Application Of Interval Analysis Research Articles

An Interval Parameter Calculation Method of Asphalt Pavement Structure Design Based on Point Numerical Algorithm

In the asphalt pavement structure design method, the structural analysis and design are generally performed in the form of point values. However, determining the point value form of design parameters based on the statistical analysis theory cannot fully reflect the complex properties such as variability and uncertainty of parameters. In order to further improve the reliability and practicability of pavement design parameters, in this article, we have introduced the interval number representation that can better reflect the complex nature of parameters; but the interval number algorithm is too complicated and common calculation tools and software are difficult to adopt, which limit the wide application of interval analysis to some extent. The article analyzes the algorithm of interval numbers, focusing on the analysis of interval numbers of unary and binary functions. In this way, the point number operation can be used to obtain the interval number result of the function consistent with the interval number algorithm, which avoids the complicated interval number operation process and the interval expansion. The point numerical function algorithm of interval numbers is verified by design parameters and the calculation of asphalt pavement structure such as axle load conversion, cumulative equivalent axis calculation, calculation of foundation layer tensile stress of each structure layer, calculation of mixture penetration strength, fatigue cracking check of asphalt mixture layer, permanent deformation check, and vertical pressure strain test of roadbed top surface. In conclusion, this research provides a simple and easy way to implement the application of mathematical tools for interval analysis, which is suitable for direct use for existing point numerical calculation tools and software.

Efficiency in uncertain variational control problems

In this paper, considering the applications of interval analysis in various fields (such as artificial intelligence, neural computation, genetic algorithms, information theory or fuzzy logic), a new class of interval-valued variational control problems governed by multiple integral functionals, first-order PDE and inequality constraints is studied. More precisely, efficiency conditions for the considered uncertain variational control problem are formulated and proved. The sufficiency of Karush–Kuhn–Tucker conditions is established under some invexity and $$(\rho, b)$$ -quasiinvexity assumptions of the involved functionals. In addition, the paper is completed with illustrative applications (describing the controlled behavior of an artificial neural system) and the corresponding algorithm.

Application of Interval Approach to Spectral Analysis. Investigation of Solvability of a Problem with the Use of a Recognizing Functional

The application of interval analysis in a linear formulation (linear problem of tolerances) to a concrete problem of spectral analysis is considered. The principal procedural objective is to demonstrate the use of a recognizing functional as an investigatory tool. The method of intervalization of the right sides of an interval system of linear algebraic equations without recourse to models of the form of the background under a peak along with the form of united solution sets of an interval system of linear algebraic equations for a particular problem are studied. The solvability of an interval system of linear algebraic equations relative to the sign of the recognizing functional is investigated, and it is shown that in the initial formulation the system is not solvable. Methods of attaining solvability by weakening the tolerances of the right side of the system and through a modification of the matrix are considered and results of the use of both methods are discussed.

Nonlinear Triangular Intuitionistic Fuzzy Number and Its Application in Linear Integral Equation

In this paper we introduce the different arithmetic operations on nonlinear intuitionistic fuzzy number (NIFN). All the arithmetic operations are done by max-min principle method which is nothing but the application of interval analysis. We also define the nonlinear intuitionistic fuzzy function which is used for finding the values, ambiguities, and ranking of nonlinear intuitionistic fuzzy number. The de-i-fuzzification of the corresponding intuitionistic fuzzy solution is done by average of (α,β)-cut method. Finally we solve integral equation with NIFN by the help of intuitionistic fuzzy Laplace transform method.

Simple application of interval analyses to structural safety: standard versus parameterised versions

In this paper the implications of the use of interval analysis are studied in regards to the decision making process on the safety of the structure. It is shown that the conclusions derived by application of the standard interval analysis about the structural safety may be misleading in some cases. Parameterised interval analysis is demonstrated to yield rigorous results.

Linear and Quasi-linear Spaces of Interval Maps

Interval-valued functions, or briefly interval functions, are traditionally associated with numerical analysis and validated computing, see for instance [4, 5]. Such maps also occur naturally as mathematical tools for modeling phenomena where uncertainty is present, such as in biological dynamical systems [3]. Recently interval functions have also been applied to problems in pure mathematics, such as the Dedekind order completion of spaces of continuous functions [1]. In [2] the structure of a linear space was introduced on the set of finite Hausdorff continuous interval functions with domain an open subset of Euclidean space R^n. In this paper we generalize this result to the case of functions defined on an arbitrary topological space. It should be noted that the problem of defining algebraic operations on spaces of interval functions is nontrivial, due to the fact that the set of compact intervals in R is not a group with respect to the Minkowski sum. [1] R. Anguelov R, S. Markov, B. Sendov, The set of Hausdorff continuousfunctions — the largest linear space of interval functions, ReliableComputing 12 (2006) 337 – 363. [2] R. Anguelov R, S. Markov, B. Sendov, The set of Hausdorff continuous functions - the largest linear space of interval functions, Reliable Computing 12 (2006) 337 - 363. [3] S. Markov, Biomathematics and interval analysis: a prosperous marriage, In: M. D. Todorov, Ch. I. Christov, Eds., AIP ConferenceProceedings, Vol. 1301, Amitans’2010, American Institute of Physics,Melville, New York, 2010, pp. 26–36. [4] R. E. Moore, Methods and Applications of Interval Analysis, SIAM,Philadelphia, 1979. [5] B. Sendov, Hausdorff Approximations, Kluwer, Dordrecht, 1990.

Application of interval analysis and evidence theory to fault location

The technique of fast fault clearance for railway power transmission lines is an urgent and noticeable problem. According to the segmented structure characteristic of the lines, a novel fault location scheme based on the interval analysis and evidence theory is proposed. First, the interval model is addressed. The interval analysis is applied to deduce three interval algorithms, which implement interval location. Then, in terms of the characteristic of interval set, the difference matrix of deviation degree is used to represent quantitatively the degree of similarity between interval numbers, and construct an expression of basic probability assignment function. Finally, the evidence theory is introduced to combine the results of three interval algorithms and infer the fault segmented interval. The simulated results and practical application have effectively verified the accuracy and reliability of the proposed location scheme, which is suitable for all segmented structure lines.

An Application of Interval Analysis and Optimization to Electric Energy Markets

In this paper, we address the issue of quantifying model uncertainties in a deregulated power market. We are interested in a worst-case analysis of effects of network parameter uncertainties on optimization problems that are solved on the level of an independent system operator (ISO). The focus is on linear programming (LP) formulations because of their practical relevance. After briefly reviewing applications of different interval analysis tools (interval matrix inversion and various contractors and solvers for systems of interval equations), we propose two tailored algorithms. In the first (basic LP approach), we propose another solver for a system of linear interval equations, while in the second (power system-customized LP approach), we use features of the power system model to reduce the width of the interval solution. The interval-based results are compared with a deterministic robust noninterval matrix inversion-based solution of a model of the benchmark 68-node New England/New York interconnected power system. Presented analyzes demonstrate that even small parameter variations can have significant impact on optimization.

A New Subdivision Strategy for Range Computations

We present a new subdivision strategy in interval analysis for computing the ranges of functions. We show through several real-world examples that the proposed subdivision strategy is more efficient than the widely used uniform and adaptive subdivision strategies of Moore (Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979).

Consistency Techniques in Ordinary Differential Equations

This paper takes a fresh look at the application of interval analysis to ordinary differential equations and studies how consistency techniques can help address the accuracy problems typically exhibited by these methods, while trying to preserve their efficiency. It proposes to generalize interval techniques into a two-step process: a forward process that computes an enclosure and a backward process that reduces this enclosure. Consistency techniques apply naturally to the backward (pruning) step but can also be applied to the forward phase. The paper describes the framework, studies the various steps in detail, proposes a number of novel techniques, and gives some preliminary experimental results to indicate the potential of this new research avenue.

Nonlinear bounded-error state estimation of continuous-time systems

This paper presents a first study on the application of interval analysis and consistency techniques to state estimation of continuous-time systems described by nonlinear ordinary differential equations. The approach is presented in a bounded-error context and the resulting methodology is illustrated by an example.

Application of interval analysis to impulsive differential equations

Impulsive differential equations with continuous or discontinuous right hand side are studied by interval analysis. The existence of solutions is characterized and bounds are obtained.

On solving the equation Ax=b with more variables than equations

A recently derived mathematical tool is applied directly to solving Ax=b with more variables than equations. This problem arises in the application of interval analysis to circuit design. An arbitrary number of the variables are chosen as parameters, and the complete zero set in the remaining variables multidimensional plane is explicitly determined. >