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Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals. [ABSTRACT FROM AUTHOR]

This paper deals with variational inclusions of the form 0∈K−f(x) where f:Rn→Rm is a semismooth function and K is a nonempty closed convex cone in Rm. We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of f. The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341-347, 1972). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence. [ABSTRACT FROM AUTHOR]

The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion. The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation. Thus, the second motivation of this paper is to discuss some 'mysterious' configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion. [ABSTRACT FROM AUTHOR]

The Conway-Maxwell-Poisson distribution is a two-parameter generalization of the Poisson distribution that can be used to model data that are under- or over-dispersed relative to the Poisson distribution. The normalizing constant Z(λ,ν) is given by an infinite series that in general has no closed form, although several papers have derived approximations for this sum. In this work, we start by using probabilistic argument to obtain the leading term in the asymptotic expansion of Z(λ,ν) in the limit λ→∞ that holds for all ν>0. We then use an integral representation to obtain the entire asymptotic series and give explicit formulas for the first eight coefficients. We apply this asymptotic series to obtain approximations for the mean, variance, cumulants, skewness, excess kurtosis and raw moments of CMP random variables. Numerical results confirm that these correction terms yield more accurate estimates than those obtained using just the leading-order term. [ABSTRACT FROM AUTHOR]

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently by Svensson. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new technique for the asymmetric approximation of discrete functions arising in seasonal customer demand extrapolation. We adapt the technique for two different settings, the so-called pull and push models. Our main goal here is to find effectively extrapolations minimizing the loss. For bothmodels, we discuss several features related to sampling, approximation, and extrapolation. [ABSTRACT FROM AUTHOR]

A typical interpolation strategy for line segments consists of a transition scheme, a Look-ahead ACC/DEC scheduling, and an interpolation algorithm. In these three parts, the main computation occurs in the first and second part. Some research work has been carried out to decrease the computation in the previous literatures, but these methods occupy a lot of computing resources for the optimization process during the calculation of transition curve parameters and feed rates. Consequently, the computational efficiency of interpolation strategy is greatly reduced. To deal with the issue, a real-time interpolation strategy is proposed in this paper. In the transition scheme, a Bézier curve is utilized to smooth the line segments. Based on the relationship among the approximation error, the approximation radius, and the transition curve, the curve can be directly generated when the approximation error is given. In the ACC/DEC scheduling, a 3-segment feed rate profile with jerk continuity is constructed. Meanwhile, a Look-ahead planning based on Backward Scanning and Forward Revision (BSFR) algorithm is utilized to eliminate redundant computation. Compared with Zhao’s and Shi’s strategy, the proposed strategy has the merits of C2 and G2 continuity for the tool path, jerk continuity for the tool movement, and distinguished real-time performance for interpolation. The experiments of 3D pentagram and 2D butterfly are carried out with different strategies and their results demonstrate that the interpolation efficiency can be greatly improved with the proposed strategy. [ABSTRACT FROM AUTHOR]

In this paper and for the first time in the literature, we build a new hybrid symmetric two-step method with the following properties: (1) the new scheme is of symmetric type, (2) the new scheme is of two-step, (3) the new scheme is of five-stages, (4) the new scheme is of twelfth-algebraic order, (5) the new scheme has eliminated the phase-lag and its first, second, third, fourth and fifth derivatives, (6) the new scheme has improved stability characteristics for the general problems, (7) the new scheme is P-stable [with interval of periodicity equal to 0,∞] and (8) the new scheme builded based on the following approximations:the first stage is approximation on the point xn-1,the second stage is approximation on the point xn-1,the third stage is approximation on the point xn-1,the fourth stage is approximation on the point xn and finally,the fifth stage is approximation on the point xn+1, For the new builded scheme we give a full numerical analysis (local truncation error and stability analysis). The efficiency of the new builded scheme is examined with the numerical solution of systems of coupled differential equations of the Schrödinger type. [ABSTRACT FROM AUTHOR]

The volume under a surface (VUS) is an effective measure for evaluating the discriminating power of a diagnostic test with three ordinal diagnostic groups. In this paper, we investigate the difference of two correlated VUS’s to compare two treatments for discrimination of three-class classification data. A jackknife empirical likelihood (JEL) procedure is employed to avoid the variance estimation in the existing methods. We prove that the limiting distribution of the empirical log-likelihood ratio statistic follows a χ2 distribution. Extensive numerical studies show that the JEL confidence intervals outperform those based on the normal approximation method. The proposed method is also applied to the Alzheimer’s disease data. [ABSTRACT FROM AUTHOR]

Krylov subspace methods for approximating a matrix function f( A) times a vector v are analyzed in this paper. For the Arnoldi approximation to e v, two reliable a posteriori error estimates are derived from the new bounds and generalized error expansion we establish. One of them is similar to the residual norm of an approximate solution of the linear system, and the other one is determined critically by the first term of the error expansion of the Arnoldi approximation to e v due to Saad. We prove that each of the two estimates is reliable to measure the true error norm, and the second one theoretically justifies an empirical claim by Saad. In the paper, by introducing certain functions ϕ( z) defined recursively by the given function f( z) for certain nodes, we obtain the error expansion of the Krylov-like approximation for f( z) sufficiently smooth, which generalizes Saad's result on the Arnoldi approximation to e v. Similarly, it is shown that the first term of the generalized error expansion can be used as a reliable a posteriori estimate for the Krylov-like approximation to some other matrix functions times v. Numerical examples are reported to demonstrate the effectiveness of the a posteriori error estimates for the Krylov-like approximations to e v, cos( A) v and sin( A) v. [ABSTRACT FROM AUTHOR]

In this work an improved numerical solution of the singular boundary integral equation of the 2D compressible fluid flow around obstacles is obtained by a boundary element method based on modified shape functions and cubic boundary elements. The singular boundary integral equation with sources distribution is considered in this paper, and for its discretization cubic boundary elements are used. The integrals of singular kernels are evaluated using modified shape functions which are deduced by using series expansions for the basis functions we choose for the local approximation models. A computer code is made using Mathcad programming language and, based on it, some particular cases are solved. In order to validate the proposed method, comparisons between numerical solutions and exact ones are performed for the considered test problems. The advantage of using modified shape functions for evaluating the singularities is pointed out through a comparison study between the numerical solution obtained by the method proposed in this paper and the one obtained by using a truncation method for evaluating the singularities. [ABSTRACT FROM AUTHOR]

Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. [21] for solving time fractional partial differential equation. We prove that this method has the convergence order O(τ3−α) for all α ∈ (0, 1) when the first and second derivatives of the solution are vanish at t = 0, where τ is the time step size and α is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. [21]. We show that this new method also has the convergence order O(τ3−α) for all α ∈ (0, 1). The proofs of the error estimates are based on the energy method developed recently by Lv and Xu [26]. We also consider the space discretization by using the finite element method. Error estimates with convergence order O(τ3−α + h2) are proved in the fully discrete case, where h is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

The paper is concerned with the problem of generalized spline interpolation of functions having large-gradient regions. Splines of the class C2, represented on each interval of the grid by the sum of a second-degree polynomial and a boundary layer function, are considered. The existence and uniqueness of the interpolation L-spline are proven, and asymptotically exact two-sided error estimates for the class of functions with an exponential boundary layer are obtained. It is established that the cubic and parabolic interpolation splines are limiting for the solution of the given problem. The results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

In this paper, we first present a new finite difference scheme to approximate the time fractional derivatives, which is defined in the sense of Caputo, and give a semidiscrete scheme in time with the truncation error O((Δt)3-α), where Δt is the time step size. Then a fully discrete scheme based on the semidiscrete scheme for the fractional Cattaneo equation in which the space direction is approximated by a local discontinuous Galerkin method is presented and analyzed. We prove that the method is unconditionally stable and convergent with order O(hk+1 + (Δt)3-α), where k is the degree of piecewise polynomial. Numerical examples are also given to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

Recently, a mixed hybrid operator, generalizing the well-known Phillips operators and Baskakov–Szász type operators, was introduced. In this paper, we study Bézier variant of these new operators. We investigate the degree of approximation of these operators by means of the Lipschitz class function, the modulus of continuity, and a weighted space. We study a direct approximation theorem by means of the unified Ditzian–Totik modulus of smoothness. Furthermore, the rate of convergence for functions having derivatives of bounded variation is discussed. [ABSTRACT FROM AUTHOR]

Older and contemporary CAD-based interpolations, either for surfaces or for volume blocks, are capable of creating sets of basis functions on which finite-element (Galerkin–Ritz) and global collocation procedures can be supported. For some of these interpolations, this paper investigates the quality of the relevant numerical solution in several 2D and 3D engineering problems. It is shown that the global character of all these CAD interpolations ensures excellent numerical solution, although somewhere the boundary may be slightly violated. The study deals with several benchmark tests that span a large part in the spectrum of engineering analysis, from potential problems (Poisson equation-electrostatics and acoustics) to elasticity ones (beam in torsion, plate bending: statics and dynamics). [ABSTRACT FROM AUTHOR]

The paper presents a numerical solution of the oblique derivative boundary value problem on and above the Earth's topography using the finite volume method (FVM). It introduces a novel method for constructing non-uniform hexahedron 3D grids above the Earth's surface. It is based on an evolution of a surface, which approximates the Earth's topography, by mean curvature. To obtain optimal shapes of non-uniform 3D grid, the proposed evolution is accompanied by a tangential redistribution of grid nodes. Afterwards, the Laplace equation is discretized using FVM developed for such a non-uniform grid. The oblique derivative boundary condition is treated as a stationary advection equation, and we derive a new upwind type discretization suitable for non-uniform 3D grids. The discretization of the Laplace equation together with the discretization of the oblique derivative boundary condition leads to a linear system of equations. The solution of this system gives the disturbing potential in the whole computational domain including the Earth's surface. Numerical experiments aim to show properties and demonstrate efficiency of the developed FVM approach. The first experiments study an experimental order of convergence of the method. Then, a reconstruction of the harmonic function on the Earth's topography, which is generated from the EGM2008 or EIGEN-6C4 global geopotential model, is presented. The obtained FVM solutions show that refining of the computational grid leads to more precise results. The last experiment deals with local gravity field modelling in Slovakia using terrestrial gravity data. The GNSS-levelling test shows accuracy of the obtained local quasigeoid model. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

In this paper, we present a new sharp approximation for the gamma function via the tri-gamma function. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the inequalities related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation. [ABSTRACT FROM AUTHOR]

In this paper, we present an alternative approach to a priori $$L^\infty $$ -error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all. [ABSTRACT FROM AUTHOR]

A new scheme is developed in this paper, for the first time in the literature. The new scheme: (1) is a symmetric two-step method, (2) is of three-stages scheme, (3) is a high order method (i.e of eight-algebraic order), (4) the approximations of the layers are taken place as follows: first layer on the point $$x_{n-1}$$ , second layer on the point $$x_{n}$$ , third layer on the point $$x_{n+1}$$ , (5) has vanished the phase-lag and its derivatives up to order four, (6) has good interval of periodicity properties [i.e. interval of periodicity equal to (0, 9.8)]. A detailed theoretical analysis is also presented. More specifically we present: (1) the development of the new method, (2) comparative error analysis (3) stability analysis. The effectiveness of the new scheme is tested via the solution of systems of coupled differential equations of the Schrödinger type. [ABSTRACT FROM AUTHOR]

In a recent paper (Appl. Math. Comput. 215:1622-1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the convergence acceleration theorem in the case of summation of generalized hypergeometric series. Some new theoretical results and illustrative numerical examples are given. [ABSTRACT FROM AUTHOR]

In the current paper, a powerful approximate analytical approach namely the global residue harmonic balance method (GRHBM) is proposed for obtaining higher-order approximate frequency and periodic solution of nonlinear conservative oscillatory systems arising in engineering problems. The proposed method has a main difference with other traditional harmonic balance methods such that the residual errors obtained in pervious order approximation are used in the present one. Comparison of the obtained results with the exact and numerical solution as well as well-known analytical methods such as Hamiltonian approach, Max-Min approach, variational approach, and He's amplitude-frequency formulation reveals the correctness and usefulness of the GRHBM. It is shown that the results are valid for different values of system parameters and both small and large amplitudes. Hence, the method can be easily applied to other strongly nonlinear conservative oscillatory systems. Furthermore, using the obtained analytical expressions, the effect of amplitude and system parameters on nonlinear frequency is studied. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an efficient demosaicking method based on an advanced nonlocal mean (NLM) filter using adaptive weight with consideration of both neighborhood similarity and patch distance. The to-be-interpolated missing color information is considered to be populated by the weight average of directional estimation in six directions: north, south, west, east, pro-diagonal, and anti-diagonal. The NLM sets adaptive weight by obtaining neighborhood similarity between a missing color pixel and the given neighbor color pixels, and similar local neighborhoods are selected to calculate the adaptive weight. The proposed advanced NLM filter is designed to assign weight as a combination of NLM and spatial location distance. The nearer a pixel to a missing pixel, the larger is its chance of having a local structure similar to that of the to-be-interpolated pixel. Considering this evidence and from the viewpoint of reducing computational complexity, we employ location distance to control weights. The new advanced NLM filter is better parametrized, and the weights therein are impacted both by patch similarity and location distance. We present experimental results in terms of CPSNR, S-CIELAB $$\Delta $$ E*, and FSIM to clarify the performance of the proposed algorithm objectively and subjectively. The experimental results indicate that the proposed method outperforms existing approaches both objectively and subjectively. [ABSTRACT FROM AUTHOR]

This paper studies level set topology optimization of structures predicting the structural response by the eXtended Finite Element Method (XFEM). In contrast to Ersatz material approaches, the XFEM represents the geometry in the mechanical model by crisp boundaries. The traditional XFEM approach augments the approximation of the state variable fields with a fixed set of enrichment functions. For complex material layouts with small geometric features, this strategy may result in interpolation errors and non-physical coupling between disconnected material domains. These defects can lead to numerical instabilities in the optimized material layout, similar to checker-board patterns found in density methods. In this paper, a generalized Heaviside enrichment strategy is presented that adapts the set of enrichment functions to the material layout and consistently interpolates the state variable fields, bypassing the limitations of the traditional approach. This XFEM formulation is embedded into a level set topology optimization framework and studied with 'material-void' and 'material-material' design problems, optimizing the compliance via a mathematical programming method. The numerical results suggest that the generalized formulation of the XFEM resolves numerical instabilities, but regularization techniques are still required to control the optimized geometry. It is observed that constraining the perimeter effectively eliminates the emergence of small geometric features. In contrast, smoothing the level set field does not provide a reliable geometry control but mainly improves the convergence rate of the optimization process. [ABSTRACT FROM AUTHOR]

In this paper, we present an existence of solution for a functional integro-differential equation with an integral boundary condition arising in chemical engineering, underground water flow and population dynamics, and other field of physics and mathematical chemistry. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo's theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a powerful numerical approach based on Sinc approximation to solve the equation. Then convergence of this technique is discussed by preparing a theorem which shows exponential type convergence rate and guarantees the applicability of that. Finally, some numerical examples are given to confirm efficiency and accuracy of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min–max problems which arise in the context of semi-infinite programming. Here, our main motivation are nonlinear inequality constrained robust optimization problems. In the first part of the paper, we propose a conservative approximation strategy for such nonlinear and non-convex robust optimization problems: under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given, we show how to formulate a lower-level concave min–max problem which approximates the robust counterpart in a conservative way. This approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties. In the second part of the paper, we review existing theory on optimality conditions for nonlinear lower-level concave min–max problems which arise in the context of semi-infinite programming. Regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem, we end up with a structured mathematical program with complementarity constraints (MPCC). The special hierarchical structure of this MPCC can be exploited in a novel sequential convex bilevel programming algorithm. We discuss the surprisingly strong global and locally quadratic convergence properties of this method, which can in this form neither be obtained with existing SQP methods nor with interior point relaxation techniques for general MPCCs. Finally, we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, a kinetic method to compute approximative solutions of the one-dimensional open channel flow equations with a varying cross-sectional area and a varying bottom profile is proposed. The scheme preserves the steady states at rest, keeps the water height non-negative and is able to handle dry channel beds. These three properties are essential indicators for the quality of a numerical scheme. The ability of the scheme to treat varying cross-sectional areas along the channel is an important advancement in comparison to previous papers. Further the same equations are considered on a network by coupling the equations at the nodes. The main difference to the majority of common numerical hydraulic models is the usage of the energy equality as coupling condition which seems to be more realistic than conditions based on the water height. Moreover, we set up a numerical method for computing subcritical flow on such networks. Several examples are treated to illustrate the results. [ABSTRACT FROM AUTHOR]

In the paper, the verification of the limit load solutions proposed by EPRI procedures for single edge notched plate under tension (SEN(T)) is presented. For the concept of limit load of the component containing a crack, the force (or torque or pressure) which causes a full plasticity of the uncracked ligament of the structural component must be understood. It should be noted that the value of the limit load is determined under the assumption of elastic–perfectly plastic material. Numerical calculations presented in the paper (FEM) and analysis of the obtained FEM results were used to recalculate existing limit load formulas proposed by EPRI procedures for plane strain and plane stress states. On the basis of numerical calculations and verifications of the present solutions (EPRI solutions), in the paper new analytical formulas for better estimating the limit load value for SEN(T) specimen are presented. The measurable effect of the paper is a catalog of the numerical solutions and their approximation, which may be useful in engineering analysis. [ABSTRACT FROM AUTHOR]

In this paper we investigate the degree of approximation of a function belonging to the generalized Zygmund class $Z_{p}^{(\omega)}$ ( $p \ge1$ ) by Hausdorff means of its Fourier series. We also deduce a corollary and mention a few applications of our main results. [ABSTRACT FROM AUTHOR]

Today artificial neural networks are applied in various fields-engineering, data analysis, robotics. While they represent a successful tool for a variety of relevant applications, mathematically speaking they are still far from being conclusive. In particular, they suffer from being unable to find the best configuration possible during the training process (local minimum problem). In this paper, we focus on this issue and suggest a simple, but effective, post-learning strategy to allow the search for improved set of weights at a relatively small extra computational cost. Therefore, we introduce a novel technique based on analogy with quantum effects occurring in nature as a way to improve (and sometimes overcome) this problem. Several numerical experiments are presented to validate the approach. [ABSTRACT FROM AUTHOR]

Tensor-compressed numerical solution of elliptic multiscale-diffusion and high frequency scattering problems is considered. For either problem class, solutions exhibit multiple length scales governed by the corresponding scale parameter: the scale of oscillations of the diffusion coefficient or smallest wavelength, respectively. As is well-known, this imposes a scale-resolution requirement on the number of degrees of freedom required to accurately represent the solutions in standard finite-element (FE) discretizations. Low-order FE methods are by now generally perceived unsuitable for high-frequency coefficients in diffusion problems and high wavenumbers in scattering problems. Accordingly, special techniques have been proposed instead (such as numerical homogenization, heterogeneous multiscale method, oversampling, etc.) which require, in some form, a-priori information on the microstructure of the solution. We analyze the approximation properties of tensor-formatted, conforming first-order FE methods for scale resolution in multiscale problems without a-priori information. The FE methods are based on the dynamic extraction of principal components from stiffness matrices, load and solution vectors by the quantized tensor train (QTT) decomposition. For prototypical model problems, we prove that this approach, by means of the QTT reparametrization of the FE space, allows to identify effective degrees of freedom to replace the degrees of freedom of a uniform 'virtual' (i.e. never directly accessed) mesh, whose number may be prohibitively large to realize computationally. Precisely, solutions of model elliptic homogenization and high-frequency acoustic scattering problems are proved to admit QTT-structured approximations whose number of effective degrees of freedom required to reach a prescribed approximation error scales polylogarithmically with respect to the reciprocal of the target Sobolev-norm accuracy ε with only a mild dependence on the scale parameter. No a-priori information on the nature of the problems and intrinsic length scales of the solution is required in the numerical realization of the presently proposed QTT-structured approach. Although only univariate model multiscale problems are analyzed in the present paper, QTT structured algorithms are applicable also in several variables. Detailed numerical experiments confirm the theoretical bounds. As a corollary of our analysis, we prove that for the mentioned model problems, the Kolmogorov n-widths of solution sets are exponentially small for analytic data, independently of the problems' scale parameters. That implies, in particular, the exponential convergence of reduced basis techniques which is scale-robust, i.e., independent of the scale parameter in the problem. [ABSTRACT FROM AUTHOR]

Although numerical methods of nonlinear stochastic differential delay equations (SDDEs) have been discussed by several authors, there is so far little work on the numerical approximation of SDDE with coefficients of polynomial growth. The main aim of the paper is to investigate convergence in probability of the Euler-Maruyama (EM) approximate solution for SDDE with one-sided polynomial growing drift coefficient and polynomial growing diffusion coefficient. Moreover, we prove the existence-and-uniqueness of almost surely exponentially stable global solution for this nonlinear stochastic delay system. Finally, a computer simulation confirms the efficiency of our numerical method. [ABSTRACT FROM AUTHOR]

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Itô or the Stratonovich sense. The proof of convergence of the Euler scheme, which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge. [ABSTRACT FROM AUTHOR]

The aim of the paper is to determine the degree of approximation of functions by matrix means of their Fourier series in a new space of functions introduced by Das, Nath, and Ray. In particular, we extend some results of Leindler and some other results by weakening the monotonicity conditions in the results obtained by Singh and Sonker for some classes of numerical sequences introduced by Mohapatra and Szal and present new results by using matrix means. [ABSTRACT FROM AUTHOR]

This paper build a structure of fuzzy neural network, which is well sufficient to gain a fuzzy interpolation polynomial of the form $$y_{p}=a_{n}x_{p}^n+ \cdots +a_{1}x_{p}+a_{0}$$ where $$a_{j}$$ is crisp number (for $$j=0,\ldots ,n)$$ , which interpolates the fuzzy data $$(x_{j},y_{j})\,(for\,j=0,\ldots ,n)$$ . Thus, a gradient descent algorithm is constructed to train the neural network in such a way that the unknown coefficients of fuzzy polynomial are estimated by the neural network. The numeral experimentations portray that the present interpolation methodology is reliable and efficient. [ABSTRACT FROM AUTHOR]

We study the effect of the approximation viscosity of Godunov-type difference schemes in a model problem on the accretion disk in a binary star system. Computations are carried out by schemes of the first and increased approximation order. In the higher-order scheme, we take into account the viscous stress tensor, which is used to model the influence of the approximation viscosity observed in the first-order scheme. It is noteworthy that the approximation viscosity can lead to global qualitative changes in the flows to be modeled. The numerical results given in the paper are of independent interest in that they illustrate some new specific features of gasdynamic flows in this classical problem of modern astrophysics. [ABSTRACT FROM AUTHOR]

In literature, there is no exact analytical solution available for determining the radius of Roche equipotential surfaces of distorted close binary systems in synchronous rotation. However, Kopal (Roche Model and Its Application to Close Binary Systems, Advances in Astronomy and Astrophysics, Academic Press, New York ) and Morris (Publ. Astron. Soc. Pac. 106:154, ) have provided the approximate analytical solutions in the form of infinite mathematical series. These series expressions have been commonly used by various authors to determine the radius of the Roche equipotential surfaces, and hence the equilibrium structures of rotating stars and stars in the binary systems. However, numerical results obtained from these approximating series expressions are not very accurate. In the present paper, we have expanded these series expressions to higher orders so as to improve their accuracy. The objective of this paper is to check, whether, there is any effect on the accuracy of these series expressions when the terms of higher orders are considered. Our results show that in most of the cases these expanded series give better results than the earlier series. We have further used these expanded series to find numerically the volume radius of the Roche equipotential surfaces. The obtained results are in good agreement with the results available in literature. We have also presented simple and accurate approximating formulas to calculate the radius of the primary component in a close binary system. These formulas give very accurate results in a specified range of mass ratio. [ABSTRACT FROM AUTHOR]

The increasing resolution of contemporary regional numerical weather prediction (NWP) models, reaching horizontal grid sizes of O(1 km), requires robust and reliable dynamical cores, working well beyond the approximation of quasi-horizontal flows. That stimulates an interest in an application for NWP purposes of dynamical cores based on the anelastic, or - more generally - sound-proof flow equations, and characterized by appropriate robustness and reliability. The paper presents results from testing the dynamical core of EULAG, the anelastic research model for multi-scale flows, as a prospective NWP dynamical core. The model simulates the semi-realistic frictionless and adiabatic flow over realistic steep Alpine topographies, employing horizontal grid sizes of 2.2, 1.1, and 0.55 km. The paper demonstrates not only the numerical robustness of EULAG, but also studies the influence of the varying horizontal resolution on the simulated flow. Results show that the increased horizontal resolution increases orographic drag on the flow. While the general flow pattern remains the same, increased resolution influences the flow on scales from hundreds of kilometers to meso-gamma scales. The differences are especially apparent in the near-surface layer of 1.5 to 3 km deep, and in the distribution and amplitudes of the orographically-induced gravity waves. [ABSTRACT FROM AUTHOR]

The kernel method suffers from the following problem: the computational efficiency of the feature extraction procedure is inversely proportional to the size of the training sample set. In this paper, from a novel viewpoint, we propose a very simple and mathematically tractable method to produce the computationally efficient kernel-method-based feature extraction procedure. We first address the issue that how to make the feature extraction result of the reformulated kernel method well approximate that of the naïve kernel method. We identify these training samples that statistically contribute much to the feature extraction results and exploit them to reformulate the kernel method to produce the computationally efficient kernel-method-based feature extraction procedure. Indeed, the proposed method has the following basic idea: when one training sample has little effect on the feature extraction result and statistically has the high correlation with regard to all the training samples, the feature extraction term associated with this training sample can be removed from the feature extraction procedure. The proposed method has the following advantages: First, it proposes, for the first time, to improve the kernel method through formal and reasonable evaluation on the feature extraction term. Second, the proposed method improves the kernel method at a low extra cost and thus has a much more computationally efficient training phase than most of the previous improvements to the kernel method. The experimental comparison shows that the proposed method performs well in classification problems. This paper also intuitively shows the geometrical relation between the identified training samples and other training samples. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical approximate solutions of singular fourth order four-point boundary value problems using reproducing kernel method (RKM). The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the RKM can not be used directly to solve singular fourth order four-point boundary value problems (BVPs), since there is no method of obtaining reproducing kernel (RK) satisfying four-point boundary conditions. The aim of this paper is to fill this gap. A method for obtaining RK satisfying four-point boundary conditions is proposed so that RKM can be used to solve singular fourth order four-point BVPs. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular fourth order four-point BVPs. [ABSTRACT FROM AUTHOR]

tenth algebraic order eight-step method is developed in this paper. For this method we require the phase-lag and its first and second derivatives to be vanished. A comparative error analysis and a comparative stability analysis are also presented in this paper. The new proposed method is applied for the numerical solution of the one-dimensional Schrödinger equation. The efficiency of the new methodology is proved via the theoretical analysis and the numerical applications. General conclusions about the importance of several properties on the construction of numerical algorithms for the approximate solution of the radial Schrödinger equation are also presented. [ABSTRACT FROM AUTHOR]

In this paper, Homotopy Analysis Method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator $\mathcal{L}$. It can be seen in this paper that the auxiliary parameter $\hbar,$ which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions. [ABSTRACT FROM AUTHOR]

In this paper, we shall introduce and study a family of multivariate interpolating refinable function vectors with some prescribed interpolation property. Such interpolating refinable function vectors are of interest in approximation theory, sampling theorems, and wavelet analysis. In this paper, we characterize a multivariate interpolating refinable function vector in terms of its mask and analyze the underlying sum rule structure of its generalized interpolatory matrix mask. We also discuss the symmetry property of multivariate interpolating refinable function vectors. Based on these results, we construct a family of univariate generalized interpolatory matrix masks with increasing orders of sum rules and with symmetry for interpolating refinable function vectors. Such a family includes several known important families of univariate refinable function vectors as special cases. Several examples of bivariate interpolating refinable function vectors with symmetry will also be presented. [ABSTRACT FROM AUTHOR]

The paper is devoted to Koosis' approach to the Carleson theorem. We show that this approach also works for other related questions. The main emphasis in this paper is on the method. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

To meet the requirement of exploring a new generation of CNC systems based on STEP-NC, NURBS interpolation has been studied. In contrast to existing NURBS interpolation based on the Taylor’s expansion, this paper proposes a new method for NURBS interpolation which is based on exponential smoothing forecasting. The new method is very simple; it decreases the computational load observably, and speed and precision control are also achieved. This paper presents relative formulas and main computational processes, including speed and precision control, computation of real-time mapping, computation of real-time track, etc. The simulation results indicate that the method is feasible and has high-precision and better real-time performance, which is sufficient for the STEP-NC system. [ABSTRACT FROM AUTHOR]

Abstract  This paper deals with the Perron root of nonnegative irreducible matrices, all of whose entries are continuous bijective functions of some parameter vector. It is known that if the functions are log-convex, then the Perron root is a convex function of the parameter vector. In this paper, we strengthen this result by showing that the log-convexity property is also necessary when the Perron root is required to be convex for any nonnegative irreducible matrix. Furthermore, we show that a less restrictive requirement is sufficient when the matrix is confined to belong to two subsets of irreducible matrices. In particular, in case of positive semidefinite matrices, convexity is sufficient for the Perron root to be convex. [ABSTRACT FROM AUTHOR]

In this paper we describe the implementation of the charged current decays of the type t→bl+νl(γ) in the framework of the SANC system. All calculations are done taking into account the one-loop electroweak correction in the standard model. The emphasis of this paper is on the presentation of numerical results. Various distributions are produced by means of a Monte Carlo integrator and event generator. Comparison with the results of the CompHEP and PYTHIA packages are presented for the Born and hard photon contributions. The validity of the cascade approximation at one-loop level is also studied. [ABSTRACT FROM AUTHOR]

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by $${2\frac{kd^2}{n}}$$ , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation $${U_{\mu +\nu} \subset U_\mu \otimes U_\nu}$$ of the unitary group U( d), for highest weights μ, ν and μ + ν. Let ξμ be the state obtained by tracing out U ν. Then ξμ is close to a convex combination of the coherent states $${U_\mu(g)|{v_\mu\rangle}}$$ , where $${g\in U(d)}$$ and $${|v_\mu\rangle}$$ is the highest weight vector in U μ. For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general. [ABSTRACT FROM AUTHOR]

The paper discusses several versions of the method of shortest residuals, a specific variant of the conjugate gradient algorithm, first introduced by Lemaréchal and Wolfe and discussed by Hestenes in a quadratic case. In the paper we analyze the global convergence of the versions considered. Numerical comparison of these versions of the method of shortest residuals and an implementation of a standard Polak–Ribière conjugate gradient algorithm is also provided. It supports the claim that the method of shortest residuals is a viable technique, competitive to other conjugate gradient algorithms. [ABSTRACT FROM AUTHOR]