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This paper proposes an operating performance index (OPI) based directly on the Six Sigma process quality index. We considered the following case: the number of customers arriving at a store follows a Poisson process of rate λ and the sequence of inter-arrival times is an exponential random variable with mean τ , where τ = 1/ λ. In operating performance assessments, using the statistical testing to evaluate the OPI can lower the risk of misjudgment caused by sampling error. A smaller sample size entails a larger sampling error, while a larger sample size entails a smaller sampling error. Thus, using critical value for statistical testing can result in inconsistencies for different sample size. We therefore referred to the one-tailed fuzzy method proposed by Buckley (2005) for the parameters of population distributions and developed a two-tailed fuzzy test for the OPI to assess operating performance. The advantage of this approach is that conventional measurement methods can be used during data collection, and then the confidence interval of the OPI can be used to construct a fuzzy membership function for fuzzy testing. This not only lowers the risk of misjudgment caused by sampling error, but also enhances testing accuracy. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method is proposed to solve the nonlinear Bitsadze–Samarskii boundary value problems with multi-point boundary conditions. The algorithm is based on the reproducing kernel Hilbert space method. We use an iterative scheme to overcome the nonlinearity of the problem. The convergence and error estimate of the iterative scheme are established. The reproducing kernel Hilbert space method is used to generate an approximation of the linearized problem. In fact, the reproducing kernel Hilbert space method is combined with an iterative scheme to approximate the solution and an error estimate of the approximate solution is derived. In order to show the efficiency and versatility of the proposed method, some numerical results are reported. The comparison of numerical results with the analytical solution and the best results reported in the literature confirms the good accuracy and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

We consider singularly perturbed differential-difference equation, which contains negative shift in the differentiated term with boundary layer behavior. RKHSM (Reproducing Kernel Hilbert Space Method) without Gram–Schmidt orthogonalization process, is considered in the present paper. We decompose the domain of the problem into two subintervals. One of them has not the boundary layer and the other one has. The side of the interval in which the boundary layer exists is important. If the boundary layer of this problem exists on the left side of interval, the RKHSM will provide a proper approximation of solution, otherwise for the implement of RKHSM, we need to change the variable of the singularly perturbed problem to shift the boundary layer region. [ABSTRACT FROM AUTHOR]

Different sources of imprecision and uncertainty are encountered in practical problems and, thus, many elements may need to be imprecisely observed, defined or treated. In this setting, one of the most successfully applied techniques to describe possible relationships between fuzzy variables is the regression methodology. In this paper, we introduce a fuzzy regression procedure involving a class a fuzzy numbers defined by some level sets called finite fuzzy numbers. We give a characterization of the image of a finite fuzzy number in terms of the extremes of its level sets and we present a parametric family of fuzzy semidistances between them that let us to consider a total fuzzy error of estimation (described as a fuzzy sum of squares of residuals in particular cases). The estimation process consists in finding a regression model that minimizes, in a fuzzy sense, such fuzzy error. Although spreads of finite fuzzy numbers can take some values very close to zero, which complicate the task of finding nonnegative models, the presented algorithm is able to guarantee that the predicted response is a fuzzy variable. Finally a numerical example based on fuzzy economic data of China is given to illustrate the use of the proposed method. [ABSTRACT FROM AUTHOR]

For finite difference matrices that are monotone, a discretization error estimate in maximum norm follows from the truncation errors of the discretization. It enables also discretization error estimates for derivatives of the solution. These results are extended to convergent operator splittings of the difference matrix where the major, preconditioning part is monotone but the whole operator is not necessarily monotone. [ABSTRACT FROM AUTHOR]

Knowing the time of changes, called change-point (CP), in a process is crucial for engineers to recognize the root cause fast and accurately. Since special causes may induce simultaneous changes in mean and variance, detecting changes in both at once is required. Many methodologies in quality control were developed for detecting changes in either mean or variance only, and process parameters were assumed known often. However, they are rarely known exactly and a small estimation error may lead to unfavorable CP estimates. Fuzzy partitioning is better suited to cases of vague boundaries between two segments which appear very often in reality. A new mechanism, called fuzzy classification maximum likelihood change-point (FCML-CP) algorithm, is proposed to detect shifts in mean and variance simultaneously. A CP framework is transferred into a mixture model and then a FCML-CP algorithm is created through fuzzy classification maximum likelihood procedures. The proposed FCML-CP can be applied to phase I and II processes without knowledge of in-control process parameters; it can estimate multiple CPs of process mean or/and variance simultaneously. The effectiveness and superiority of FCML-CP are shown by extensive experiments with numerical and real data sets. Specifically, the proposed FCML-CP is superior to the commonly used statistical mixture likelihood approach using expectation–maximization (EM) algorithm; it is much more time-saving especially. The remarkable performance of FCML-CP in detecting CPs for small changes is particularly important and helpful for engineers to recognize the special cause fast and correctly since an out-of-control signal resulted from small changes is usually delayed long. [ABSTRACT FROM AUTHOR]

In this paper, a new conservative difference scheme is proposed for solving the nonlinear space-fractional Schrödinger equation. The Riesz space-fractional derivative is approximated by the second-order accurate weighted and shifted Grünwald difference operator. Building on a careful analysis of this operator, the conservation properties including the mass and energy are analyzed and some norm equivalences are established. Then the numerical solution is shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme in the l 2 norm, semi- H α / 2 norm and l ∞ norm are derived. The uniform bounds on the numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed. Numerical tests are performed to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we study the a priori error estimates of finite element method for the system of time-dependent Poisson–Nernst–Planck equations, and for the first time, we obtain its optimal error estimates in L ∞ ( H 1 ) and L 2 ( H 1 ) norms, and suboptimal error estimates in L ∞ ( L 2 ) norm, with linear element, and optimal error estimates in L ∞ ( L 2 ) norm with quadratic or higher-order element, for both semi- and fully discrete finite element approximations. Numerical experiments are also given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we first present a combined finite element-upwind finite volume method for a fully nonlinear convection-dominated diffusion reaction equation and derive its a priori optimal error estimate in H 1 -norm and sub-optimal error estimate in L 2 -norm for piecewise linear finite element combining with the first order upwind finite volume scheme. Then we study a type of two-grid method for the nonlinear convection-dominated transport equation together with the combined finite element-upwind finite volume method on the fine grid T h and the streamline diffusion finite element scheme on the coarse grid T H , which not only significantly reduces the computational cost on nonlinear iterations but also remains the numerical computation stabilized and the approximation accuracy unchanged. A priori error estimate of such designed two-grid method in H 1 -norm is proved to be O ( h + H 3 2 ) , showing that the two-grid method achieves the optimal approximation as long as the mesh sizes satisfy h = O ( H 3 2 ) . Finally, a numerical example is carried out to verify the accuracy and efficiency of the present numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a backward space-fractional diffusion problem. We propose an a posteriori parameter choice rule for the regularization method given in Zheng and Wei (2010), where the authors proposed a regularization method called convolution regularization method, and gave an a priori parameter choice strategy. In this paper, we study the same problem but give a new a posteriori parameter choice based on a modified version of the discrepancy principle, and obtain a log -type error estimate under an additional source condition. Numerical results show that our method is feasible. [ABSTRACT FROM AUTHOR]

In this paper, a novel L ∞ analysis for finite volume approximations of the Stokes problem is presented. Optimal order estimates in the L ∞ -norm for the velocity gradient and pressure of this problem are obtained by using new technical results. In particular, these optimal error estimates for the finite volume approximations are obtained without the presence of a usual logarithmic factor O ( | log h | ) for the stationary Stokes problem. Several numerical tests are performed to check these theoretical expectations. [ABSTRACT FROM AUTHOR]

The dual weighted residual method (DWR) and its localization for mesh adaptivity applied to elliptic partial differential equations are investigated. The contribution of this paper is twofold: first, we introduce a novel localization technique based on the introduction of a partition of unity. This new technique is very easy to apply, as neither strong residuals nor jumps over element edges are required. Second, we compare and analyze (theoretically and numerically) different localization techniques used for mesh adaptivity with respect to their effectivity. Here, we focus on localizations in variational formulations that do not require the evaluation of the corresponding differential operator in the classical strong formulation. In our mathematical analysis, we show for different localization techniques (established methods and our new approach) that the local error indicators used for mesh adaptivity converge with proper order in the error functional. Several numerical tests substantiate our theoretical investigations. [ABSTRACT FROM AUTHOR]

A formula of error estimation of Mann iteration is given in the case of strongly demicontractive mappings. Based on this estimation, a condition of strong convergence is obtained for the same class of mappings. T -stability for a particular case of strongly demicontractive mappings is proved. Some numerical examples showing the accuracy of the proposed estimations are also given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the solution of parabolic equation using the modified weak Galerkin finite element procedure, which is named as MWG-FEM, based on the conception of the modified weak derivative over discontinuous functions with heterogeneous properties, in which the classical gradient operator is replaced by a modified weak gradient operator. Optimal order error estimates in a discrete norm and norm are established for the corresponding modified weak Galerkin finite element solutions. Finally, we numerically verify the convergence theory for the MWG-FEM through some examples. [Copyright &y& Elsevier]

Abstract: In this paper we obtain a priori and a posteriori error estimates for stabilized low-order mixed finite element methods for the Stokes eigenvalue problem. We prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues. We define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove that, up to higher order terms, the estimator is equivalent to the energy norm of the error. We also present some numerical tests which show the performance of the adaptive scheme. [Copyright &y& Elsevier]

In this paper, we propose an efficient and practical implementation of the ensemble Kalman filter (EnKF) via the distribution-free Ledoit and Wolf (LW) covariance matrix estimator. Initially, we develop a tractable implementation of the LW estimator in high-dimensional probability spaces such as those found in the context of operational data assimilation. We employ this well-conditioned, full-rank covariance matrix estimator to approximate background error covariances and to mitigate the impact of spurious correlations during assimilation steps. The proposed formulation can be coupled within the EnKF framework to derive a matrix-free implementation via an iterative Woodbury formula. Experimental tests are performed by using an Atmospheric General Circulation Model. The numerical results are compared with those of the EnKF based on the Rao–Blackwell Ledoit, and Wolf covariance matrix estimator (EnKF-RBLW) wherein Gaussian assumptions are a must on prior members. The outcomes reveal that the use of the proposed filter can mitigate the impact of spurious correlations during assimilation stages, and even more, the proposed method can improve on the results of the EnKF-RBLW as a consequence of the Gaussian relaxation on prior errors. [ABSTRACT FROM AUTHOR]