A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method

In References 1 and 2 we showed that the error in the finite-element solution has two parts, the local error and the pollution error, and we studied the effect of the pollution error on the quality of the local error-indicators and the quality of the derivatives recovered by local post-processing. Here we show that it is possible to construct a posteriori estimates of the pollution error in any patch of elements by employing the local error-indicators over the mesh outside the patch. We also give an algorithm for the adaptive control of the pollution error in any patch of elements of interest.     

hp Optimization of finite element approximations: Analysis of the optimal mesh sequences in one dimension

This paper addresses the problem of the construction of the optimal hp meshes for the singular solution x, >1/2, in the one-dimensional setting. First, a review of the asymptotic analysis of Gui and Babu ka [1] is given and is used to characterize the optimal hp meshes which are geometrically graded toward the singular point. The asymptotically optimal hp meshes are compared with the hp meshes which are constructed by employing the hp strategy of Rachowicz et al. [2], which employs the principle of maximum change of error per change in the number of degrees of freedom, and the true optimal hp meshes which minimize the error for fixed number of degrees of freedom and which are determined by searching through the entire sequence of the admissible hp meshes. It is shown that, for the singular solution, the meshes constructed by the hp strategy of Rachowicz coincide with the true optimal meshes; nevertheless, there exist various solutions for which the algorithm which employs the principle of maximum change of error may not converge. For this reason the hp strategy of Rachowicz is combined with the classical feedback algorithm to obtain a robust hp strategy which always converges. Further, it is shown that the asymptotically optimal hp meshes are very close to the true optimal hp meshes in the entire range of practical computation. The effect of the accuracy of the error estimator on the optimality of the constructed hp meshes is also analyzed. It is shown that unless the estimator is sufficiently accurate in the element adjacent to the singular point, the meshes constructed by the adaptive hp strategy can differ substantially from the true optimal meshes.     

Hardware Architectures for the Efficient Implementation of Multi-Service Broadband Access and Multimedia Home Networks

In multimedia applications, the stringent requirements for balancing transmission capacity, flexible service provisioning and cost reduction lead the manufactures to provide highly integrated System-on Chip (SoC) solutions. This paper analyzes the application of high-bandwidth-networking SoCs to improve on the cost efficiency of multimedia service distribution in home networks. We present a case study, where we utilize the inherent protocol processing capabilities and high bandwidth interfaces of a modern network processor, scaled down to match the performance targets and low cost requirements of the home networking environment. An efficient, low cost Residential Gateway architecture results by mapping the home services onto the processing and memory blocks of this SoC. This revised version was published online in September 2006 with corrections to the Cover Date.     

The generalized finite element method for Helmholtz equation. Part II: Effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment

In Part I [T. Strouboulis, I. Babuška, R. Hidajat, The generalized finite element method for Helmholtz equation: theory, computation, and open problems, Comput. Methods Appl. Mech. Engrg. 195 (2006) 4711-4731] we introduced the q-version of the generalized finite element method (GFEM) for the Helmholtz equation and we addressed its: (a) pollution error due to the wave number; (b) exponential q-convergence; (c) robustness to perturbations of the mesh, the roundoff and numerical quadrature errors; and (d) a-posteriori error estimation. Here we continue the development of the GFEM for Helmholtz and we address the effects of: (a) alternative handbook functions and mesh types; (b) the error due to the artificial truncation boundary conditions and its assessment. The conclusions are: (1) the employment of plane-wave, wave-band, and Vekua handbook functions lead to equivalent results; and (2) for high q, the most significant component of error may be the one due to the artificial truncation boundary conditions. A rather straightforward approach for assessing this error is proposed.     

Theory-relational belief revision

Annals of Mathematics and Artificial Intelligence - The prominent formal framework for belief change established by Alchourrón, Gärdenfors and Makinson circumscribes the territory of all...     

The generalized finite element method: an example of its implementation and illustration of its performance

The generalized finite element method (GFEM) was introduced in Reference 1 as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM. Copyright © 2000 John Wiley & Sons, Ltd.     

Testing equality of autocovariance operators for functional time series

We consider strictly stationary stochastic processes of Hilbert space-valued random variables and focus on fully functional tests for the equality of the lag-zero autocovariance operators of several independent functional time series. A moving block bootstrap (MBB)-based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomessome common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can be potentially applied to a broad range of test statistics of the hypotheses of interest. As an example, we establish validity for approximating the distribution under the null of a test statistic based on the Hilbert–Schmidt distance of the corresponding sample lag-zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the MBB procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested MBB-based testing procedure is illustrated through simulations and an application to a real-life dataset is discussed.     

Initialization of the Benders master problem using valid inequalities applied to fixed-charge network problems

Network problems concern the selection of arcs in a graph in order to satisfy, at minimum cost, some flow requirements, usually expressed in the form of node–node pair demands. Benders decomposition methods, based on the idea of partitioning of the initial problem to two sub-problems and on the generation of cuts, have been successfully applied to many of these problems. This paper presents a novel way to reinitialize the Benders master problem for this group of problems using a series of valid inequalities. A generic presentation of the developed valid inequalities is presented as well as a case study of a refinery system is used in order to illustrate the advantage of the proposed procedure. The valid inequalities significantly restrict the solution space of the Benders master problem from the first iteration of the algorithm leading to improved convergence.