Быстрый прямой алгоритм реализации метода конечных элементов высокого порядка на прямоугольниках для краевых задач для уравнения Пуассона
This paper deals with the technique known as the periodic synchronous averaging. The exact analytical expression for the fast Fourier transform (FFT) representing the digital spectrum of the signal undergoing periodic synchronous averaging is derived using the general signal and spectral framework. This formula connects the coefficient of Fourier series of the original continuous-time signal with the FFT of the averaged sampled version taking into consideration all the effects such as difference between the true and averaging periods, the attenuation and the leakage. The results of the numerical simulation are presented for the case of periodic signal, which was chosen a train of triangle pulses, the spectrum of which possesses a closed form and whose Fourier series coefficients rapidly decrease with the index. The chosen example allows the authors to illustrate that the waveform of the recovered signal can vary significantly, despite a rather slight difference in values between the true and averaging periods. Another important effect emphasized in the presented paper is that overall distinction between the original and averaged signals measured by means of relative mean square error raises if the total observation length increases while the other parameters remain fixed.
The boundary value problem for the Poisson and Helmholtz equations with a piecewise constant coefficient with a jump on a triangle is studied numerically. At the jump of the coefficient (at the boundary of the media), the docking conditions are set. A compact difference scheme with high accuracy with a relatively small number of calculations is proposed.
Parallel program for the fast Fourier transform is implemented on the basis of MPI (Message Passing Interface) technology. Radix-4 algorithm is chosen as a basic method to use. The dependence of parallel calculation acceleration on the number of processors is studied for two supercomputers. The formula describing the dependence of the calculation time on the number of processors is proposed for the range of the input data volume and supercomputer characteristics. The number of nodes providing you with maximum acceleration of calculation is estimated.
Modern Elbrus-4S and Elbrus-8S processors show floating point performance comparable to the popular Intel processors in the field of high-performance computing. Tasks oriented to take advantage of the VLIW architecture show even greater efficiency on Elbrus processors. In this paper the efficiency of the most popular materials science codes in the field of classical molecular dynamics and quantum-mechanical calculations is considered. A comparative analysis of the performance of these codes on Elbrus processor and other modern processors is carried out
A new fast direct algorithm for implementing a finite element method (FEM) of order on rectangles as applied to boundary value problems for Poisson-type equations is described that extends a well-known algorithm for the case of difference schemes or bilinear finite elements (n = 1). Its core consists of fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for an nth-order FEM based on the fast discrete Fourier transform. The amount of arithmetic operations is logarithmically optimal in the theory and is rather attractive in practice. The algorithm admits numerous further applications (including the multidimensional case).
Poisson equation in the whole space was studied earlier for so called ergodic generators L corresponding to homogeneous Markov diffusions. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenisation. Here a similar equation with a potential is considered, firstly because it is natural for PDEs, and secondly with a hope that it may be also useful for some extensions related to homogenization and averaging.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.