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  1. 4 International Journal of Applied Mathematics and Computer Science
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  1. 4 Landowski M.
  2. 4 Pawlik P.
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  4. 3 Marciniak A.
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1
Content available A realistic tolerant solution of a system of interval linear equations with the use of multidimensional interval arithmetic
Piegat Andrzej, Pluciński Marcin
International Journal of Applied Mathematics and Computer Science
|
2023
|
Vol. 33, no. 2
229--247
EN
The paper presents a method of determining the robustness of solutions of systems of interval linear equations (ILEs). The method can be applied also for the ILE systems for which it has been impossible to find solutions so far or for which solutions in the form of improper intervals have been obtained (which cannot be implemented in practice). The research conducted by the authors has shown that for many problems it is impossible to arrive at ideal solutions that would be fully robust to data uncertainty. However, partially robust solutions can be obtained, and those with the highest robustness can be selected and put into practice. The paper shows that the degree of robustness to the uncertainty of the entire system can be calculated on the basis of the degrees of robustness of individual equations, which greatly simplifies calculations. The presented method is illustrated with a series of examples (also benchmark ones) that facilitate its understanding. It is an extension of the authors’ previously published method for first-order ILEs.
2
Content available A decomposition approach to type 2 interval arithmetic
Piegat Andrzej, Dobryakova Larisa
International Journal of Applied Mathematics and Computer Science
|
2020
|
Vol. 30, no. 1
185--201
EN
The classic interval has precise borders A = [a, ā] . Therefore, it can be called a type 1 interval. Because of great practical importance of such interval data, several versions of type 1 interval arithmetic have been created. However, sometimes precise borders a and ā of intervals cannot be determined in practice. If the borders are uncertain, then we have to do with type 2 intervals. A type 2 interval can be denoted as AT2 = [aL, aR], [āL, āR]. The paper presents multidimensional decomposition RDM type 2 interval arithmetic (D-RDM-T2-I arithmetic), where RDM means relative-distance measure. The decomposition approach considerably simplifies calculations and is transparent for users. Apart from this arithmetic, examples of its applications are also presented. To the authors’ best knowledge, no papers on this arithmetic exist. D-RDM-T2-I arithmetic is necessary to create type 2 fuzzy arithmetic based on horizontal µ-cuts, which the authors aim to do.
3
Content available Application of the interval Lattice Boltzmann method for a numerical modeling of 2D thin metal films irradiated by ultrashort laser pulses
Piasecka-Belkhayat Alicja, Korczak Anna
Journal of Applied Mathematics and Computational Mechanics
|
2019
|
Vol. 18, nr 4
63--71
EN
In the paper, the two-dimensional numerical modelling of heat transfer in thin metal films irradiated by ultrashort laser pulses using the D2Q9 scheme is considered. In the mathematical description, the relaxation times and the boundary conditions for phonons and electrons are given as interval numbers. The problem has been formulated using the interval coupled lattice Boltzmann equations for electrons and phonons. The solution has been obtained by means of the interval lattice Boltzmann method using the rules of directed interval arithmetic. Examples of numerical computations are presented in the final part of the paper.
4
Content available remote Interval Runge-Kutta Methods with Variable Step Sizes
Marciniak A., Szyszka B.
Computational Methods in Science and Technology
|
2019
|
Vol. 25, No. 1
17--29
EN
In a number of our previous papers we have presented interval versions of Runge-Kutta methods (explicit and implicit) in which the step size was constant. Such an approach has required to choose manually the step size in order to ensure an interval enclosure to the solution with the smallest width. In this paper we propose an algorithm for choosing automatically the step size which guarantees the best (i.e., the tiniest) interval enclosure. This step size is determined with machine accuracy.
5
Content available Modelowanie czasu inicjacji korozji zbrojenia z zastosowaniem arytmetyki przedziałowej
Krykowski T.
Roczniki Inżynierii Budowlanej
|
2017
|
z. 17
43--52
EN
In this paper, the application of interval arithmetic in the affine formulation to the determination of time to initialization of corrosion cracking of reinforced concrete structures was formulated. The calculations ware made with the use of the incremental FEM formulation (backward Euler scheme) and the library of numerical procedures for the interval-affine computations INTALB.
6
Content available remote RDM interval method for solving quadratic interval equation
Landowski M.
Przegląd Elektrotechniczny
|
2017
|
R. 93, nr 1
65--68
EN
The main task of uncertainty theory is to find the solution with uncertain variable. The ways of uncertainty description are probability density distribution, possibility distribution or interval. To solve the problem with uncertainty variable the calculation on interval is needed. The article presents the usage of RDM interval arithmetic for solving quadratic interval equation. The results obtained from examples are compared with Moore’s standard interval arithmetic solutions.
PL
Głównym zadaniem teorii niepewnos´ci jest znalezienie rozwiazania ze zmienna˛ niepewna˛. Niepewnos´c´ moz˙na zapisac´ w postaci rozkładu ge˛stos´ci prawdopodobien´stwa, rozkładu moz˙liwos´ci lub przedziału. Do rozwia˛zania zadania ze zmienna˛ niepewna˛ potrzebne sa˛ obliczenia na przedziałach. Artykuł przedstawia wykorzystanie arytmetyki interwałowej RDM do rozwia˛zania interwałowych równan´ kwadratowych. Wyniki otrzymane z przykładów porównano z rozwia˛zaniami standardowej arytmetyki interwałowej Moore’a.
7
Content available Is an interval the right result of arithmetic operations on intervals?
Piegat A., Landowski M.
International Journal of Applied Mathematics and Computer Science
|
2017
|
Vol. 27, no. 3
575--590
EN
For many scientists interval arithmetic (IA, I arithmetic) seems to be easy and simple. However, this is not true. Interval arithmetic is complicated. This is confirmed by the fact that, for years, new, alternative versions of this arithmetic have been created and published. These new versions tried to remove shortcomings and weaknesses of previously proposed options of the arithmetic, which decreased the prestige not only of interval arithmetic itself, but also of fuzzy arithmetic, which, to a great extent, is based on it. In our opinion, the main reason for the observed shortcomings of the present IA is the assumption that the direct result of arithmetic operations on intervals is also an interval. However, the interval is not a direct result but only a simplified representative (indicator) of the result. This hypothesis seems surprising, but investigations prove that it is true. The paper shows what conditions should be satisfied by the result of interval arithmetic operations to call it a “result”, how great its dimensionality is, how to perform arithmetic operations and solve equations. Examples illustrate the proposed method of interval computations.
8
Content available Application of Interval Arithmetic to Production Planning in a Foundry
Duda J., Stawowy A.
Archives of Foundry Engineering
|
2017
|
Vol. 17, iss. 1
41--44
EN
A novel approach for treating the uncertainty about the real levels of finished products during production planning and scheduling process is presented in the paper. Interval arithmetic is used to describe uncertainty concerning the production that was planned to cover potential defective products, but meets customer’s quality requirement and can be delivered as fully valuable products. Interval lot sizing and scheduling model to solve this problem is proposed, then a dedicated version of genetic algorithm that is able to deal with interval arithmetic is used to solve the test problems taken from a real world example described in the literature. The achieved results are compared with a standard approach in which no uncertainty about real production of valuable castings is considered. It has been shown that interval arithmetic can be a valuable method for modeling uncertainty, and proposed approach can provide more accurate information to the planners allowing them to take more tailored decisions.
9
Content available Sensitivity Analysis of the Estimation of the Single-Number Sound Absorption Evaluation Index DLα
Batko W., Pawlik P., Wszołek G.
Archives of Acoustics
|
2017
|
Vol. 42, No. 4
689--696
EN
Acoustic barriers are assigned to the respective categories of sound absorbing properties on the basis of a single-number sound absorption evaluation index. Categories of absorbing properties play a significant role in selecting the barrier type for the given localisation. The estimation of the single-number sound absorption evaluation index is performed, among others, by means of measuring the sound absorption coefficient of the analysed acoustic barrier sample in the reverberation chamber. The sensitivity analysis of the determination of the single-number sound absorption evaluation index was performed in this work. The estimation of the input parameters uncertainty contribution to the expanded uncertainty of the sound absorption evaluation index, was done first. The Monte Carlo method and the reduction interval arithmetic were used for this aim. The relative sensitivity coefficients were determined by means of the author’s method based on the interval arithmetic. These coefficients contain information concerning the quantitative influence of the given input value on the final result.
10
Content available remote Solving the Generalized Poisson Equation in Proper and Directed Interval Arithmetic
Hoffmann T., Marciniak A.
Computational Methods in Science and Technology
|
2016
|
Vol. 22, No. 4
225--232
EN
In the paper some interval methods for solving the generalized Poisson equation (GPE) are presented. The main aim of this work is focused on providing such algorithms for solving this type of equation that are able to store information about potentially made numerical errors inside the results. In order to cope with these assumptions the floating-point interval arithmetic is used. We proposed to use interval versions of the central-difference method for two types of interval arithmetic: proper and directed. In the experimental part of this paper both arithmetics for three examples of GPE are compared
11
Content available On an application of an interval backward finite difference method for solving the one-dimensional heat conduction problem
Jankowska M., Marciniak A., Hoffmann T.
Control and Cybernetics
|
2015
|
Vol. 44, no. 4
463--480
EN
The paper concerns the interval method for solving the one-dimensional heat conduction problem. It is based on the conventional backward finite difference scheme with the appropriate local truncation error terms that are also taken into account. For the theoretical formulation of the interval approach we can show that the exact solution is included in the interval one. In practice, there are problems, for which we cannot determine the endpoints of the error term intervals exactly. Nevertheless, if we use the appropriate approximation, related to the endpoints considered, then the numerical experiments confirm that the interval solution includes the exact one.
12
Content available remote Uogólnienie metody TOPSIS w warunkach niepewnosci rozmytej
Tikhonenko-Kedziak A.
Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie
|
2014
|
Nr 1(7)
13--38
PL
Technika obliczania odległości od rozwiązania idealnego (TOPSIS) jest jedną z najbardziej znanych klasycznych metod wielokryterialnego podejmowania decyzji (MCDM). W klasycznej metodzie TOPSIS wartości i wagi kryteriów są zwykłymi liczbami. Czasami jednak rozwiązanie zagadnienia dokładnego wyznaczenia wartości kryteriów jest trudne, dlatego w konsekwencji ich wartości są przedstawione w postaci liczb rozmytych. Istnieje kilka publikacji dotyczących zastosowania metody TOPSIS w ramach niepewności rozmytej, lecz autorzy zazwyczaj wprowadzają rozmaite ograniczenia oraz uproszczenia sformułowanego problemu, które mogą prowadzić do otrzymania niepoprawnych wyników. W niniejszym opracowaniu przedstawiono nowe podejście oparte na matematyce przedziałowej.
EN
The TOPSIS method is a technique for establishing order preference by similarity to the ideal solution and was primarily developed for dealing with real-valued data. This technique is currently one of most popular methods for Multiple Criteria Decision Making (MCDM). In many cases, it is hard to present precisely exact ratings of alternatives with respect to local criteria and as a result these ratings are seen as fuzzy values. A number of papers have been devoted to fuzzy extensions of the TOPSIS method in the literature, but in most of them, a defuzzification of elements of the fuzzy decision matrix is used, that leads inevitably to a loss of important information and may even produce the wrong results. In this paper a new direct approach to the fuzzy extension of the TOPSIS based on interval arithmetic had proposed.
13
Content available Wpływ liczby punktów pomiarowych na niepewność wyznaczania izolacyjności akustycznej przegród budowlanych
Pawlik P., Przysucha B.
Pomiary Automatyka Kontrola
|
2014
|
R. 60, nr 12
1124--1126
PL
W artykule przedstawiono problem doboru liczby punktów pomiarowych do prawidłowego wyznaczenia izolacyjności akustycznej przegród budowlanych. Obliczenia pozwoliły na wyznaczenie zależności pomiędzy niepewnością wyznaczania izolacyjności akustycznej przegród budowlanych a liczbą wykonanych pomiarów.
EN
The article presents the problem of adjustment of the number of measurement points to correct designation of sound insulation of building partitions. The calculations allowed to determine the relationship between the uncertainty of the determination of sound insulation of building partitions and the number of measurements (Fig. 2, 3, 4).
14
RDM interval arithmetic for solving economic problem with uncertain variables
Landowski M.
Logistyka
|
2014
|
nr 6
EN
Uncertain variables are often used for solving realistic problems. To find the solution of a realistic problem the model with uncertain variable has to be built. Based on the model with uncertain variables operations on intervals are necessary. The article presents the multidimensional RDM interval arithmetic and its application to solving an economic problem. Obtained solutions are compared with results from the one dimensional Moore interval arithmetic and global optimization.
15
Content available Nowa metoda oceny niepewności identyfikacji źródeł hałasu
Batko W., Pawlik P.
Pomiary Automatyka Kontrola
|
2013
|
R. 59, nr 6
529--531
PL
W artykule przedstawiono nową metodę oceny niepewności estymacji poziomu dźwięku pochodzącego od identyfikowanych źródeł oraz tła akustycznego. Wykorzystano metodę eliminacji w celu zbadania wpływu poszczególnych źródeł na sumaryczny poziom hałasu. Operacje matematyczne opisujące wykorzystaną metodykę obliczeniową, przeprowadzono w formalizmie redukcyjnej arytmetyki przedziałowej, w celu oceny wpływu niepewności pomiarowej na wyniki obliczeń. W artykule przedstawiono ogólny schemat oceny niepewności, dający możliwość uwzględnienia informacji probabilistycznej związanej z wynikami pomiaru. Zaproponowane rozwiązanie oparto na pomiarach zrealizowanych w warunkach laboratoryjnych.
EN
A new method for uncertainty assessment of the sound level originated from identified noise sources and their acoustic background is presented in the paper. The elimination method [1] was applied in order to investigate influence of individual sources on the cumulative noise level. Mathematical operations describing the used computational method were performed in the reduction interval arithmetic formalism [6] to assess the influence of the measuring uncertainty on the calculation results. The measurement values of the total noise level and the noise levels characteristic for disconnections of successive noise sources were presented in the interval numbers. These numbers contain the measurement values and the uncertainty (Tab. 2). The authors determined the ranges of variation for estimates of the noise levels by individual sources and the background noise using the reductive interval arithmetic (Tab. 3). The general uncertainty estimation scheme presented in the paper provides the possibility of taking into account probabilistic information related to the obtained results. The proposed solution was based on measurements realised under laboratory conditions.
16
Content available Interval arithmetic-based fuzzy discrete-time crane control scheme design
Smoczek J.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2013
|
Vol. 61, nr 4
863--870
EN
In many manufacturing segments, container terminals and shipping yards the automation of material handling systems is an important element of enhancing productivity, safety and efficiency. The fast, precise and safe transfer of goods in crane operations requires a control application solving the problems, including non-collision trajectory planning and limitation of payload oscillations. The paper presents the interval arithmetic-based method of designing a discrete-time closed-loop anti-sway crane control system based on the fuzzy interpolation of linear controller parameters. The interval analysis of a closed-loop control system characteristic polynomial coefficients deviation from their nominal values is proposed to define a minimum number of fuzzy sets on the scheduling variables universe of discourse and to determine the distribution of triangular-shaped membership functions parameters, which satisfy the acceptable range of performances deterioration in the presence of the system’s parameters variation. The effectiveness of this method was proved in experiments conducted using the PAC system on the laboratory scaled overhead crane.
17
Is the conventional interval arithmetic correct?
Piegat A., Landowski M.
Journal of Theoretical and Applied Computer Science
|
2012
|
Vol. 6, nr 2
27-44
EN
Interval arithmetic as part of interval mathematics and Granular Computing is unusually important for development of science and engineering in connection with necessity of taking into account uncertainty and approximativeness of data occurring in almost all calculations. Interval arithmetic also conditions development of Artificial Intelligence and especially of automatic thinking, Computing with Words, grey systems, fuzzy arithmetic and probabilistic arithmetic. However, the mostly used conventional Moore-arithmetic has evident weak-points. These weak-points are well known, but nonetheless it is further on frequently used. The paper presents basic operations of RDM-arithmetic that does not possess faults of Moore-arithmetic. The RDM-arithmetic is based on multi-dimensional approach, the Moore-arithmetic on one-dimensional approach to interval calculations. The paper also presents a testing method, which allows for clear checking whether results of any interval arithmetic are correct or not. The paper contains many examples and illustrations for better understanding of the RDM-arithmetic. In the paper, because of volume limitations only operations of addition and subtraction are discussed. Operations of multiplication and division of intervals will be presented in next publication. Author of the RDM-arithmetic concept is Andrzej Piegat.
18
Content available Computer Assisted Proofs in Dissipative Partial Differential Equations
Cyranka J.
Challenges of Modern Technology
|
2012
|
Vol. 3, no. 3
3--6
EN
The goal of this paper is to present a brief survey of our research that has focused on studying the dynamics of dissipative partial differential equations by performing computer as sisted proofs. We provide a description of the main ideas behind the computer assisted proofs that we have performed, along with related topics. The emphasis is given to the case of the vis cous Burgers equation with constant forcing, for which the existence of globally attracting fixed points has been established. To achieve this goal, we used a combination of analytical results with computer assistance.
19
Content available New Approach to the Uncertainty Assessment of Acoustic Effects in the Environment
Batko W., Pawlik P.
Archives of Acoustics
|
2012
|
Vol. 37, No. 1
57-61
EN
The acoustic climate assessment needed for the selection of solutions (technical, legal and organisa- tional), which will help to minimise the acoustic hazards in the analysed areas, is realised on the basis of acoustic maps. The reference computational algorithms, assigned to them, require very thorough prepa- ration of input data for the considered noise source model representing – in the best possible way – the acoustic climate. These input data are burdened with certain uncertainties in this class of computational tasks. The uncertainties are related to the problem of selecting proper argument values (from the inter- val of their possible variability) for the modelled processes. This situation has a direct influence on the uncertainty of acoustic maps. The idea of applying the interval arithmetic for the assessment of acoustic models uncertainty is formulated in this paper. The computational formalism assigned to the interval arithmetic was discussed. The rules of interval estimations for the model solutions determining the sound level distribution around the analysed noise source – caused by possible errors in the input data – were presented. The application of this formalism was illustrated in uncertainty assessments of modelling acoustic influences of the railway noise linear source on the environment.
20
Content available Evaluation of the Accuracy of the Solution to the Heat Conduction Problem with the Interval Method of Crank-Nicolson Type
Jankowska M. A., Sypniewska-Kamińska G., Kamiński H.
Acta Mechanica et Automatica
|
2012
|
Vol. 6, no. 1
36-43
EN
The paper deals with the interval method of Crank-Nicolson type used for some initial-boundary value problem for the onedimensional heat conduction equation. The numerical experiments are directed at a short presentation of advantages of the interval solutions obtained in the floating-point interval arithmetic over the approximate ones. It is also shown how we can deal with errors that occur during computations in terms of interval analysis and interval arithmetic.
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