Application of the parallel diagonal dominant algorithm for the incompressible Navier-Stokes equations

Three parallel algorithms, namely the parallel partition LU (PPT) algorithm, the parallel partition hybrid (PPH) algorithm, and the parallel diagonal dominant (PDD) algorithm are proposed for solving tridiagonal linear systems on multicomputers. These algorithms are based on the divide-and-conquer parallel computation model. The PPT and PPH algorithms support both pivoting and non-pivoting. The PPT algorithm is good when the number of processors is small; otherwise, the PPH algorithm is better. When the system is diagonal dominant, the PDD algorithm is highly parallel and provides an approximate solution which equals to the exact solution within machine accuracy. Both computation and communication complexities of the three algorithms are presented. All three methods proposed in this paper outperform other known parallel algorithms and have been implemented on a 64-node Ncube multicomputer. The analytic results matches closely with the results measured from the Ncube machine.

Various tridiagonal solvers have been proposed in recent years for different parallel platforms. In this paper, the performance of three tridiagonal solvers, namely, the parallel partition LU algorithm, the parallel diagonal dominant algorithm, and the reduced diagonal dominant algorithm, is studied. These algorithms are designed for distributed-memory machines and are tested on an Intel Paragon and an IBM SP2 machines. Measured results are reported in terms of execution time and speedup. Analytical study are conducted for different communication topologies and for different tridiagonal systems. The measured results match the analytical results closely. In addition to address implementation issues, performance considerations such as problem sizes and models of speedup are also discussed.

A new compact scheme for parallel computing using domain decomposition

Fully sequential ranking-and-selection (R&S) procedures to find the best from a finite set of simulated alternatives are often designed to be implemented on a single processor. However, parallel computing environments, such as multi-core personal computers and many-core servers, are becoming ubiquitous and easily accessible for ordinary users. In this paper, we propose two types of fully sequential procedures that can be used in parallel computing environments. We call them vector-filling procedures and asymptotic parallel selection procedures, respectively. Extensive numerical experiments show that the proposed procedures can take advantage of multiple parallel processors and solve large-scale R&S problems.

The demand for processing power has been ever increasing with the growth of high-performance computing (HPC) applications and so have the constraints restricting the solutions to such requirements. High-performance distributed and parallel computing and custom-built, hardware-based computing have attempted to address this problem with some success but at a substantial cost. Recently, systems augmented with Field-Programmable Gate Arrays (FPGAs) offering a fusion of traditional parallel and distributed machines with customizable and dynamically reconfigurable hardware have emerged as a cost-effective alternative to traditional systems. However, providing a robust runtime environment for such systems to which HPC users have become accustomed has been fraught with numerous challenges. Dynamic scheduling of large-scale HPC applications in such parallel reconfigurable computing (RC) environments is one such challenge and has not been sufficiently studied to our knowledge. In this paper, we simulatively analyze the performance of several common HPC scheduling heuristics that can be used by an automated job management service to schedule application tasks on a parallel RC system. We also present a performance prediction model which the scheduling heuristics employ to schedule several common HPC applications on a collection of typical FPGA processing platforms.

In this work, we study the performance of some local projection-based solvers in the Large Eddy Simulation (LES) of laminar and turbulent flows governed by the incompressible Navier–Stokes Equations (NSE). On one side, we focus on a high-order term-by-term stabilization Finite Element (FE) method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard Local Projection Stabilization (LPS) methods is replaced by an interpolation-stabilized structure. The interest of LPS methods is that they ensure a self-adapting high accuracy in laminar regions of turbulent flows, which turns to be of overall optimal high accuracy if the flow is fully laminar. On the other side, we propose a new Reduced Basis (RB) Variational Multi-Scale (VMS)-Smargorinsky turbulence model, based upon an empirical interpolation of the sub-grid eddy viscosity term. This method yields dramatical improvements of the computing time for benchmark flows. An overview about known results from the numerical analysis of the proposed methods is given, by highlighting the used mathematical tools. In the numerical study, we have considered two well known problems with applications in industry: the (3D) turbulent flow in a channel and the (2D/3D) recirculating flow in a lid-driven cavity.

The need of the aerospace industry, at national or European level, of faster yet reliable computational fluid dynamics models is the main drive for the application of model reduction techniques. This need is linked to the time cost of high-fidelity models, rendering them inefficient for applications like multi-disciplinary optimization. With the goal of testing and applying model reduction to computational fluid dynamics models applicable to lifting surfaces, a bibliographical research covering reduction of nonlinear, dynamic, or steady models was conducted. This established the prevalence of projection and least mean squares methods, which rely on solutions of the original high-fidelity model and their proper orthogonal decomposition to work. Other complementary techniques such as adaptive sampling, greedy sampling, and hybrid models are also presented and discussed. These projection and least mean squares methods are then tested on simple and documented benchmarks to estimate the error and speed-up of the reduced order models thus generated. Dynamic, steady, nonlinear, and multiparametric problems were reduced, with the simplest version of these methods showing the most promise. These methods were later applied to single parameter problems, namely the lid-driven cavity with incompressible Navier–Stokes equations and varying Reynolds number, and the elliptic airfoil at varying angles of attack for compressible Euler flow. An analysis of the performance of these methods is given at the end of this article, highlighting the computational speed-up obtained with these techniques, and the challenges presented by multiparametric problems and problems showing point singularities in their domain.

A numerical method is developed for solving parabolic partial differential equations with integral boundary conditions. The method is moderately sixth-order accurate due to merging of sixth order finite difference scheme and fifth order Pade’s approximation. Simpson’s 1/3 rule is used to approximate integral conditions. The method does not involve the use of complex arithmetic and optimizes the results. It is observed that this numerical method can be easily coded on serial as well as parallel computers.

Doubly diagonally dominant matrices

This article presents a review of various hardware and software parallel computing environments so far developed for the implementation of parallel algorithms to solve complex and computationally dominated optimization problems. These parallel environments are composed of parallel programming tools, performance evaluation tools, debuggers and optimization libraries. The environments can be classified based on the ease of implementation, hardware independence, ease to understand, and guarantee of performance.

In this article, a cost-efficient explicit third-order upwind compact difference (UCD3) method with adjustable dissipation (EUCD3-AD) is proposed for the two-dimensional (2D) transient incompressible Navier–Stokes equations (INSEs) based on the entropically damped artificial compressibility method. First of all, the averaged UCD3 scheme (AUCD3), which is the averaged results obtained by the upwind and downwind terms in the existing standard UCD3 scheme, is introduced for the first derivatives. Afterward, a new explicit UCD3 scheme with adjustable dissipation (EUCD3-AD) is proposed for the first derivative within a three-point stencil, which have advantage in cost-efficiency and avoid reducing order of accuracy of the standard UCD3. Moreover, an explicit fourth-order compact difference scheme for calculating the second derivative (ESCD4-S) is proposed based on the AUCD3. The proposed EUCD3-AD with adjusted dissipative nature and the AUCD3 for the first derivative are utilized to approximate the convection terms and the pressure gradient and velocity divergence terms, respectively. The ESCD4-S presnted for the second derivative is used to discretize the viscous and diffusion terms in INSEs. For temporal discretization, the third-order total-variation-diminishing Runge–Kutta method is adopted. Finally, numerical validations of the performance of the newly proposed method are conducted with three benchmark problems involving Taylor–Green vortex, doubly periodic shear layer flows, and lid-driven square cavity. Numerical results demonstrate that EUCD3-AD not only preserves the third-order accuracy and overcomes the over numerical dissipation of the standard UCD3 but also greatly improves the computational efficiency in numerical simulation.

Theory and implementation of a semi-implicit finite element method for viscous incompressible flow

A highly efficient parallelization scheme of direct numerical simulation (DNS) for two-dimensional Rayleigh-Bénard convection is presented. By introducing the parallel diagonal dominant (PDD) algorithm to solve the pressure Poisson equation and adjusting the domain decomposition accordingly, all-to-all communication as the usual obstacle to parallel computing can be eliminated. Excellent strong scaling and weak scaling for the parallel efficiency are achieved. Numerical results show that very complex structures in flow exist at very high Ra numbers. The required high resolution both in space and in time can be obtained by the present method at low parallel overhead.

Chapter 50 - Security Considerations in the Distributed Parallel Computation Environment

The problem of finding a $k\\times k$ submatrix of maximum volume of a matrix A is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of A. We show that such a submatrix can always be chosen to be a principal submatrix if A is symmetric semidefinite or diagonally dominant. Then we analyze the low-rank approximation error returned by a greedy method for volume maximization, cross approximation with complete pivoting. Our bound for general matrices extends an existing result for symmetric semidefinite matrices and yields new error estimates for diagonally dominant matrices. In particular, for doubly diagonally dominant matrices the error is shown to remain within a modest factor of the best approximation error. We also illustrate how the application of our results to cross approximation for functions leads to new and better convergence results.