Hausdorff Approximations Research Articles

A Newton Method for Hausdorff Approximations of the Pareto Front Within Multi-objective Evolutionary Algorithms

A Newton Method for Hausdorff Approximations of the Pareto Front Within Multi-objective Evolutionary Algorithms

Hausdorff approximations and volume of tubes of singular algebraic sets

We prove bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This generalizes previous work of Lotz (Proc Am Math Soc 143(5):1875–1889, 2015) on smooth complete intersections in the euclidean space and of Bürgisser et al. (Math Comp 77(263):1559–1583, 2008) on hypersurfaces in the sphere, and gives a complete solution to Bürgisser and Cucker (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 349, Springer, Heidelberg, 2013, Problem 17).

A Bounded Archiver for Hausdorff Approximations of the Pareto Front for Multi-Objective Evolutionary Algorithms

Multi-objective evolutionary algorithms (MOEAs) have been successfully applied for the numerical treatment of multi-objective optimization problems (MOP) during the last three decades. One important task within MOEAs is the archiving (or selection) of the computed candidate solutions, since one can expect that an MOP has infinitely many solutions. We present and analyze in this work ArchiveUpdateHD, which is a bounded archiver that aims for Hausdorff approximations of the Pareto front. We show that the sequence of archives generated by ArchiveUpdateHD yields under certain (mild) assumptions with a probability of one after finitely many steps a Δ+-approximation of the Pareto front, where the value Δ+ is computed by the archiver within the run of the algorithm without any prior knowledge of the Pareto front. The knowledge of this value is of great importance for the decision maker, since it is a measure for the “completeness” of the Pareto front approximation. Numerical results on several well-known academic test problems as well as the usage of ArchiveUpdateHD as an external archiver within three state-of-the-art MOEAs indicate the benefit of the novel strategy.

A scalar optimization approach for averaged Hausdorff approximations of the Pareto front

This article presents a novel method to compute averaged Hausdorff () approximations of the Pareto fronts of multi-objective optimization problems. The underlying idea is to utilize directly the scalar optimization problem that is induced by the performance indicator. This method can be viewed as a certain set based scalarization approach and can be addressed both by mathematical programming techniques and evolutionary algorithms (EAs). In this work, the focus is on the latter where a first single objective EA for such approximations is proposed. Finally, the strength of the novel approach is demonstrated on some bi-objective benchmark problems with different shapes of the Pareto front.

On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions

We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the logistic and the Gompertz functions. The limiting case of the interval-valued В step function is В discussed В using В Hausdorff metric. В Various expressions for the error estimates of the corresponding uniform and Hausdorff approximations are obtained. Numerical examples are presented using CAS MATHEMATICA.

Topography Influence on the Lake Equations in Bounded Domains

We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and $L^p$ perturbations of the depth. As a byproduct, we obtain the existence of a weak solution to the lake equations in the case of singular domains and rough bottoms. Our result thus extends earlier works by Bresch and M\'etivier treating the lake equations with a fixed topography and by G\'erard-Varet and Lacave treating the Euler equations in singular domains.

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.

Dedekind Order Completion of C (X) by Hausdorff Continuous Functions

The concept of Hausdorff continuous interval valued functions, developed within the theory of Hausdorff approximations and originaly defined for interval valued functions of one real variable is extended to interval valued functions defined on a topological space X. The main result is that the set Hft (X) of all finite Hausdorff continuous functions on any topological space X is Dedekind order complete. Hence it contains the Dedekind order completion of the set C(X) of all continuous real functions defined on X as well as the Dedekind order completion of the set C b(X) of all bounded continuous functions on X. Under some general assumptions about the topological space X the Dedekind order completions of both C(X) and C b(X) are characterised as subsets of Hft(X). This solves a long outstanding open problem about the Dedekind order completion of C(X). In addition, it has major applications to the regularity of solutions of large classes of nonlinear PDEs.

Approximations with a sign-sensitive weight. Stability, applications to the theory of snakes and Hausdorff approximations

Sign-sensitive approximations take into account not only the absolute value of the approximation error but also its sign. In the previous paper with the same title and the subtitle “existence and uniqueness theorems” we studied the problems of existence, uniqueness and plurality for the element of best uniform approximation with a sign-sensitive weight (, ) by some (in particular, Chebyshev) family of bounded functions on a set . An important role was played by the notions of rigidity and freedom of the system . Here we consider the stability of this process of approximation, that is, whether the least deviations and the best approximations by elements depend continuously on if the variation of is measured in the so-called -metric. The results are applied to the theory of snakes and Hausdorff approximations of special multivalued functions.

Hausdorff approximations on Hadamard manifolds and their ideal boundaries

Hausdorff approximations on Hadamard manifolds and their ideal boundaries

Best Hausdorff approximations by algebraic polynomials and porosity

For anyb ∈ (a, a′), a sharp lower bound for the coefficient ofv-uniformity of the set of discontinuity points of a functionf on the interval (−b, b) is obtained.

Rigidity of compact ideal boundaries of manifolds joined by Hausdorff approximations

Rigidity of compact ideal boundaries of manifolds joined by Hausdorff approximations

Measured Hausdorff convergence of Riemannian manifolds and Laplace operators

K. Fukaya introduced in [11] a topology on a set of metric spaces equipped with Borel measures, called the measured Hausdorff topology and discussed the continuity of the eigenvalues of the Laplace operators of Riemannian manifolds with uniformly bounded curvature. The purpose of the present paper is to study more closely the Laplace operators of Riemannian manifolds which collapse in this topology to a space of lower dimension while keeping their curvature bounded. 0.1. According to [16], a map h:X—>Y of metric spaces is said to be an £-Hausdorff approximation if |dis(#, x')—dis(A(#),h(x')) ^£ for all x,x' ^X and the ^-neighborhood of the image h(X) coincides with Y. A sequence of compact metric spaces {Xf} converges, by definition, to a compact metric space Y in the Hausdorff distance if there are a sequence of positive numbers {£(&')} going to zero as i tends to infinity and £(/)-Hausdorff approximations hi: Xi~+ Y of Xi into Y. Moreovre when each metric space Xs is equipped with a Borel measure μi of unit mass, according to [11], we say that {(Xg, μ, )} converges to Y with a Borel measure μ^ of unit mass in the measured Hausdorff topology, if in addition, these maps kg: Xg-* Y are Borel measurable and the push-forward measure h^μ{ converges to μ^ in the weak* topology. Now we shall consider a sequence of compact Riemannian manifolds {(Mgygg)} of dimension m whose sectional curvature KM. is bounded uniformly in its absolute value by a constant, say 1, and assume that tihs sequence converges to a compact metric space Moo in the Hausdorff distance. When the volume of Mf is bounded uniformly away from zero by a positive constant, Gromov's

Best Hausdorff approximations by algebraic polynomials and continuity of a function

Best Hausdorff approximations by algebraic polynomials and continuity of a function