Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations

In this paper, we prove that the Legendre tau method has the optimal rate of convergence in L ²-norm, H ¹-norm and H ²-norm for one-dimensional second-order steady differential equations with three kinds of boundary conditions and in C([0,T];L ²(I))-norm for the corresponding evolution equation with the Dirichlet boundary condition. For the generalized Burgers equation, we develop a Legendre tau-Chebyshev collocation method, which can also be optimally convergent in C([0,T];L ²(I))-norm. Finally, we give some numerical examples.     

The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Optimality of the IIPG Method for Odd Degrees of Polynomial Approximation in 1D

The paper deals with a numerical analysis of the incomplete interior penalty Galerkin (IIPG) method applied to one dimensional Poisson problem. Based on a particular choice of the interior penalty parameter σ (order of O(h ⁻¹)), we derive the optimal error estimate in the L ²-norm for odd degrees of polynomial approximation for locally quasi-uniform meshes. Moreover, we show that only the mentioned choice of the penalty parameter leads to optimal orders of convergence. Finally, presented numerical experiments verify the theoretical results.     

Radial Basis Function Interpolation on Irregular Domain through Conformal Transplantation

In this paper, Radial Basis Function (RBF) method for interpolating two dimensional functions with localized features defined on irregular domain is presented. RBF points located inside the domain and on its boundary are chosen such that they are the image of conformally mapped points on concentric circles on a unit disk. On the disk, a fast RBF solver to compute RBF coefficients developed by Karageorghis et al. (Appl. Numer. Math. 57(3):304–319, 2007) is used. Approximation values at desired points in the domain can be computed through the process of conformal transplantation. Some numerical experiments are given in a style of a tutorial and MATLAB code that solves RBF coefficients using up to 100,000 RBF points is provided.     

A Fast Explicit Operator Splitting Method for Passive Scalar Advection

The dispersal and mixing of scalar quantities such as concentrations or thermal energy are often modeled by advection-diffusion equations. Such problems arise in a wide variety of engineering, ecological and geophysical applications. In these situations a quantity such as chemical or pollutant concentration or temperature variation diffuses while being transported by the governing flow. In the passive scalar case, this flow prescribed and unaffected by the scalar. Both steady laminar and complex (chaotic, turbulent or random) time-dependent flows are of interest and such systems naturally lead to questions about the effectiveness of the stirring to disperse and mix the scalar. The development of reliable numerical methods for advection-diffusion equations is crucial for understanding their properties, both physical and mathematical. In this paper, we extend a fast explicit operator splitting method, recently proposed in (A. Chertock, A. Kurganov, G. Petrova, Int. J. Numer. Methods Fluids 59:309–332, 2009), for solving deterministic convection-diffusion equations, to the problems with random velocity fields and singular source terms. A superb performance of the method is demonstrated on several two-dimensional examples.     

Fast and highly scalable parallel computations for fundamental matrix problems on distributed memory systems

We present fast and highly scalable parallel computations for a number of important and fundamental matrix problems on distributed memory systems (DMS). These problems include matrix multiplication, matrix chain product, and computing the powers, the inverse, the characteristic polynomial, the determinant, the rank, the Krylov matrix, and an LU- and a QR-factorization of a matrix, and solving linear systems of equations. Our highly scalable parallel computations for these problems are based on a highly scalable implementation of the fastest sequential matrix multiplication algorithm on DMS. We show that compared with the best known parallel time complexities on parallel random access machines (PRAM), the most powerful but unrealistic shared memory model of parallel computing, our parallel matrix computations achieve the same speeds on distributed memory parallel computers (DMPC), and have an extra polylog factor in the time complexities on DMS with hypercubic networks. Furthermore, our parallel matrix computations are fully scalable on DMPC and highly scalable over a wide range of system size on DMS with hypercubic networks. Such fast (in terms of parallel time complexity) and highly scalable (in terms of our definition of scalability) parallel matrix computations were rarely seen before on any distributed memory systems.     

SENFIS: a Sensor Node File System for increasing the scalability and reliability of Wireless Sensor Networks applications

In the last years the Wireless Sensor Networks’ (WSN) technology has been increasingly employed in various application domains. The extensive use of WSN posed new challenges in terms of both scalability and reliability. This paper proposes Sensor Node File System (SENFIS), a novel file system for sensor nodes, which addresses both scalability and reliability concerns. SENFIS can be mainly used in two broad scenarios. First, it can transparently be employed as a permanent storage for distributed TinyDB queries, in order to increase the reliability and scalability. Second, it can be directly used by a WSN application for permanent storage of data on the WSN nodes. The experimental section shows that SENFIS implementation makes an efficient use of resources in terms of energy consumption, memory footprint, flash wear levelling, while achieving execution times similarly with existing WSN file systems.     

The hybrid dynamic parallel scheduling algorithm for load balancing on Chained-Cubic Tree interconnection networks

The Chained-Cubic Tree (CCT) interconnection network topology was recently proposed as a continuation for the extended efforts in the area of interconnection networks’ performance improvement. This topology, which promises to exhibit the best properties of the hypercube and tree topologies, needs to be deeply investigated in order to evaluate its performance among other interconnection networks’ topologies. This work comes as a complementary effort, in which the load balancing technique is investigated as one of the most important aspects of performance improvement. This paper proposes a new load balancing algorithm on CCT interconnection networks. The proposed algorithm, which is called Hybrid Dynamic Parallel Scheduling Algorithm (HD-PSA), is a combination of two common load balancing strategies; dynamic load balancing and parallel scheduling. The performance of the proposed algorithm is evaluated both, analytically and experimentally, in terms of various performance metrics; including, execution time, load balancing accuracy, communication cost, number of tasks hops, and tasks locality.     

Buffer management in a real-time shared disks cluster

A great deal of research indicates that the shared disks (SD) cluster is suitable to high performance transaction processing. However, the aggregation of SD cluster with real-time processing has not been investigated. By adopting cluster technology, the real-time services will be highly available and can exploit internode parallelism. In this paper, we consider buffer management issues in a real-time SD cluster. We first propose a real-time buffer coherency algorithm that exploits inherent characteristics of real-time applications and SD cluster. Then we extend traditional buffer replacement algorithms to the real-time SD cluster. They emphasize specific attributes of buffer pages to capitalize locality of references, transaction deadline, and affinity-based routing. We evaluate the performance of the proposed algorithms under a wide variety of transaction workloads and system configurations.     

Non-Parametric Mixture Model Based Evolution of Level Sets and Application to Medical Images

We present a novel region-based curve evolution algorithm which has three primary contributions: (i) non-parametric estimation of probability distributions using the recently developed NP windows method; (ii) an inequality-constrained least squares method to model the image histogram with a mixture of nonparametric probability distributions; and (iii) accommodation of the partial volume effect, which is primarily due to low resolution images, and which often poses a significant challenge in medical image analysis (our primary application area). We first approximate the image intensity histogram as a mixture of non-parametric probability density functions (PDFs), justifying its use with respect to medical image analysis. The individual densities in the mixture are estimated using the recent NP windows PDF estimation method, which builds a continuous representation of discrete signals. A Bayesian framework is then formulated in which likelihood probabilities are given by the non-parametric PDFs and prior probabilities are calculated using an inequality constrained least squares method. The non-parametric PDFs are then learnt and the segmentation solution is spatially regularised using a level sets framework. The log ratio of the posterior probabilities is used to drive the level set evolution. As background to our approach, we recall related developments in level set methods. Results are presented for a set of synthetic and natural images as well as simulated and real medical images of various anatomical organs. Results on a range of images show the effectiveness of the proposed algorithm.     

On the Length and Area Regularization for Multiphase Level Set Segmentation

In this paper we introduce novel regularization techniques for level set segmentation that target specifically the problem of multiphase segmentation. When the multiphase model is used to obtain a partitioning of the image in more than two regions, a new set of issues arise with respect to the single phase case in terms of regularization strategies. For example, if smoothing or shrinking each contour individually could be a good model in the single phase case, this is not necessarily true in the multiphase scenario.