Showing papers in "Numerical Functional Analysis and Optimization in 2003"
TL;DR: In this paper, the proximinality and best proximity pair theorems in hyperconvex metric spaces and in Hilbert spaces have been studied in the context of finding the optimal solution in the case that the mapping T does not have fixed points.
Abstract: This note is concerned with proximinality and best proximity pair theorems in hyperconvex metric spaces and in Hilbert spaces. Given two subsets A and B of a metric space and a mapping best proximity pair theorems provide sufficient conditions that ensure the existence of an such that Thus such theorems provide optimal approximate solutions in the case that the mapping T does not have fixed points.
220 citations
TL;DR: In this paper, the authors consider self-mappings of a closed convex subset of a reflexive Banach space and show that almost all of them share the property that they have a fixed point z T such that, for any x ∈ K, the orbit converges weakly to z T.
Abstract: Let K be a closed convex subset of a reflexive Banach space X. We consider self-mappings of K which are bounded on bounded subsets of K and satisfy a relaxed form of nonexpansivity with respect to a given convex function f. The family of these operators is endowed with the topology of uniform convergence on bounded subsets of K. We show that “almost all” such operators T share the property that they have a fixed point z T such that, for any x ∈ K, the orbit converges weakly to z T . Here the meaning of “almost all” is in the sense of Baire's categories: the collection of all those operators which do not have this property is contained in a countable union of nowhere dense sets.
167 citations
TL;DR: In this article, the authors considered the nonlocal boundary value problem in an arbitrary Banach space E with the positive operator A and established the well-posedness of this boundary-value problem in the spaces of smooth functions.
Abstract: The nonlocal boundary value problem in an arbitrary Banach space E with the positive operator A is considered. The well-posedness of this boundary value problem in the spaces of smooth functions is established. The new exact Schauder's estimates of solutions of the boundary value problems for elliptic equations are obtained.
69 citations
TL;DR: In this paper, the analytical properties of r-limit set, relation to other convergence notions, and the dependence of the rlimit set on the roughness degree r are investigated. And the minimal value of r such that an arbitrary ρ-Cauchy sequence in X is certainly r-convergent.
Abstract: For given r, ρ ≥ 0, a sequence (xi ) in some normed linear space X is said to be r-convergent if the r-limit set defined by is nonempty, and it is called a ρ-Cauchy sequence if This article investigates different aspects of this rough convergence, especially in infinite dimensional spaces, such as analytical properties of r-limit set, relation to other convergence notions, and the dependence of the r-limit set on the roughness degree r. Moreover, by using the Jung constant we find the minimal value of r such that an arbitrary ρ-Cauchy sequence in X is certainly r-convergent.
52 citations
TL;DR: The notion of a *-none-expansive multivalued map is different from that of a continuous map as discussed by the authors, and some Ky Fan type best approximation theorems for mappings defined on closed convex unbounded subsets of a Hilbert space are given.
Abstract: The notion of a *-nonexpansive multivalued map is different from that of a continuous map. We give some Ky Fan type best approximation theorems for *-nonexpansive mappings defined on closed convex unbounded subsets of a Hilbert space. As applications of our theorems, we derive fixed point results under many boundary conditions. Approximating sequences to the fixed points are also constructed.
29 citations
TL;DR: In this paper, a smoothing method is proposed for the nonsmooth nonlinear penalty function, and an algorithm for the constrained optimization problem based on the smoothed non-linear penalty method and prove the convergence of the algorithm.
Abstract: In this article, we discuss a nondifferentiable nonlinear penalty method for an optimization problem with inequality constraints. A smoothing method is proposed for the nonsmooth nonlinear penalty function. Error estimations are obtained among the optimal value of smoothed penalty problem, the optimal value of the nonsmooth nonlinear penalty optimization problem and that of the original constrained optimization problem. We give an algorithm for the constrained optimization problem based on the smoothed nonlinear penalty method and prove the convergence of the algorithm. The efficiency of the smoothed nonlinear penalty method is illustrated with a numerical example.
28 citations
TL;DR: In this article, the authors survey the attempts in literature to cover the terrain incognita between the convergence region for the Newton-Kantorovich method and the convergence regions for the modified Newton K-means method and give a further step improving all previous results.
Abstract: Let an operator with f′ Holder continuous. An existence result on the solutions of the equation f(x) = 0 was obtained by Vertgeim (Vertgeim, B. A. (1956). On conditions for the applicability of Newton's method. Dokl. Akad. Nauk. SSSR 110:719–722 (in Russian); Vertgeim, B. A. (1960). On some methods of the approximate solution of nonlinear functional equations in banach spaces. Uspekhi Mat. Nauk. 12:166–169 (in Russian) [(1960). Engl. Transl.: Amer. Math. Soc. Transl. 16(2):378–382.]) by means of the convergence of the modified Newton–Kantorovich method. There is an unknown land (“terra incognita”), left by earlier results on this topic, between the convergence region for the Newton–Kantorovich method and the convergence region for the modified Newton-Kantorovich method. We survey the attempts in literature to cover this “terra incognita” and we give a further step improving all previous results.
27 citations
TL;DR: In this paper, a new approach to generalizations of Stirling numbers of the first and the second kind in terms of fractional calculus analysis by using differences and differentiation operators was presented.
Abstract: The purpose of this article and companion ones is to present a new approach to generalizations of Stirling numbers of the first and the second kind in terms of fractional calculus analysis by using differences and differentiation operators of fractional order. Such an approach allows us to extend the classical Stirling numbers of the first and the second kind in a natural way not only to any positive, but also to any negative order. Moreover, an application of the fractional approach gives us the opportunity to extend the classical Stirling numbers to more general complex functions. In the present article we extend the classical Stirling numbers of the second kind, S(n, k), for the first parameter from a nonnegative integer number n to any complex α. Such constructions, S(α, k), will be defined for any complex α and by when k >0, while S(0, 0) = 1 and when k = 0. We show that S(α, k) with positive α can be represented by the Liouville and Marchaud fractional derivatives of the exponential functio...
26 citations
TL;DR: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings has been studied in this paper, where convergence properties were examined in some cases.
Abstract: In this paper, we consider, in a finite dimensional real Hilbert space , the variational inequality problem VIP : find , where is nonexpansive mapping with bounded and is paramonotone and Lipschitzian over . The nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping is a typical example of such a variational inequality problem. We show that the hybrid steepest descent method, of which convergence properties were examined in some cases for example (Yamada, I. (2000). Convex projection algorithm from POCS to Hybrid steepest descent method. The Journal of the IEICE (in Japanese) 83:616–623; Yamada, I. (2001). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithm for Feasibility and Optimization. Elsevier; Ogura, N., Yamada, I. (2002). Non-strictly convex minimization over the fixed point set of an asymp...
25 citations
TL;DR: In this paper, a nonseparable multiresolution structure based on frames which is defined by radial frame scaling functions is presented, which can be carried out in any number of dimensions and for a big variety of dilation matrices.
Abstract: In this article we present a nonseparable multiresolution structure based on frames which is defined by radial frame scaling functions. The Fourier transform of these functions is the indicator (characteristic) function of a measurable set. We also construct the resulting frame multiwavelets, which can be isotropic as well. Our construction can be carried out in any number of dimensions and for a big variety of dilation matrices.
23 citations
TL;DR: In this article, the authors studied the problem of identifying the solution of linear ill-posed problems where instead of exact data y noisy data y ∈ X are given satisfying with known noise level δ.
Abstract: In this article we study the problem of identifying the solution x † of linear ill-posed problems Ax = y in a Hilbert space X where instead of exact data y noisy data y δ ∈ X are given satisfying with known noise level δ. Regularized approximations are obtained by the method of Lavrentiev regularization in Hilbert scales, that is, is the solution of the singularly perturbed operator equation where B is an unbounded self-adjoint strictly positive definite operator satisfying . Assuming the smoothness condition we prove that the regularized approximation provides order optimal error bounds (i) in case of a priori parameter choice for and (ii) in case of Morozov's discrepancy principle for s ≥ p. In addition, we provide generalizations, extend our study to the case of infinitely smoothing operators A as well as to nonlinear ill-posed problems and discuss some applications.
TL;DR: In this paper, the conjugate gradient method for nonsymmetric linear operators in Hilbert space is investigated and conditions on the coincidence of the full and truncated versions, known from the finite-dimensional case, are extended to the Hilbert space setting.
Abstract: The conjugate gradient method for nonsymmetric linear operators in Hilbert space is investigated. Conditions on the coincidence of the full and truncated versions, known from the finite-dimensional case, are extended to the Hilbert space setting. The focus is on preconditioning by the symmetric part of the operator, in which case estimates are given for the resulting condition number. An important motivation for this study is given by differential operators, for which the obtained estimates yield mesh independent conditioning properties of the full CG method, and are in fact achieved by the simpler truncated version. †Dedicated to the memory of Jean-Jacques Lions: a source of inspiration for rigour in Applied Mathematics.
TL;DR: In this paper, a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a convex set constraint is studied, where all the involved functions are, at least, directionally differentiable.
Abstract: We study a multiobjective optimization problem in with a feasible set defined by inequality and equality constraints and a convex set constraint. All the involved functions are, at least, directionally differentiable. Quasiconvex inequalities are considered. Given these assumptions, an expression for the contingent cone to the feasible set using an extended Mangasarian-Fromovitz constraint qualification is provided. As application, necessary and sufficient optimality conditions of Kuhn-Tucker type are established for a local Pareto minimal point.
TL;DR: Li et al. as mentioned in this paper studied superconvergence of solution derivatives for the Shortley-Weller difference approximation of Poisson's equation, Part I. Smoothness problems.
Abstract: This is a continued analysis on superconvergence of solution derivatives for the Shortley–Weller approximation in Li (Li, Z. C., Yamamoto, T., Fang, Q. ([2003]): Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math. 152(2):307–333), which is to explore superconvergence for unbounded derivatives near the boundary. By using the stretching function proposed in Yamamoto (Yamamoto, T. ([2002]): Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math. 140:849–866), the second order superconvergence for the solution derivatives can be established. Moreover, numerical experiments are provided to support the error analysis made. The analytical approaches in this article are non-trivial, intriguing, and different from Li, Z. C., Yamamoto, T., Fang, Q. ([2003]). This article also provides the superconvergence analysis for the bilinea...
TL;DR: In this article, the uniqueness of and numerical techniques for the inverse Sturm-Liouville problem with eigenparameter dependent boundary conditions are discussed, and the potential q in the spectral data can be uniquely determined using a Gel'fand-Levitan technique.
Abstract: Uniqueness of and numerical techniques for the inverse Sturm-Liouville problem with eigenparameter dependent boundary conditions will be discussed. We will use a Gel'fand-Levitan technique to show that the potential q in can be uniquely determined using spectral data. In the presence of finite spectral data, q can be reconstructed using a successive approximation method that involves solving a hyperbolic boundary value problem that arises in the the Gel'fand-Levitan analysis. We also consider a shooting method where the right endpoint boundary condition is used in conjunction with a quasi-Newton scheme to recover the unknown potential, q.
TL;DR: In this article, a method based on projections for approximate solution of eigenvalue problems associated with a compact linear operator is proposed for an integral operator with a smooth kernel using the projection method.
Abstract: We propose here a new method based on projections for approximate solution of eigenvalue problems associated with a compact linear operator. For an integral operator with a smooth kernel using the ...
TL;DR: In this article, a discontinuous finite element approximation for a model of quasi-static growth of brittle fractures in linearly elastic bodies formulated by Francfort and Marigo, and based on the classical Griffith's criterion was proposed.
Abstract: We propose a discontinuous finite element approximation for a model of quasi-static growth of brittle fractures in linearly elastic bodies formulated by Francfort and Marigo, and based on the classical Griffith's criterion. We restrict our analysis to the case of anti-planar shear and we consider discontinuous displacements which are piecewise affine with respect to a regular triangulation.
TL;DR: Lindner and Silbermann as mentioned in this paper showed that the sequence of finite sections is stable if and only if some associated operator is invertible at infinity, which is the norm-limits of band operators on.
Abstract: We present an approach to the finite section method for band-dominated operators—the norm-limits of band operators on . We hereby show that the sequence of finite sections is stable if and only if some associated operator is invertible at infinity. By means of the theory in Lindner and Silbermann (Lindner, M., Silbermann, B. (2003). Invertibility at infinity of band-dominated operators in the space of essentially bounded functions, (accepted at) Integral Equations and Operator Theory.) and Lindner (Lindner, M. (2003). Classes of multiplication operators and their limit operators (submitted to) Zeitschrift fur Analysis und ihre Anwendungen), we study this invertibility at infinity using limit operators. Having the mentioned criterion at our disposal, we will give some applications in an algebra of convolution and multiplication operators: one for the usual finite section method and one for an approximation method of operators on the space of continuous functions.
TL;DR: An efficient numerical method for the solution of Fredholm integral equations involving boundary-value problems of the Helmholtz equation corresponding to (general) regular (boundary) surfaces is discussed in more detail.
Abstract: By means of the limit and jump relations of classical potential theory with respect to the Helmholtz equation a wavelet approach is established on a regular surface. The multiscale procedure is constructed in such a way that the emerging potential kernels act as scaling functions, wavelets are defined via a canonical refinement equation. A tree algorithm for fast computation of a function discretely given on a regular surface is developed based on numerical integration rules. By virtue of the tree algorithm, an efficient numerical method for the solution of Fredholm integral equations involving boundary-value problems of the Helmholtz equation corresponding to (general) regular (boundary) surfaces is discussed in more detail.
TL;DR: In this article, the authors considered an abstract differential inclusion of the form where is a linear, continuous, symmetric and monotone operator defined over a separable Banach space V, and ∆ is the subdifferential of a proper, convex, l.s.c, positive real function.
Abstract: This work is concerned with an abstract differential inclusion of the form where is a linear, continuous, symmetric and monotone operator defined over a separable Banach space V, and ∂φ is the subdifferential of a proper, convex, l.s.c, positive real function. We consider an approximation of the previous equation by a backward Euler method with variable time-step. Under suitable hypothesis of coercivity we prove that the discrete solution converges uniformly to a strong solution of the equation, in the seminorm induced by B, as the maximum of the time steps goes to 0. We derive computable estimates of the discretization error, which are optimal w.r.t. the order and impose no constrains between consecutive time steps. In addition we prove some regularity and uniqueness results for the solution. Finally we extend some of the previous results to the case in which ∂φ is perturbed by a Lipschitz map.
TL;DR: In this paper, a generalization of the notion of invexity, called (p, r)-type I and (p r)-Type II with respect to η, is introduced.
Abstract: New two classes of real differentiable functions, called (p, r)-Type I and (p, r)-Type II with respect to η, which represent a generalization of the notion of invexity, are introduced. Some examples of these functions are derived. The sufficient optimality conditions are obtained for a nonlinear programming problem involving (p, r)-Type I functions with respect to η and for an associated Wolfe dual problem in which involved functions are (p, 0)-Type II. It is also shown that the optimization problems possesing the considered notion of invexity need not to be equivalent to the class of optimization problems for which some notion of invexity guarantees that Karush–Kuhn–Tucker necessary conditions for optimality are also sufficient and the sufficiency for optimality in associated Wolfe dual problems.
TL;DR: In this article, the authors studied the approximation of solutions to a class of second order semilinear integrodifferential equations in a Hilbert space, which arise in the study of viscoelastic materials with memory.
Abstract: In the present work we study the approximation of solutions to a class of second order semilinear integrodifferential equations in a Hilbert space. These equations arise in the study of viscoelastic materials with memory. Using a pair of associated nonlinear integral equations and projection operators we consider a pair of approximate nonlinear integral equations. We first show the existence and uniqueness of solutions to this pair of approximate integral equations. We then establish the convergence of the sequences of the approximate solutions and the pair of approximate integral equations to the solution and the pair of associated integral equations, respectively. We finally consider the Faedo–Galerkin approximation of the solution and prove some convergence results.
TL;DR: In this paper, the existence of time optimal controls for the Boussinesq equation and Pontryagin's maximum principle of the time-optimal control problem governed by this equation were derived.
Abstract: In this article, we obtain the existence of time optimal controls for the Boussinesq equation and derive Pontryagin’s maximum principle of time optimal control problem governed by the Boussinesq equation.
TL;DR: In this paper, an external field which realises the vorticity of micropolar fluids is determined and an existence result for the proposed control problem is obtained and the necessary conditions of optimality are derived.
Abstract: In the theory of micropolar fluids, a special case appears when the microrotation is equal to the vorticity of the fluid. The aim of this article is to determine an external field which realises this case. An existence result for the proposed control problem is obtained and the necessary conditions of optimality are derived. For solving the optimality system, an iterative algorithm is proposed and its convergence is obtained. The discretization of the approximation is studied; stability and convergence theorems are proved.
TL;DR: In this article, Chen and Yamamoto this article introduced more general Chen-Yamamoto-type conditions to generate a Newton-like method which converges to a locally unique solution of a nonlinear equation in a Banach space containing a non-differentiable term.
Abstract: In this study we introduce more general Chen–Yamamoto-type conditions to generate a Newton-like method which converges to a locally unique solution of a nonlinear equation in a Banach space containing a non-differentiable term. Using new and more precise majorizing sequences we provide local and semilocal results, first under the same and secondly, under weaker sufficient convergence conditions than before. In both cases we show that our results can be reduced to the ones by Chen and Yamamoto (Chen, X., Yamamoto, T. (1989). Convergence domains of certain iterative methods for solving nonlinear equations. Numer. Funct. Anal. Optimiz. 10(1&2):37–48.), whereas the error bounds and the information on the location of the solution can be more precise, and under more general conditions. Finally some numerical examples are provided where our results compare favorably with earlier ones in both the local and semilocal case.
TL;DR: In this paper, the convergence rate of Riemann-type sums to their scalar products in L 2 is analyzed for the space-time domain and W(L 2,l 1) for the frequency domain.
Abstract: In signal processing, discrete convolutions are usually involved in fast calculating coefficients of time-frequency decompositions like wavelet and Gabor frames. Depending on the regularity of the mother analyzing functions, one wants to detect the right resolution in order to achieve good approximations of coefficients. Local–global conditions on functions in order to get the convergence rate of Riemann-type sums to their scalar products in L 2 are presented. Wiener amalgam spaces, in particular for the space-time domain and W(L 2,l 1) for the frequency domain, give natural norms in order to estimate errors. In particular, relations between the rate of convergence of these series to integrals by increasing resolution and the (minimal) required Besov regularity are presented by means of functional and harmonic analysis techniques.
TL;DR: In this paper, a domain decomposition method for the numerical simulation of a two-body contact problem in two-dimensional linear elasticity is studied, and the problem is restated as a decomposition-coordination problem by introducing a smooth fictitious rigid surface between the two elastic bodies.
Abstract: We study a domain decomposition method for the numerical simulation of a two-body contact problem in two-dimensional linear elasticity. The problem is restated as a decomposition-coordination problem by introducing a smooth fictitious rigid surface between the two elastic bodies. In the decomposition step, two independent problems are solved: a Signorini-like problem and a prescribed displacement problem. In the coordination step, the fictitious rigid surface is adjusted to minimize a auxiliary functional.
TL;DR: In this article, the adaptive finite difference methods for the Dirichlet boundary value problems of Poisson-type equations on a sector or a disk were studied and it was shown that under some assumptions, under some conditions, the solutions are convergent and the convergence can be accelerated by varying parameters in the stretching functions.
Abstract: This article treats of adaptive finite difference methods for the Dirichlet boundary value problems of Poisson-type equations on a sector or a disk. It is assumed that the exact solutions have singular derivatives on a part or the whole of the boundary. Some stretching functions are used to generate nonuniform grid points. It is then shown that, under some assumptions, the adaptive finite difference solutions are convergent and the convergence can be accelerated by varying parameters in the stretching functions. Numerical examples are given to illustrate how the accuracy of numerical solutions depends on the parameters.
TL;DR: In this article, a quasivariational inequality in R d, d = 2, 3 with perturbed input data is solved by means of a worst scenario (anti-optimization) approach, using a stability result for the solution set of perturbed QVI-problems.
Abstract: A quasivariational inequality (QVI) in R d , d = 2, 3, with perturbed input data is solved by means of a worst scenario (anti-optimization) approach, using a stability result for the solution set of perturbed QVI-problems. The theory is applied to the dual finite element formulation of the Signorini problem with Coulomb friction and uncertain coefficients of stress-strain law, friction, and loading.
TL;DR: In this article, the authors consider systems of nonlinear difference equations arising when convergence analysis of an iterative method for solving operator equations in Banach spaces is carried out via Kantorovich's technique of majorization.
Abstract: We consider systems of nonlinear difference equations arising when convergence analysis of an iterative method for solving operator equations in Banach spaces is carried out via Kantorovich's technique of majorization. The main challenge in this context is to determine the convergence domain of the corresponding majorant generator. As it turns out, dealing with this task leads to solution of functional equations of a certain kind. After considering several examples, we formulate two generic models and develop an approach to their solution.