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1
Content available A new approach to hull consistency
Kolev L.
Archives of Electrical Engineering
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2016
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Vol. 65, nr 2
305--314
EN
Hull consistency is a known technique to improve the efficiency of iterative interval methods for solving nonlinear systems describing steady-states in various circuits. Presently, hull consistency is checked in a scalar manner, i.e. successively for each equation of the nonlinear system with respect to a single variable. In the present poster, a new more general approach to implementing hull consistency is suggested which consists in treating simultaneously several equations with respect to the same number of variables.
2
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
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2015
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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.
3
Content available remote Interval versions of central-difference method for solving the poisson equation in proper and directed interval arithmetic
Hoffmann T., Marciniak A., Szyszka B.
Foundations of Computing and Decision Sciences
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2013
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Vol. 38, No. 3
193--206
EN
To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.
4
Content available remote Interval multistep predictor-corrector methods of Adams type for solving the initial value problem for ordinary differential equations
Jankowska M. A.
Vibrations in Physical Systems
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2010
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Vol. 24
185--193
EN
In the paper we propose the interval multistep predictor-corrector methods of Adams type for solving the initial value problem (IVP) for ordinary differential equations (ODEs). These methods are based on the explicit interval methods of Adams-Bashforth type and the implicit interval methods of Adams-Moulton type. The interval methods considered belong to a class of algorithms that allow to obtain the guaranteed result, i.e. the interval solution that contain the exact solution of the problem.
PL
W pracy zaproponowane zostały przedziałowe metody wielokrokowe predyktor-korektor typu Adamsa rozwiązywania zagadnienia początkowego dla równań różniczkowych zwyczajnych. Metody te oparte są na jawnych przedziałowych metodach typu Adamsa-Bashfortha oraz niejawnych przedziałowych metodach typu Adamsa-Moultona. Metody przedziałowe należą do klasy algorytmów, które pozwalają otrzymać rozwiązanie danego problemu w postaci przedziału-rozwiązania, który zawiera w sobie rozwiązanie dokładne.
5
Content available remote Numerical modelling of solidification process using interval finite difference method
Piasecka Belkhayat A.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2010
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Vol. 9, nr 1
155-163
EN
The numerical modelling of steel cast solidification process in sand mould is considered. The problem analyzed is described by the system of partial differential equations supplemented by adequate boundary and initial conditions. The latent heat appearing in the model of a casting sub-domain is treated as directed interval value. The problem formulated has been solved by means of interval finite difference method with the approach of directed interval arithmetic. In the final part of the paper, results of numerical computations are shown.
6
Comparison of Interval C/C++ Libraries in Global Optimization
Dąbrowski R., Kubica B.J.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2009
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z. 169
51-56
EN
This article contains results of examination and comparison of some most popular interval libraries. Comparative analysis was based on investigating the efficiency of computations of simple functions and efficiency of the Interval-Branch-And-Bound global optimization method.
7
Content available remote Multistep Interval Methods of Nyström and Milne-Simpson Types
Marciniak A.
Computational Methods in Science and Technology
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2007
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Vol. 13, nr 1
23-39
EN
The paper is dealt with two kinds of multistep intervals methods which can be used to solve the initial value problem in the form of intervals containing all possible numerical errors. The interval methods of Nyström type are explicit, while the methods of Milne- Simpson are implicit. It appears that we can get two families of interval methods of the second kind. For both kinds of interval methods numerical examples are presented and compared with other interval multistep method considered in previous papers of the author.
8
Content available remote On Two Families of Implicit Interval Methods of Adams-Moulton Type
Jankowska M., Marciniak A.
Computational Methods in Science and Technology
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2006
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Vol. 12, nr 2
109-113
EN
In our previous paper [1] we have presented implicit interval methods of Adams-Moulton type. It appears that two families of these types of methods exist. We compare both families of methods and present a numerical example.
9
Content available remote Application of the interval methods for solving linear thermal diffusion problems
Belkhayat-Piasecka A.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2005
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Vol. 4, nr 1
204-210
EN
The example of two-dimensional non-steady state heat flow using the interval arithmetic and the 1st scheme of the boundary element method is presented. This example is a typical linear task, where the boundary of the homogeneous domain is an interval. In the final part of the paper, results of numerical computations are shown with the different boundary conditions.
10
Unconstrained and constrained global optimization using interval analysis
Kubica B., Niewiadomska-Szynkiewicz E.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2002
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z. 139
185--199
EN
This chapter presents techniques adressed to continuous, unconstrained and constrained optimization problems. Global optimization is defined as the problem of finding points on a bounded subset of X of Rn in which some real valued function f reaches its optimal (minimum or maximum) value. The algorithms considered are based on the branch-and-bound principles where the problem domain is partitioned iteratively. They apply interval arithmetic tools. The main objective is to discuss the advantages and disadvantages of existing algorithms and to provide some modifications for increasing their efficiency. The numerical results for several test problems are presented in the final part of the chapter.
11
Content available remote On explicit interval methods of Adams - Bashforth type
Jankowska M., Marciniak A.
Computational Methods in Science and Technology
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2002
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Vol. 8, nr 2
46--57
EN
In our previous paper [1] we have considered implicit interval multistep methods of Adams-Moulton type for solving the initial value problem. On the basis of these methods and the explicit ones introduced by Sokin [2] we wanted to construct predictor-corrector (explicit-implicit) interval methods. However, it turned out that the formulas given by Šokin are incorrect even in the simplest case. Therefore, in this paper we direct our attention to the explicit interval methods of Adams-Bashforth type and modify the formulas of Šokin. For the modified explicit interval methods it is proved, like f o r the implicit interval methods considered in [1], that the exact solution of the problem belongs to interval-solutions obtained by these methods. Moreover, it is shown an estimation of the widths of such interval-solutions.
12
Content available remote Przedziałowe metody całkowania równań dynamiki konstrukcji
Pownuk A.
Zeszyty Naukowe Politechniki Rzeszowskiej. Mechanika
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1999
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z. 52 [174]
77-84
EN
In this paper the interval methods for validated solution of ordinary differential equation of dynamics of structures are presented. The interval Euler's method for solving system of ordinary differential equations with automatic determination of guaranteed estimation and high order Taylor series method are described. Using these methods two problems of free vibration are solved.
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