Solving interval systems of equations obtained during the numerical solution of boundary value problems

Zieniuk, Eugeniusz; Kapturczak, Marta; Kużelewski, Andrzej

doi:10.1007/s40314-014-0209-9

Cited by 12 publications

(4 citation statements)

References 17 publications

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“…Presented way of uncertainty modeling, using directed interval arithmetic, significantly reduce overestimation. However, during the research studies of applications of such numbers for modeling and solving uncertainly defined boundary value problems many problems appeared (Zieniuk et al, 2016. As a result of the research on the problems, the modification of directed interval arithmetic has been proposed.…”

Section: Interval Numbersmentioning

confidence: 99%

Modification of interval arithmetic in modeling uncertainty of boundary conditions in boundary value problems

2019

El Sawah, S. (Ed.) MODSIM2019, 23rd International Congress on Modelling and Simulation.

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Nowadays, the finite element method (FEM) is the most known and the most commonly used method for solving boundary value problems defined exactly (without uncertainty). This method is based on the division of the domain into finite elements. These elements are connected to each other and produces a mesh. The second method, more recent, but also often used, is the boundary element method (BEM). This method is based on the division of the boundary only on the so-called boundary elements. This solution significantly reduces the amount of data required to solve the problem. Therefore, the amount of computer resources, needed to obtain a solution, is also significantly reduced. However, both FEM and BEM require a large number of points to define the shape of the boundary. It was the main reason to look for new methods. This is how the parametric integral equations system (PIES) appeared. In this paper, the application of interval arithmetic for modeling uncertainly defined boundary conditions (in solving boundary value problems) is presented. In the literature, for solving such problems, different modifications of mentioned FEM and BEM methods appeared. Many different ways of modeling uncertainty have been used. However, the most important problem is to reduce the number of calculations. This is the main reason to choose interval arithmetic. Direct application of classical or directed interval arithmetic, into interval modifications of FEM and BEM methods, leads to significant overestimation. Therefore, it was decided to propose interval modification of the parametric integral equations system (PIES) method. Recently, the PIES method has been thoroughly tested on examples of boundary problems, where input data were defined without uncertainty. During tests, the number of input data (necessary to define the problem) was always smaller (comparing with well-known methods). This also allows reducing the number of equations in the obtained system of equations. This gives a significant advantage (over other methods) in modeling uncertainty of input data. The application of the interval PIES method resulted in a significant reduction of overestimation. However, obtained solutions (by direct application of interval arithmetic known from literature) are not exactly in line with expectations. Therefore, the modification of directed interval arithmetic has been proposed. It has been applied for calculations in the interval PIES method. The proposed method has been tested on examples of boundary value problems (modeled by Laplace's equation) with uncertainly defined boundary conditions. The uncertainty of boundary conditions has been defined as interval constant boundary conditions as well as interval linear function. Obtained solutions have been compared with the solutions of exactly defined (without uncertainty) PIES.

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Section: Interval Numbersmentioning

confidence: 99%

Modification of interval arithmetic in modeling uncertainty of boundary conditions in boundary value problems

2019

El Sawah, S. (Ed.) MODSIM2019, 23rd International Congress on Modelling and Simulation.

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“…Most approaches in the literature deal with the problem of solving Linear Interval Systems of Equations (LISE), due to the complexity that arises when using interval arithmetic [15], [13]. When using interval arithmetic to solve LISE, the solution interval is inherently an overestimation, thus many usually iterative methods have been proposed to approximate the IH solution, such as the Gauss Elimination method, LU decomposition method and the iterative Jacobi method [16].…”

Section: B Measurement and Parameter Uncertaintymentioning

confidence: 99%

Interval State Estimation of Hydraulics in Water Distribution Networks

Vrachimis

Timotheou

Ηλιάδης

et al. 2018

2018 European Control Conference (ECC)

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Water distribution networks are critical infrastructure systems, which are required to reliably deliver water to consumers. Hydraulic state estimation of flows and pressures is key for the efficient and dependable operation of water systems. However, in practice, hydraulic state estimation is a challenging task due to the scarcity of sensors compared to system states and the presence of several modeling uncertainties. It is a common practice for historical data to be used in the place of missing measurements, however the lack of a statistical characterization for the error of these measurements results in having no knowledge of the state estimate error magnitude. This paper presents a methodology for generating hydraulic state bounding estimates by modeling measurement and parametric uncertainties as intervals. The resulting nonlinear interval hydraulic equations are re-formulated using bounding linearization, a technique that restricts the nonlinearities within a convex set, and solved using linear optimization. An iterative procedure improves the bounding linearization, converging to the tightest possible bounds. Simulation results demonstrate that the proposed methodology produces tight state bounds and can be successfully used to detect water leakages in the network.

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“…Therefore, the choice of method of solving the interval system of equations is strictly connected with directed interval arithmetic. The interval methods appearing in the literature include Jacobi iterative method described by Markov (1999); Zieniuk et al (2016), LU decomposition method (Goldsztejn and Chambert 2007) and the Gauss elimination method (Neumaier 1990;Zieniuk et al 2016), which is the most commonly used method. During the tests we decided to present the results of the Gauss elimination method only.…”

Section: Interval System Of Algebraic Equationsmentioning

confidence: 99%

The influence of interval arithmetic on the shape of uncertainly defined domains modelled by closed curves

2016

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The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C 2 continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves.

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Solving interval systems of equations obtained during the numerical solution of boundary value problems

Cited by 12 publications

References 17 publications

Modification of interval arithmetic in modeling uncertainty of boundary conditions in boundary value problems

Modification of interval arithmetic in modeling uncertainty of boundary conditions in boundary value problems

Interval State Estimation of Hydraulics in Water Distribution Networks

The influence of interval arithmetic on the shape of uncertainly defined domains modelled by closed curves

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