roman Y., Dulski Kamil

Bio-Algorithms and Med-Systems

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2023

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Vol. 19, no. 1

54--63

EN

Positronium imaging is a new technique complementary to positron emission tomography (PET) based on the histogramming of time delay between the emission of a de-excitation photon, and a consequent electron-positron annihilation, to estimate the mean lifetime of orthopositronium (o-Ps), which depends on the local size of the voids, concentration of oxygen and bioactive molecules. We improve the resolution and reduce noise in positronium imaging by building time-delay spectra from the PET activity reconstructed by a 3-photon time-of-flight maximum likelihood expectation maximisation. The method was tested on the data measured for four human-tissue samples injected by 22Na and put in the Jagiellonian PET “Big barrel” scanner. Due to an ill-posed problem of fitting time-delay histograms, a multistage optimisation procedure was explored along with inferential analysis of the solution space. Run in parallel for multiple sets of initial guesses, we compared the second-order LevenbergMarquardt algorithm (LMA) and the direct search Nelder-Mead simplex (NMS) method. The LMA proved to be faster and more precise, but the NMS was more stable with a higher convergence rate. The estimated mean o-Ps lifetimes in the 1.9 ns - 2.6 ns range were consistent with the reference results, while other fitting parameters allowed differentiation between the two patients who provided the tissue samples.

2

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.

3

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.

4

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.

5

Uncertainty in structure dynamics resulting from matrix ill-conditioning

Iwaniec J.

Archives of Acoustics

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2005

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Vol. 30, No. 4

473-482

EN

In the paper there are discussed certain issues concerning ill-posed problems that frequently appear in inverse problems, modal analysis, acoustics and other methods making use of matrix algebra. There is presented the mathematical definition, application of the singular value decomposition method to ill-posed problems detecting as well as a new method of improving such problems conditioning by the use of the Tikhonov regularization method. In the paper are presented some results of solution estimation of the Fredholm integral equation of the first kind that is the classical ill-posed problem. Analysis was carried out in the Matlab environment by means of the least squares and Tikhonov regularization methods for both the noiseless and noisy cases.

6

The application of regularisation methods to analysis of structure dynamics

Iwaniec J.

Diagnostyka

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2004

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Vol. 30, T. 1

199--203

EN

In the paper there are discussed issues concerning ill-posed problems. Mathematical definition and a method of detecting ill-posed problems as well as a method of improving such problems conditioning by the use of the Tikhonov regularisation are presented. The results of transfer function noise reduction by the use of the Tikhonov regularisation method are shown.

PL

W pracy omówiono zagadnienia dotyczące zagadnień źle zdefiniowanych. Przedstawiono definicję matematyczną, metodę wykrywania zagadnień źle zdefiniowanych oraz metodę poprawiania uwarunkowania tych zagadnień przy użyciu regularyzacji Tikhonova. Zaprezentowano również możliwość zastosowania metody regularyzacji Tikhonova do redukcji szumów widmowych funkcji przejścia.

7

On determining the temperature dependent heat transfer coefficient

Orzechowski T.

Archives of Thermodynamics

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2003

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Vol. 24, no 2

69-88

EN

The paper deals with analysis of a thermal field measured by means of a thermovision camera. A two-dimensional ill-posed problem of heat transfer was solved using a fin as an example. The quantity that needs to be found is the heat flux leaving the invisible inner surface of the fin. The method of heat functions was used for the calculations. The results of the numerical analysis of the problem are some local values of the transfer coefficients. The convergence of the calculation procedures was chacked by applying the heat balance to the considered area. Then, the optimal degree of the heat polynomials for such a problem was determined and the range of the method convergence was analysed. During the experiments, the heat transfer from the surface of a fin submerged in boiling water was observed. Values of heat transfer coefficients were determined and compared with the calculation results. A reasonable degree of congruence was found to exist between the calculation and experimental results.

8

New criterion of regularisation parameter choice in Tikhonov's method

Kojdecki M. A.

Biuletyn Wojskowej Akademii Technicznej

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2000

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Vol. 49, nr 1

47-126

EN

This work presents methods of solving linear operator equations of the first kind in real Hilbert's spaces and is aimed to present a new criterion of parameter choice in Tikhonov's regularisation mthod. The Tikhonov method is developed here from smoothing functional with zero-order stabilising term. The properties of such regularisation method are briefly analysed. The convergence rates for sequences of regularised solutions of test source problem are estimated. The criteria of a-priori regularisation parameter choice are analysed.On this background the new criterion of a-posteriori parameter choice is presented and studied in detail. The numerical algorithm for practical application of the method is also proposed and analysed. The theoretical considerations are updated with an algorithm of olving Fredholm's integral equations of the first kind and with examples of numerical experiments with such equations.

PL

W niniejszej pracy przedstawiono metody rozwiązywania liniowych równań operatorowych pierwszego rodzaju w przestrzeniach Hilberta; jej celem jest przedstawienie nowego kryterium wyboru parametru w metodzie regularyzacji Tichonowa. Metoda Tichonowa wyprowadzona jest tu od funkcjonału wygładzającego z członem stabilizującym zerowego rzędu. Oszacowano rzędy zbieżności dla ciągów rozwiązań zregulayzowanych testowego zadania o postaci źródłowej. Przeanalizowano kryteria wyboru parametru regularyzacji z góry. Na tym tle przedstawiono i zbadano szczegółowo nowe kryterium wyboru parametru z dołu. Zaproponowano i przeanalizowano numeryczny algorytm dla praktycznego zastosowania metody. Rozważania teoretyczne uzupełniono algorytmem rozwiązywania równań całkowych Fredholma pierwszego rodzaju i przykładami eksperymentów numerycznych z takimi równaniami.