An Improved Interval Linearization for Solving Nonlinear Problems

Kolev, Lubomir V.

doi:10.1023/b:numa.0000049468.03595.4c

Cited by 32 publications

(33 citation statements)

References 13 publications

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“…{The component x i too narrow for bisection} 19: return x i , "found a pseudo-solution" 20: end if 21: bisect x i , obtaining x (1) i and x (2) i 22: if (left narrow (x (1) , f, i, ε) results in (x * , "found a pseudo-solution")) then 23: return x * , "found a pseudo-solution" 24: end if 25: if (left narrow (x (2) , f, i, ε) results in (x * , "found a pseudo-solution")) then 26: return x * , "found a pseudo-solution" 27: end if 28: return "no solution"…”

Section: Algorithm 2 Procedures Left Narrowmentioning

confidence: 99%

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

Kubica

2015

Numer Algor

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The paper summarizes author's investigations in tuning a multithreaded interval branch-and-prune algorithm for nonlinear systems and presents the developed solver. New results for using the box-consistency enforcing operator and a new variant of the initial exclusion phase are presented. Also, a new heuristic to choose the coordinate for bisection is considered. Extensive numerical experiments are analyzed to provide the satisfying version of the algorithm.

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Section: Algorithm 2 Procedures Left Narrowmentioning

confidence: 99%

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

Kubica

2015

Numer Algor

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“…Inspired by the ideas in [18,19,30], we propose a revised affine form similar to (4) but the new term z k ε k is replaced by an accumulative error [−e z , e z ] which represents the maximum absolute error z k of non-affine operations. In other words, the revised affine form of a real-valued quantityx is defined aŝ…”

Section: Revised Affine Arithmeticmentioning

confidence: 99%

“…We now compare the proposed technique with a recent mathematical solution technique, called A2, in [19], which was specially designed to solve a nonlinear equation system f (x) = 0. The A2 algorithm converts this system into separable form g(x) = 0, and then uses Kolev affine arithmetic to evaluate g(x) and get a linear form L(x, y) = −Ax + By + b , x ∈ x, y ∈ y; where A and B are real matrices, b is a real vector, and x and y are interval vectors.…”

Section: Comparisons With Linear Relaxation Based Techniquesmentioning

confidence: 99%

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Enhancing numerical constraint propagation using multiple inclusion representations

Sam-Haroud

Faltings

2009

Ann Math Artif Intell

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Building tight and conservative enclosures of the solution set is of crucial importance in the design of efficient complete solvers for numerical constraint satisfaction problems (NCSPs). This paper proposes a novel generic algorithm enabling the cooperative use, during constraint propagation, of multiple enclosure techniques. The new algorithm brings into the constraint propagation framework the strength of techniques coming from different areas such as interval arithmetic, affine arithmetic, and mathematical programming. It is based on the directed acyclic graph (DAG) representation of NCSPs whose flexibility and expressiveness facilitates the design of fine-grained combination strategies for general factorable systems. The paper presents several possible combination strategies for creating practical instances of the generic algorithm. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic, and linear programming illustrate the flexibility and efficiency of the approach.

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“…This is achieved using interval linearization (Kolev, 2004b). Given a range of values for the state x in which interval linearization will be performed, each of the nonlinear functions is enclosed between two lines and an interval term represents the linearization uncertainty.…”

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confidence: 99%

Real-Time Hydraulic Interval State Estimation for Water Transport Networks: a Case Study

Vrachimis¹,

Ηλιάδης²,

Polycarpou³

2017

Preprint

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Abstract. Hydraulic state estimation in water distribution networks is the task of estimating water flows and pressures in the pipes and nodes of the network based on some sensor measurements. This requires a model of the network, as well as knowledge of demand outflow and tank water levels. Due to modeling and measurement uncertainty, standard state-estimation may result in inaccurate hydraulic estimates without any measure of the estimation error. This paper describes a methodology for generating hydraulic state bounding estimates based on interval bounds on the parametric and measurement uncertainties. 5The estimation error bounds provided by this method can be applied to estimate the unaccounted-for water in water distribution networks. As a case study, the method is applied to a transport network in Cyprus, using actual data in real-time.

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An Improved Interval Linearization for Solving Nonlinear Problems

Cited by 32 publications

References 13 publications

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

Enhancing numerical constraint propagation using multiple inclusion representations

Real-Time Hydraulic Interval State Estimation for Water Transport Networks: a Case Study

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