Hausdorff methods for approximating the convex Edgeworth–Pareto hull in integer problems with monotone objectives

Adaptive methods for the polyhedral approximation of the convex Edgeworth–Pareto hull in multiobjective monotone integer optimization problems are proposed and studied. For these methods, theoretical convergence rate estimates with respect to the number of vertices are obtained. The estimates coincide in order with those for filling and augmentation H-methods intended for the approximation of nonsmooth convex compact bodies.     

Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls

The estimate refinement method for the polyhedral approximation of convex compact bodies is analyzed. When applied to convex bodies with a smooth boundary, this method is known to generate polytopes with an optimal order of growth of the number of vertices and facets depending on the approximation error. In previous studies, for the approximation of a multidimensional ball, the convergence rates of the method were estimated in terms of the number of faces of all dimensions and the cardinality of the facial structure (the norm of the f-vector) of the constructed polytope was shown to have an optimal rate of growth. In this paper, the asymptotic convergence rate of the method with respect to faces of all dimensions is compared with the convergence rate of best approximation polytopes. Explicit expressions are obtained for the asymptotic efficiency, including the case of low dimensions. Theoretical estimates are compared with numerical results.     

Simultaneous Interpolation and Approximation by a Class of Multivariate Positive Operators

The purpose of this paper is to show that the interpolation positive operators of a wide class satisfy also the approximation property. Such a situation of simultaneous interpolation and approximation may be very desirable, but is rather unusual. Our attention is focused on the convergence problem, giving the conditions under which a sequence of operators of the considered class converges to a continuous function in a convex compact set in R ^m (m∈N). It must be recalled that many of these operators are very interesting in applications and that suitable algorithms can be devised for parallel, multistage and iterative computation.     

Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions

We study the long-time asymptotic behavior of solutions u of the Hamilton-Jacobi equation u _t (x, t)?+?H(x, Du(x, t))?=?0 in ?? × (0, ??), where ?? is a bounded open subset of ${\mathbb{R}^n}$ , with Hamiltonian H?=?H(x, p) being convex and coercive in p, and establish the uniform convergence of u to an asymptotic solution as t ?? ??.     

On polyhedral approximations in an <Emphasis Type="Italic">n</Emphasis>-dimensional space

The polyhedral approximation of a positively homogeneous (and, in general, nonconvex) function on a unit sphere is investigated. Such a function is presupporting (i.e., its convex hull is the supporting function) for a convex compact subset of Rⁿ. The considered polyhedral approximation of this function provides a polyhedral approximation of this convex compact set. The best possible estimate for the error of the considered approximation is obtained in terms of the modulus of uniform continuous subdifferentiability in the class of a priori grids of given step in the Hausdorff metric.     

The initial convergence rate of adaptive methods for polyhedral approximation of convex bodies

The convergence rate at the initial stage is analyzed for a previously proposed class of asymptotically optimal adaptive methods for polyhedral approximation of convex bodies. Based on the results, the initial convergence rate of these methods can be evaluated for arbitrary bodies (including the case of polyhedral approximation of polytopes) and the resources sufficient for achieving optimal asymptotic properties can be estimated.     

Modelling and mathematical results arising from ferromagnetic problems

In this article,we investigate the equations of magnetostatics for a configuration where a ferromagnetic material occupies a bounded domain and is surrounded by vacuum.Furthermore,the ferromagnetic law takes the form B=μ0μr(|H|)H,i.e.,the magnetizing field H and the magnetic induction B are collinear,but the relative permeability μr is allowed to depend on the modulus of H.We prove the well-posedness of the magnetostatic problem under suitable convexity assumptions,and the convergence of several iterative methods,both for the original problem set in the Beppo-Levi space W1(R3),and for a finite-dimensional approximation.The theoretical results are illustrated by numerical examples,which capture the known physical phenomena.     

Compact holomorphic mappings on Banach spaces and the approximation property

Let $\text{H}$(E) be the space of complex valued holomorphic functions on a complex Banach space E. The approximation property for $\text{H}$(E), endowed with various natural locally convex topologies, is studied. For example, $\text{H}$(E) with the compact-open topology has the approximation property if and only if E has the approximation property. In order to characterize when $\text{H}$(E) has the approximation property for topologies other than the compact-open, the notion of a compact holomorphic map between Banach spaces in introduced and studied.     

On Multivariate Incomplete Polynomials on Starlike Domains

Multivariate incomplete polynomials are considered on compact 0-symmetric starlike domains. Problems of density and quantitative approximation properties of such polynomials are investigated. It is shown that density holds for a certain class of starlike domains which includes both convex and some nonconvex domains. On the other hand, a family of nonconvex starlike domains is also found for which density fails. In addition, it is also shown that on 0-symmetric convex bodies in $\mathbb{R}^{d}$ , continuous functions can be approximated by θ-incomplete polynomials with the rate O(ω ₂(n ^?1/(d+3))). Moreover, if the convex body is the intersection of simplexes with vertex at the origin, then this order improves to $O (\omega_{2}(f,1/\sqrt{n}) )$ .     

A Geometric Lower Bound Theorem

We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for C ²-convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.     

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto–Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto–Straus’s original result holds for a large class of compact convex sets.     

Fuchsian convex bodies: basics of Brunn–Minkowski theory

The hyperbolic space ${\mathbb{H}^d}$ can be defined as a pseudo-sphere in the (d + 1) Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that ${\mathbb{H}^d/\Gamma}$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn–Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov–Fenchel and Brunn–Minkowski inequalities. Here the inequalities are reversed.     

Uniform approximation of Poisson integrals of functions from the class H ω by de la Vallée Poussin sums

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C _?? ^q H _?? of Poisson integrals of functions from the class H _?? generated by convex upwards moduli of continuity ??(t) which satisfy the condition ??(t)/t ?? ?? as t ?? 0. As an implication, a solution of the Kolmogorov-Nikol??skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H ^??, 0 < ?? < 1, is obtained.     

Strong convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258??271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140??152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400??411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372??79 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336??46 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334??345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835??842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77??8 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766??76 (2007)].     

Mann type iterative methods for finding a common solution of split feasibility and fixed point problems

The purpose of this paper is to study and analyze three different kinds of Mann type iterative methods for finding a common element of the solution set ?? of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. By combining Mann??s iterative method and the extragradient method, we first propose Mann type extragradient-like algorithm for finding an element of the set ${{{\rm Fix}}(S) \cap \Gamma}$ ; moreover, we derive the weak convergence of the proposed algorithm under appropriate conditions. Second, we combine Mann??s iterative method and the viscosity approximation method to introduce Mann type viscosity algorithm for finding an element of the ${{{\rm Fix}}(S)\cap \Gamma}$ ; moreover, we derive the strong convergence of the sequences generated by the proposed algorithm to an element of set ${{{\rm Fix}}(S) \cap \Gamma}$ under mild conditions. Finally, by combining Mann??s iterative method and the relaxed CQ method, we introduce Mann type relaxed CQ algorithm for finding an element of the set ${{{\rm Fix}}(S)\cap \Gamma}$ . We also establish a weak convergence result for the sequences generated by the proposed Mann type relaxed CQ algorithm under appropriate assumptions.     

Approximation of convex bodies by polytopes

LetC be a convex body ofE ^d and consider the symmetric difference metric. The distance ofC to its best approximating polytope having at mostn vertices is 0 (1/n ^2/(d?1)) asn→∞. It is shown that this estimate cannot be improved for anyC of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of «most» convex bodies are rather irregular and that ford=2 «most» convex bodies have unique best approximating polygons with respect to both metrics.     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

Iterative Optimierungsverfahren,die unter schwachen Voraussetzungen konvergieren

Some problems in nonlinear optimization theory can be formulated as minimization problems on a closed convex setD of a Hilbert spaceH with a functional ? satisfying min {?(u)/u∈D}=0. For the numerical approximation of a minimizing element one can consider the iteration process means the metric projection from ? ontoD.g(u) is a generalization of the gradient of ? inu and ω>0 is a relaxation parameter. Special iteration processes of this type using gradients or subgradients are due to N. Gastinel (1963) and W. Oettli (1971). In this paper the convergence of the above iteration is shown under weak assumptions on ?. Also estimates for the rate of convergence are given. The method is applied to linear and nonlinear optimization problems.     

Convexity,monotonicity, and gradient processes in Hilbert space

It is shown that in Hilbert spaces the gradient maps of convex functionals with uniformly bounded continuous second Frechét derivatives satisfy monotonicity conditions that insure that some convex combination of the identity, I, and I ? ▽f is either strictly contractive or at worst nonexpansive. This result leads to a complete resolution of the convergence question for a large class of associated gradient processes. In particular, weak convergence of the successive approximation sequence is established even in the singular case where f″ is not strictly positive at critical points of f.     

Jensen Measures in Potential Theory

It is shown that, for open sets in classical potential theory and??more generally??for elliptic harmonic spaces Y, the set J _x (Y) of Jensen measures (representing measures with respect to superharmonic functions on?Y) for a?point x????Y is a?simple union of closed faces of the compact convex set $M_x(\mathcal P(Y))$ of representing measures with respect to potentials on?Y, a?set which has been thoroughly studied a?long time ago. In particular, the set of extreme Jensen measures can be immediately identified. The results hold even without ellipticity (thus capturing also many examples for the heat equation) provided a?rather weak approximation property for superharmonic functions holds. Equally sufficient are a?certain transience property and a?weak regularity property. More important, each of these properties turns out to be necessary and sufficient for obtaining (in the classical case) that J _x (Y) coincides with the set of all compactly supported probability measures in $M_x(\mathcal P(Y))$ .