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1

The method of detection and localization of configuration defects in geodetic networks by means of Tikhonov regularization

Kadaj Roman

Reports on Geodesy and Geoinformatics

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2021

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Vol. 112

19--25

EN

In adjusted geodetic networks, cases of local configuration defects (defects in the geometric structure of the network due to missing data or errors in point numbering) can be encountered, which lead to the singularity of the normal equation system in the least-squares procedure. Numbering errors in observation sets cause the computer program to define the network geometry incorrectly. Another cause of a defect may be accidental omission of certain data records, causing local indeterminacy or lowering of local reliability rates in a network. Obviously, the problem of a configuration defect may be easily detectable in networks with a small number of points. However, it becomes a real problem in large networks, where manual checking of all data becomes a very expensive task. The paper presents a new strategy for the detection of configuration defects with the use of the Tikhonov regularization method. The method was implemented in 1992 in the GEONET system (www.geonet.net.pl).

2

Matematyczne sztuczki Tichonowa

Kadaj Roman

Geodeta : magazyn geoinformacyjny

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2021

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nr 9

31--35

3

Przykłady regularyzacji zadań wyznaczania współczynnika dyfuzji wilgoci i przejmowania masy w drewnie

Perkowski Z., Jeż K., Grygorowicz M.

Fizyka Budowli w Teorii i Praktyce

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2017

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T. 9, nr 1

33--38

PL

W pracy rozważa się sposób wyznaczania współczynnika dyfuzji wilgoci i przejmowania masy drewna na podstawie pomiarów masy całkowitej próbek rejestrowanych w procesie niestacjonarnej sorpcji, gdzie oszacowania parametrów materiałowych dokonuje się na drodze minimalizacji sumy błędów kwadratowych pomiędzy wynikami pomiaru i obliczeń modelowych. W [9,12] pokazano, że szczególnie w przypadku wyrażenia współczynnika dyfuzji, jako nieliniowej funkcji zależnej od wilgotności materiału lub wystąpienia względnie dużych wartości współczynnika przejmowania masy w procesie dyfuzji zachodzącym wzdłuż włókien drewna, w zagadnieniach tego typu jednoznaczne określenie globalnego minimum funkcji celu może stać się bardzo utrudnione. W niniejszej pracy, jako sposób, który pozwala na skuteczną regularyzację omawianego, współczynnikowego zadania odwrotnego, wykorzystano metody Tichonowa i L-curve. Rozważania zilustrowano przykładami pomiarów prowadzonych na próbkach z drewna sosnowego, dębowego i lipowego, w przypadku, których, dzięki zastosowanemu podejściu, otrzymano znaczną poprawę jakości wyników.

EN

In the paper, a method for determining the moisture diffusion coefficient and mass surface emission coefficient of wood is considered which employs measurements of total mass of samples recorded in a non-stationary sorption process where the estimation of the material parameters is realized by minimizing the sum of square errors between the measurement results and adopted calculation model. It was shown in [9,12] that in problems of this type, especially in the case of the expression of diffusion coefficient as a nonlinear function dependent on the moisture content or of a relatively large mass surface emission coefficient when diffusion process taking place along the wood fibres, unambiguous definition of the global minimum of the objective function can be very difficult. In this paper, as a way that allows for the effective regularization of this parametric inverse problem, Tikhonov method and L-curve method are used. The considerations are illustrated with examples of measurements conducted on the samples of pine, oak, and linden wood where, thanks to the used approach, one significantly improved quality of results.

4

Optimization of soft X-ray tomography on the COMPASS tokamak

Imríšek M., Mlynář J., Löffelmann V., Weinzettl V., Odstrčil T., Odstrčil M., Tomeš M.

Nukleonika

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2016

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Vol. 61, No. 4

403--408

EN

The COMPASS tokamak is equipped with the soft X-ray (SXR) diagnostic system based on silicon photodiode arrays shielded by a thin beryllium foil. The diagnostic is composed of two pinhole cameras having 35 channels each and one vertical pinhole camera with 20 channels, which was installed recently to improve tomographic inversions. Lines of sight of the SXR detectors cover almost complete poloidal cross section of the COMPASS vessel with a spatial resolution of 1–2 cm and temporal resolution of about 3 s. Local emissivity is reconstructed via Tikhonov regularization constrained by minimum Fisher information that provides reliable and robust solution despite limited number of projections and ill-conditionality of this task. Improved border conditions and numerical differentiation matrices suppressing artifacts in reconstructed radiation were implemented in the code. Furthermore, a fast algorithm eliminating iterative processes was developed, and it is foreseen to be tested in real-time plasma control.

5

3D Gravity Inversion using Tikhonov Regularization

Toushmalani R., Saibi H.

Acta Geophysica

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2015

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Vol. 63, no. 4

1044--1065

EN

Subsalt exploration for oil and gas is attractive in regions where 3D seismic depth-migration to recover the geometry of a salt base is difficult. Additional information to reduce the ambiguity in seismic images would be beneficial. Gravity data often serve these purposes in the petroleum industry. In this paper, the authors present an algorithm for a gravity inversion based on Tikhonov regularization and an automatically regularized solution process. They examined the 3D Euler deconvolution to extract the best anomaly source depth as a priori information to invert the gravity data and provided a synthetic example. Finally, they applied the gravity inversion to recently obtained gravity data from the Bandar Charak (Hormozgan, Iran) to identify its subsurface density structure. Their model showed the 3D shape of salt dome in this region.

6

Tikhonov regularization and constrained quadratic programming for magnetic coil design problems

Garda B., Galias Z.

International Journal of Applied Mathematics and Computer Science

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2014

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Vol. 24, no. 2

249--257

EN

In this work, the problem of coil design is studied. It is assumed that the structure of the coil is known (i.e., the positions of simple circular coils are fixed) and the problem is to find current distribution to obtain the required magnetic field in a given region. The unconstrained version of the problem (arbitrary currents are allowed) can be formulated as a Least-SQuares (LSQ) problem. However, the results obtained by solving the LSQ problem are usually useless from the application point of view. Moreover, for higher dimensions the problem is ill-conditioned. To overcome these difficulties, a regularization term is sometimes added to the cost function, in order to make the solution smoother. The regularization technique, however, produces suboptimal solutions. In this work, we propose to solve the problem under study using the constrained Quadratic Programming (QP) method. The methods are compared in terms of the quality of the magnetic field obtained, and the power of the designed coil. Several 1D and 2D examples are considered. It is shown that for the same value of the maximum current the QP method provides solutions with a higher quality magnetic field than the regularization method.

7

Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels

Castro L. P., Saitoh S., Tuan N. M.

Opuscula Mathematica

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2012

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Vol. 32, no. 4

633-646

EN

This paper introduces a general concept of convolutions by means of the theory of reproducing kernels which turns out to be useful for several concrete examples and applications. Consequent properties are exposed (including, in particular, associated norm inequalities).

8

Calculation of the cooling condition in the phase change problem

Słota D.

Journal of Achievements in Materials and Manufacturing Engineering

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2009

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Vol. 33, nr 1

70-77

EN

Purpose: The purpose of the paper is to present the method of calculation of the cooling condition in the phase change problem. The considered problem consists of the reconstruction of a function describing the heat transfer coefficient, when the temperature values in selected points of the solid phase are known. Design/methodology/approach: In numerical calculations, the Tikhonov regularization, the genetic algorithm and the alternating phase truncation method were used. Findings: The featured examples of calculations show a very good approximation of the exact solution and stability of the procedure. Practical implications: The paper presents an example of selection of the heat transfer coefficient given in the form of a continuous function. This method can be easily adopted also for the determination of other parameters of the problem discussed here. Originality/value: The calculations made, only part of which has been presented in this paper, show stability of the method proposed, both in terms of the input data errors and the number of control points, thus corroborating usefulness of the presented approach.

9

Modelling of the optimum cooling condition in two-dimensional solidification processes

Słota D.

Archives of Computational Materials Science and Surface Engineering

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2009

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Vol. 1, nr 1

45-52

EN

Purpose: This paper presents the method of the calculation of the cooling condition in the two-dimensional solidification processes. Design/methodology/approach: The considered problem consists in the reconstruction of the function that describes the heat transfer coefficient on the boundary, when the temperature measurements in selected points of the solid phase are well-known. In calculations the alternating phase truncation method, the genetic algorithm and the Tikhonov regularization were used. Findings: The calculations show a very good approximation of the exact solution and the stability of the procedure. Research limitations/implications: On the bases of the results that every start-up of the genetic algorithm leads to similar results, which are reflected by very low values of the standard deviation. Originality/value: The calculations point to the stability of the proposed method in view of the input data errors, number of control points, substantiating the usability of such approach.

10

Techniki tomografii gradientowej w analizie procesu chłodzenia elementów elektronicznych

Janiczek R., Pisarek J., Wojciechowski A.

Elektronika : konstrukcje, technologie, zastosowania

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2008

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Vol. 49, nr 2

29-34

PL

Większość stosowanych obecnie metod eksperymentalnej analizy procesów chłodzenia elementów elektronicznych bazuje na pomiarze rozkładów temperatur na powierzchni elementu i punktowych, kontaktowych pomiarach przepływającego gazu. W pracy zaproponowano bezkontaktowy, optyczny pomiar rozkładu temperatur medium chłodzącego w całej jego objętości. Zastosowana metoda optyczna pozwala analizować procesy wymiany ciepła, zarówno dla przepływów laminarnych, jak i turbulentnych. Metoda ta jest szczególnie przydatna do analizy zjawisk, towarzyszących chłodzeniu obiektów o niewielkich wymiarach geometrycznych. Przetestowane zostało szereg konfiguracji układu pomiarowego, zarówno w wariantach klasycznych, jak i z wykorzystaniem nowych rozwiązań. Dla przeprowadzenia weryfikacji metody fotografii smugowej zbudowano specjalne stanowisko pomiarowe.

EN

In order to obtain the information about the run of cooling process of electronic elements the method of Schlieren photography was used. This method, as optical method, permits for analysis of heat exchange processes, both for laminar and turbulent flows. This method is especially useful to analysis of effects following in cooling of no large objects. Many configurations of measuring systems were tested, both in classical variants and with using new problem resolves. For making the verification of Schlieren photography method special measuring position was constructed. New methodology of measure results elaboration was proposed, using elements of analysis theory of fringe images. The work contains also the introduction to the theory of the optical gradient tomography in the original seizure.

11

Regularization parameter selection in discrete ill-posed problems—The use of the U-curve

Krawczyk-Stańdo D., Rudnicki M.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 2

157-164

EN

To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter \alfa we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter \alfa, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.