Computational and Mathematical Methods最新文献

In this note, we exploit polynomial preconditioners for the conjugate gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X⁻¹ = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence. We provide results on very large matrices (up to 8.6 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.

Stochastic symmetric matrices with a dominant eigenvalue, α,can be written as the sum of λαα^t (where λ is the first eigenvalue), with a symmetric error matrix E. The information in the stochastic matrix will be condensed in its structured vectors, λα, and the sum of square of residues, V. When the matrices of a family correspond to the treatments of a base design, we say the family is structured. The action of the factors, which are considered in the base design, on the structure vectors of the family matrices will be analyzed. We use ANOVA (Analysis of Variance) and related techniques, to study the action under linear combinations of the components of structure vectors of the m matrices of the model. Orthogonal models with m treatments are associated to orthogonal partitions. The hypothesis to be tested, on the action of the factors in the base design, will be associated to the spaces in the orthogonal partitions. We will show how to carry out transversal and longitudinal analysis for families of stochastic symmetric matrices with dominant eigenvalue associated to orthogonal models.

This article describes a high-order well-balanced central finite volume scheme for solving the coupled Exner−shallow water equations in one dimensional channels with rectangular section and variable width. Such numerical method may solve the proposed bedload sediment transport problem without the need to diagonalize the Jacobian matrix of flow. The numerical scheme uses a Runge–Kutta method with a fourth-order continuous natural extension for time discretization. The source term approximation is designed to verify the exact conservation property. Comparison of the numerical results for two accuracy tests have proved the stability and accuracy of the scheme. The results of the laboratory tests have also been used to calibrate different expressions of the solid transport discharge in the computer code. Two experimental tests have been carried out to study the erosive phenomenon and the consequent sediment transport: one test consisting of a triangular dune, and other caused by the effect of channel contraction.

This study is devoted to solve the Chandrasekhar integral equation that it is used for modeling problems in theory of radiative transfer in a plane-parallel atmosphere, and others research areas like the kinetic theory of gases, neutron transport, traffic model, the queuing theory among others. First of all, we transform the Chandrasekhar integral equation into a nonlinear Hammerstein-type integral equation with the corresponding Nemystkii operator and the proper nonseparable kernel. Them, we approximate the kernel in order to apply an iterative scheme. This procedure it is solved in two different ways. First one, we solve a nonlinear equation with separable kernel and define an adequate nonlinear operator between Banach spaces that approximates the first problem. Second one, we introduce an approximation for the inverse of the Fréchet derivative that appears in the Newton's iterative scheme for solving nonlinear equations. Finally, we perform a numerical experiment in order to compare our results with previous ones published showing that are competitive.

In this paper, we solve linear boundary value problems of second-order in ordinary differential equations with the generalized finite difference method and compare the numerical accuracy for different orders of approximations. We develop a strategy for dealing with ill-conditioned stars based on the condition number of the matrix of derivatives. In addition, we consider a scheme implemented with parallel processing for the formation of the stars and the calculation of the derivatives. We present some examples with high gradients in irregular discretizations exaggerated on purpose, to highlight the efficiency of the proposed strategy.

Dataflow computing allows to start computations as soon as all their dependencies are satisfied. This is particularly useful in applications with irregular or complex patterns of dependencies which would otherwise involve either coarse grain synchronizations which would degrade performance, or high programming costs. A recent proposal for the easy development of performant dataflow algorithms in hybrid shared/distributed memory systems is UPC++ DepSpawn. Among the many techniques it applies to provide good performance is a software cache that minimizes the communications among the processes involved. In this article we provide the details of the implementation and operation of this cache and we present an autotuning strategy that simplifies its usage by freeing the user from having to estimate an adequate size for this cache. Rather, the runtime is now able to define reasonably sized caches that provide near optimal behavior.

The increasing demand of high-performance services has motivated the need of the 1 exaflop supercomputing performance. In systems providing such computing capacity, the interconnection network is likely the cornerstone due to the fact that it may become the bottleneck of the system degrading the performance of the entire system. In order to increase the network performance, several improvements have been introduced such as new architectures. Intel © Omni-Path architecture (OPA) is an example of the constant development of high-performance interconnection networks, which has not been largely studied because it is a recent technology and the lack of public available information. Another important aspect of all interconnect network technologies is the Quality of Service (QoS) support to applications. In this article, we present an OPA simulation model and some implementation details as well as our simulation tool OPASim. We also review the QoS mechanisms provided by the OPA architecture and present the table-based output scheduling algorithm simple bandwidth table.

In this article, we present the convergence analysis of an upwind finite difference scheme for singularly perturbed system of parabolic convection-diffusion initial-boundary-value problems with discontinuous convection coefficient and source term. The proposed numerical scheme is constructed by using the implicit-Euler scheme for the time derivative on the uniform mesh, and the upwind finite difference scheme for the spatial derivatives on a layer-resolving piecewise-uniform Shishkin mesh. It is shown that the numerical solution obtained by the proposed scheme converges uniformly with respect to the perturbation parameter. The proposed numerical scheme is of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are carried out to verify the theoretical results.

We focus on Partial Differential Equation (PDE)-based Data Assimilation problems (DA) solved by means of variational approaches and Kalman filter algorithm. Recently, we presented a Domain Decomposition framework (we call it DD-DA, for short) performing a decomposition of the whole physical domain along space and time directions, and joining the idea of Schwarz's methods and parallel in time approaches. For effective parallelization of DD-DA algorithms, the computational load assigned to subdomains must be equally distributed. Usually computational cost is proportional to the amount of data entities assigned to partitions. Good quality partitioning also requires the volume of communication during calculation to be kept at its minimum. In order to deal with DD-DA problems where the observations are nonuniformly distributed and general sparse, in the present work we employ a parallel load balancing algorithm based on adaptive and dynamic defining of boundaries of DD—which is aimed to balance workload according to data location. We call it DyDD. As the numerical model underlying DA problems arising from the so-called discretize-then-optimize approach is the constrained least square model (CLS), we will use CLS as a reference state estimation problem and we validate DyDD on different scenarios.

Every random variable (rv) X (or random vector) with finite moments generates a set of orthogonal polynomials, which can be used to obtain properties related to the distribution of X. This technique has been used in statistical inference, mainly connected to the exponential family of distributions. In this paper a review of some of its more relevant uses is provided. The first one deals with properties of expansions in terms of orthogonal polynomials for the Uniformly Minimum Variance Unbiased Estimator of a given parametric function, when sampling from a distribution in the Natural Exponential Family of distributions with Quadratic Variance Function. The second one compares two relevant methods, based on expansions in Laguerre polynomials, existing in the literature to approximate the distribution of linear combinations of independent chi-square variables.