scholarly journals Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 281
Author(s):  
Qiuyan Xu ◽  
Zhiyong Liu
Keyword(s):  
Domain Decomposition ◽  
Iterative Algorithm ◽  
Poisson Equation ◽  
Large Scale ◽  
Linear Equations ◽  
Iterative Algorithms ◽  
Computation Time ◽  
Explicit Method ◽  
Computational Process ◽  
Poisson Problem

Poisson equation is a widely used partial differential equation. It is very important to study its numerical solution. Based on the strategy of domain decomposition, the alternating asymmetric iterative algorithm for 3D Poisson equation is provided. The solution domain is divided into several sub-domains, and eight asymmetric iterative schemes with the relaxation factor for 3D Poisson equation are constructed. When the numbers of iteration are odd or even, the computational process of the presented iterative algorithm are proposed respectively. In the calculation of the inner interfaces, the group explicit method is used, which makes the algorithm to be performed fast and in parallel, and avoids the difficulty of solving large-scale linear equations. Furthermore, the convergence of the algorithm is analyzed theoretically. Finally, by comparing with the numerical experimental results of Jacobi and Gauss Seidel iterative algorithms, it is shown that the alternating asymmetric iterative algorithm based on domain decomposition has shorter computation time, fewer iteration numbers and good parallelism.

scholarly journals Parallel Multiphysics Simulation of Package Systems Using an Efficient Domain Decomposition Method

Electronics ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 158
Author(s):  
Weijie Wang ◽  
Yannan Liu ◽  
Zhenguo Zhao ◽  
Haijing Zhou
Keyword(s):  
Domain Decomposition ◽  
Antenna Arrays ◽  
Electromagnetic Interference ◽  
Decomposition Method ◽  
High Performance ◽  
Large Scale ◽  
Electromagnetic Compatibility ◽  
Domain Decomposition Method ◽  
Computation Time ◽  
Multiphysics Simulations

With the continuing downscaling in feature sizes, the thermal impact on material properties and geometrical deformations can no longer be ignored in the analysis of the electromagnetic compatibility or electromagnetic interference of package systems, including System-in-Package and antenna arrays. We present a high-performance numerical simulation program that is intended to perform large-scale multiphysics simulations using the finite element method. An efficient domain decomposition method was developed to accelerate the multiphysics loops of electromagnetic–thermal stress simulations by considering the fact that the electromagnetic field perturbations caused by geometrical deformation are small and constrained in one or a few subdomains. The multi-level parallelism of the algorithm was also obtained based on an in-house developed parallel infrastructure. Numerical examples showed that our algorithm is able to enable simulation with multiple processors in parallel and, more importantly, achieve a significant reduction in computation time compared with traditional methods.

scholarly journals Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh

2021 ◽  
Vol 2 ◽  
Author(s):  
Ali Girayhan Özbay ◽  
Arash Hamzehloo ◽  
Sylvain Laizet ◽  
Panagiotis Tzirakis ◽  
Georgios Rizos ◽  
...  
Keyword(s):  
Neural Network ◽  
Boundary Conditions ◽  
Poisson Equation ◽  
Iterative Algorithms ◽  
Initial Guess ◽  
Correct Solution ◽  
Neural Network Models ◽  
Arbitrary Boundary ◽  
Poisson Problem ◽  
The Poisson Equation

Abstract The Poisson equation is commonly encountered in engineering, for instance, in computational fluid dynamics (CFD) where it is needed to compute corrections to the pressure field to ensure the incompressibility of the velocity field. In the present work, we propose a novel fully convolutional neural network (CNN) architecture to infer the solution of the Poisson equation on a 2D Cartesian grid with different resolutions given the right-hand side term, arbitrary boundary conditions, and grid parameters. It provides unprecedented versatility for a CNN approach dealing with partial differential equations. The boundary conditions are handled using a novel approach by decomposing the original Poisson problem into a homogeneous Poisson problem plus four inhomogeneous Laplace subproblems. The model is trained using a novel loss function approximating the continuous $ {L}^p $ norm between the prediction and the target. Even when predicting on grids denser than previously encountered, our model demonstrates encouraging capacity to reproduce the correct solution profile. The proposed model, which outperforms well-known neural network models, can be included in a CFD solver to help with solving the Poisson equation. Analytical test cases indicate that our CNN architecture is capable of predicting the correct solution of a Poisson problem with mean percentage errors below 10%, an improvement by comparison to the first step of conventional iterative methods. Predictions from our model, used as the initial guess to iterative algorithms like Multigrid, can reduce the root mean square error after a single iteration by more than 90% compared to a zero initial guess.

A Large-Scale Magnetostatic Analysis Using an Iterative Domain Decomposition Method Based on the Minimal Residual Method

2012 ◽  
Vol 16 (4) ◽  
pp. 496-502 ◽  
Author(s):  
Masao Ogino ◽  
◽  
Shin-ichiro Sugimoto ◽  
Seigo Terada ◽  
Yanqing Bao ◽  
...  
Keyword(s):  
Domain Decomposition ◽  
Decomposition Method ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Domain Decomposition Method ◽  
Computation Time ◽  
Interface Problem ◽  
Stable Convergence ◽  
Magnetostatic Analysis ◽  
Minimal Residual

This paper describes a large-scale 3D magnetostatic analysis using the Domain Decomposition Method (DDM). To improve the convergence of the interface problem of DDM, a DDM approach based on the Conjugate Residual (CR) method or the MINimal RESidual (MINRES) method is proposed. The CR or MINRES method improved the convergence rate and showed more stable convergence behavior in solving the interface problem than the Conjugate Gradient (CG) method, and reduced computation time for a large-scale problem with about 10 million degrees of freedom.

scholarly journals Scalability evaluation of iterative algorithms for supercomputer simulation of physical processes

2018 ◽  
pp. 416-430
Author(s):  
Н.А. Ежова ◽  
Л.Б. Соколинский
Keyword(s):  
Large Scale ◽  
Cluster Computing ◽  
Linear Equations ◽  
Iterative Algorithms ◽  
Computing System ◽  
Jacobi Method ◽  
Map Reduce ◽  
Physical Processes ◽  
Programming Library ◽  
Cost Metrics

Статья посвящена разработке методики исследования масштабируемости ресурсоемких итерационных алгоритмов, применяемых в моделировании сложных физических процессов на суперкомпьютерных системах. В основе предлагаемой методики лежит модель параллельных вычислений BSF (Bulk Synchronous Farm), позволяющая на ранней стадии разработки итерационного алгоритма определить границу его масштабируемости. Модель BSF предполагает представление алгоритма в виде операций над списками с использованием функций высшего порядка. При этом рассматривается два класса представлений: BSF-M (Map BSF) и BSF-MR (Map-Reduce BSF). Предлагаемая методика описывается на примере решения систем линейных алгебраических уравнений методом Якоби. Для метода Якоби строится два итерационных алгоритма: Jacobi-M на основе представления BSF-M и Jacobi-MR на основе представления BSF-MR. Для указанных алгоритмов с помощью стоимостных метрик модели BSF даются аналитические оценки для ускорения, эффективности распараллеливания и верхней границы масштабируемости для многопроцессорных вычислительных систем с распределенной памятью. Приводится информация о реализации этих алгоритмов на языке C++ с использованием программного шаблона BSF и библиотеки параллельного программирования MPI. Демонстрируются результаты масштабных вычислительных экспериментов, выполненных на кластерной вычислительной системе. На основе экспериментальных результатов дается анализ адекватности оценок, полученных аналитическим путем с помощью стоимостных метрик модели BSF. This paper is devoted to the development of a methodology for evaluating the scalability of compute-intensive iterative algorithms used for simulating complex physical processes on supercomputer systems. The proposed methodology is based on the BSF (Bulk Synchronous Farm) parallel computation model, which makes it possible to predict the upper scalability bound of an iterative algorithm in early stages of its design. The BSF model assumes the representation of the algorithm in the form of operations on lists using high-order functions. Two classes of representations are considered: BSF-M (Map BSF) and BSF-MR (Map-Reduce BSF). The proposed methodology is described by the example of solving a system of linear equations by the Jacobi method. For the Jacobi method, two iterative algorithms are constructed: Jacobi-M based on the BSF-M representation and Jacobi-MR based on the BSF-MR representation. Analytical estimations of the speedup, parallel efficiency and upper scalability bound are obtained for these algorithms using the BSF cost metrics on multi-processor computing systems with distributed memory. These algorithms are implemented on C++ language using the BSF program skeleton and MPI parallel programming library. The results of large-scale computational experiments performed on a cluster computing system are discussed. Based on the experimental results, an analysis of the adequacy of estimations obtained analytically using the BSF cost metric is made.

Solving Large-Scale Asymmetric Sparse Linear Equations Based on SuperLU Algorithm

2011 ◽  
Vol 230-232 ◽  
pp. 1355-1361
Author(s):  
Pei Wang ◽  
Xu Sheng Yang ◽  
Zhuo Yuan Wang ◽  
Lin Gong Li ◽  
Ji Chang He ◽  
...  
Keyword(s):  
Large Scale ◽  
Linear Equations ◽  
Computation Time ◽  
Coefficient Matrix ◽  
Memory Space ◽  
Storage Method

This article introduces the recent research of SuperLU algorithms and the optimal storage method for [1] the sparse linear equations of coefficient matrix. How to solve large-scale non-symmetric sparse linear equations by SuperLU algorithm is the key part of this article. The advantage of SuperLU algorithm compared to other algorithms is summarized at last. SuperLU algorithm not only saves memory space, but also reduces the computation time. Because of less storage needed by this algorithm, it could solve equation with larger scale, which is much more useful.

Double Precision Is Not Needed for Many-Body Calculations: New Conventional Wisdom

2018 ◽  
Author(s):  
Pavel Pokhilko ◽  
Evgeny Epifanovsky ◽  
Anna I. Krylov
Keyword(s):  
Large Scale ◽  
Computation Time ◽  
Coupled Cluster ◽  
Double Precision ◽  
Many Body ◽  
Single Precision ◽  
Parallel Performance ◽  
Point Representation ◽  
Electron Repulsion Integrals ◽  
Cluster Methods

Using single precision floating point representation reduces the size of data and computation time by a factor of two relative to double precision conventionally used in electronic structure programs. For large-scale calculations, such as those encountered in many-body theories, reduced memory footprint alleviates memory and input/output bottlenecks. Reduced size of data can lead to additional gains due to improved parallel performance on CPUs and various accelerators. However, using single precision can potentially reduce the accuracy of computed observables. Here we report an implementation of coupled-cluster and equation-of-motion coupled-cluster methods with single and double excitations in single precision. We consider both standard implementation and one using Cholesky decomposition or resolution-of-the-identity of electron-repulsion integrals. Numerical tests illustrate that when single precision is used in correlated calculations, the loss of accuracy is insignificant and pure single-precision implementation can be used for computing energies, analytic gradients, excited states, and molecular properties. In addition to pure single-precision calculations, our implementation allows one to follow a single-precision calculation by clean-up iterations, fully recovering double-precision results while retaining significant savings.

Distributed learning with indefinite kernels

2019 ◽  
Vol 17 (06) ◽  
pp. 947-975 ◽  
Author(s):  
Lei Shi
Keyword(s):  
Large Scale ◽  
Substantial Reduction ◽  
Computation Time ◽  
Distributed Learning ◽  
Rates Of Convergence ◽  
Regression Problem ◽  
Data Set ◽  
Regularization Scheme ◽  
Original Algorithm ◽  
Indefinite Kernel

We investigate the distributed learning with coefficient-based regularization scheme under the framework of kernel regression methods. Compared with the classical kernel ridge regression (KRR), the algorithm under consideration does not require the kernel function to be positive semi-definite and hence provides a simple paradigm for designing indefinite kernel methods. The distributed learning approach partitions a massive data set into several disjoint data subsets, and then produces a global estimator by taking an average of the local estimator on each data subset. Easy exercisable partitions and performing algorithm on each subset in parallel lead to a substantial reduction in computation time versus the standard approach of performing the original algorithm on the entire samples. We establish the first mini-max optimal rates of convergence for distributed coefficient-based regularization scheme with indefinite kernels. We thus demonstrate that compared with distributed KRR, the concerned algorithm is more flexible and effective in regression problem for large-scale data sets.

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