In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel–Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the ‘branch cut’ technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for the computation of the coefficients appearing in the asymptotic expansions.

We consider a two-node queue modeled as a two-dimensional random walk. In particular, we consider the case that one or both queues have finite buffers. We develop an approximation scheme based on the Markov reward approach to error bounds in order to bound performance measures of such random walks. The approximation scheme is developed in terms of a perturbed random walk in which the transitions along the boundaries are different from those in the original model and the invariant measure of the perturbed random walk is of product-form. We then apply this approximation scheme to a tandem queue and some variants of this model, for the case that both buffers are finite. The modified approximation scheme and the corresponding applications for a two-node queueing system in which only one of the buffers has finite capacity have also been discussed.

We propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions at t = 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates under L2(I)-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

Metric regularity theory lies at the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. This paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

In this paper, we give some Łojasiewicz-type inequalities for continuous definable functions in an o-minimal structure. We also give a necessary and sufficient condition for the existence of a global error bound and the relationship between the Palais–Smale condition and this global error bound. Moreover, we give a Łojasiewicz nonsmooth gradient inequality at infinity near the fibre for continuous definable functions in an o-minimal structure.

We present generalized and unified families of (2n)-point and (2n — 1)-point p-ary interpolating subdivision schemes originated from Lagrange polynomial for any integers n ≥ 2 and p ≥ 3. Almost all existing even-point and odd-point interpolating schemes of lower and higher arity belong to this family of schemes. We also present tensor product version of generalized and unified families of schemes. Moreover error bounds between limit curves and control polygons of schemes are also calculated. It has been observed that error bounds decrease when complexity of the scheme decrease and vice versa. Furthermore, error bounds decrease with the increase of arity of the schemes. We also observe that in general the continuity of interpolating scheme do not increase by increasing complexity and arity of the scheme.

Poisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L2-energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes. The theory is illustrated by numerical examples.

Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for potential flows in affinely parametrized geometries. We review the essential ingredients: i) a Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold” in order to get a dimension reduction; ii) an efficient and effective greedy sampling method for identification of optimal and numerically stable approximations to have a rapid convergence; iii) an a posteriori error estimation procedure: rigorous and sharp bounds for the linear-functional outputs of interest and over the potential solution or related quantities of interest like velocity and / or pressure; iv) an Offline-Online computational decomposition strategies to achieve a minimum marginal computational cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present three illustrative results for inviscid potential flows in parametrized geometries representing a Venturi channel, a circular bend and an added mass problem.

We discuss, in this paper, a common flux-free method for the computation of strict errorbounds for linear and nonlinear finite-element computations. In the linear case, the errorbounds are on the energy norm of the error, while, in the nonlinear case, the concept oferror in constitutive relation is used. In both cases, the error bounds are strict in thesense that they refer to the exact solution of the continuous equations, rather than tosome FE computation over a refined mesh. For both linear and nonlinear solid mechanics,this method is based on the computation of a statically admissible stress field, which isperformed as a series of local problems on patches of elements. There is no requirement tosolve a previous problem of flux equilibration globally, as happens with othermethods.

Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we givecharacterizationsof the existence of so-called global and local error bounds, for lowersemicontinuous functions defined on complete metric spaces. We thusprovide asystematic and synthetic approach to the subject, emphasizing the specialcaseof convex functions defined on arbitrary Banach spaces (refining theabstract partof Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization of the local metric regularityof closed-graph multifunctions between complete metric spaces.

This paper studies the issue of well-posednessfor vector optimization. It is shown thatcoercivity implies well-posedness without any convexity assumptionson problem data.For convex vector optimization problems,solution sets of such problems are non-convex in general,but they are highly structured. By exploring such structures carefully via convex analysis,we are able to obtaina number of positive results, including a criterion for well-posedness in terms of that of associated scalar problems.In particularwe show that a well-known relative interiority conditioncan be used as a sufficient condition for well-posedness in convexvector optimization.

We present in this article two components: these components can in fact serve various goalsindependently, though we consider them here as an ensemble. The first component is a technique forthe rapid and reliable evaluation prediction of linear functional outputs of elliptic (andparabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent globalreduced–basis approximations — Galerkin projection onto a spaceW N spanned by solutions of the governing partial differentialequation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error–residualequation that provide inexpensive yet sharp and rigorous bounds forthe error in the outputs of interest; and (iii) off–line/on–linecomputational procedures — methods which decouple the generationand projection stages of the approximation process. This component is ideally suited — consideringthe operation count of the online stage — for the repeated and rapid evaluation required in thecontext of parameter estimation, design, optimization, andreal–time control. The second component is a framework for distributed simulations. This frameworkcomprises a library providing the necessary abstractions/concepts for distributed simulations and asmall set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of thosesimulations. While the library is the backbone of the framework and is therefore general, thevarious interfaces answer specific needs. We shall describe both components and present how theyinteract.

In this paper we consider the continuous-time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will discuss how the discrete time approximating model of the Kushner-Stratonovitch equation and the genetic type interacting particle system approximation combine. We present quenched error bounds as well as mean order convergence results.

In an earlier note the present author deduced bounds for the approximation error of stop loss premiums when the aggregate claims distribution is calculated by a method introduced by Bertram. From the error bounds of the stop loss premiums we deduced bounds for the approximation error of the cumulative distribution and the discrete density of the aggregate claims. In the present note we shall improve the bounds for the cumulative distribution and the discrete density.

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

The queues being studied here are of the type GI/G/k in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Köllerström (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.