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At least 217 records · Page 12

All sources NASA NTRS DOE OSTI

DOE OSTI2018

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Summary Preconditioned Krylov subspace (KSP) methods are widely used for solving large‐scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss–Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill‐conditioned or saddle‐point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.

Ghai, Aditi↗

DOE OSTI2024

Scalable Multiphysics Block Preconditioning for Low Mach Number Compressible Resistive MHD with Application to Magnetic Confinement Fusion

This study investigates multiphysics block preconditioners that are critical in devising scalable Newton–Krylov iterative solvers for longer time-scale fully implicit fluid plasma models. The specific model of interest is the visco-resistive, low Mach number, compressible magnetohydrodynamics (MHD) model. This model describes the dynamics of conducting fluids in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics. The specific application of interest that motivates this study is the macroscopic simulation of longer time-scale stability and disruptions of magnetic confinement fusion devices, specifically the ITER Tokamak. The computational solution of the governing balance equations for mass, momentum, heat transfer, and magnetic induction for resistive MHD systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena as well as the significant range of time and length scales that the interactions of these physical mechanisms produce. To handle the range of time and spatial scales of interest, a fully implicit unstructured variational multiscale finite element formulation is employed. For the scalable solution of the Newton linearized systems, fully coupled block preconditioners are designed to leverage algebraic multigrid subsolves. In conclusion, results are presented for the strong and weak scaling of the method as well as the robustness of these techniques for a large range of Lundquist numbers.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1982

Iterative spectral methods and spectral solutions to compressible flows

A spectral multigrid scheme is described which can solve pseudospectral discretizations of self-adjoint elliptic problems in O(N log N) operations. An iterative technique for efficiently implementing semi-implicit time-stepping for pseudospectral discretizations of Navier-Stokes equations is discussed. This approach can handle variable coefficient terms in an effective manner. Pseudospectral solutions of compressible flow problems are presented. These include one dimensional problems and two dimensional Euler solutions. Results are given both for shock-capturing approaches and for shock-fitting ones.

Hussaini, M. Y.↗

NASA NTRS1986

Efficient solutions of two-dimensional incompressible steady viscous flows

A simple, efficient, and robust numerical technique is provided for solving two dimensional incompressible steady viscous flows at moderate to high Reynolds numbers. The proposed approach employs an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function Navier-Stokes equations. Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. For this second problem, mesh refinement and Richardson extrapolation are used to obtain useful benchmark solutions in the full range of Reynolds numbers at which steady laminar flow is established.

Morrison, J. H.↗

NASA NTRS1988

Efficient solutions of two-dimensional incompressible steady viscous flows

Morrison, J. H.↗

NASA NTRS1989

An investigation of cell centered and cell vertex multigrid schemes for the Navier-Stokes equations

Two efficient and robust finite-volume multigrid schemes for solving the Navier-Stokes equations are investigated. These schemes employ either a cell centered or a cell vertex discretization technique. An explicit Runge-Kutta algorithm is used to advance the solution in time. Acceleration techniques are applied to obtain faster steady-state convergence. Accuracy and convergence of the schemes are examined. Computational results for transonic airfoil flows are essentially the same, even for a coarse mesh. Both schemes exhibit good convergence rates for a broad range of artificial dissipation coefficients.

Radespiel, R.↗

NASA NTRS1989

An efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations

A cell-vertex scheme for the three-dimensional Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing with locally varying coefficients, a multigrid method and carefully controlled dissipative terms, very good convergence rates are obtained for two- and three-dimensional flows. Details of the acceleration techniques, which are important for convergence on meshes with high aspect-ratio cells, are discussed. Emphasis is put on the analysis of the stability properties of the implicit smoothing of the explicit residuals with coefficients, which depend on cell aspect ratios.

Radespiel, R.↗

NASA NTRS1992

Viscous simulation method for unsteady flows past multicomponent configurations

The present algorithm for the numerical simulation of flows about complex configurations (whose multiple and nonsimilar components have arbitraty geometries) employs a hybridization of the domain decomposition techniques for grid generation as well as to reduce computer-memory requirements. A fully vectorized, finite-volume, upwind-biased, approximately factored multigrid method is used to solve 3D Reynolds-averaged unsteady and compressible Navier-Stokes equations simulating supersonic flows past an ogive-nose-cylinder near or within a cavity. The time-averaged surface pressures obtained compare favorably with the wind tunnel data.

Fouladi, Kamran↗

NASA NTRS1997

An Initial Multi-Domain Modeling of an Actively Cooled Structure

A methodology for the simulation of turbine cooling flows is being developed. The methodology seeks to combine numerical techniques that optimize both accuracy and computational efficiency. Key components of the methodology include the use of multiblock grid systems for modeling complex geometries, and multigrid convergence acceleration for enhancing computational efficiency in highly resolved fluid flow simulations. The use of the methodology has been demonstrated in several turbo machinery flow and heat transfer studies. Ongoing and future work involves implementing additional turbulence models, improving computational efficiency, adding AMR.

Steinthorsson, Erlendur↗

NASA NTRS1987

Relaxation schemes for spectral multigrid methods

The effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined. Iterative methods are an attractive alternative to direct methods because Fourier transform techniques enable the discrete matrix-vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations. Preconditioning is found to be essential for acceptable rates of convergence. Preconditioners based on second-order finite difference methods are used. A comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary. The investigations show that iterations based on incomplete LU decompositions provide the most efficient methods for solving these algebraic systems.

Phillips, Timothy N.↗

DOE OSTI2017

Final report for “Extreme-scale Algorithms and Solver Resilience”

This is a joint project with principal investigators at Oak Ridge National Laboratory, Sandia National Laboratories, the University of California at Berkeley, and the University of Tennessee. Our part of the project involves developing performance models for highly scalable algorithms and the development of latency tolerant iterative methods. During this project, we extended our performance models for the Multigrid method for solving large systems of linear equations and conducted experiments with highly scalable variants of conjugate gradient methods that avoid blocking synchronization. In addition, we worked with the other members of the project on alternative techniques for resilience and reproducibility. We also presented an alternative approach for reproducible dot-products in parallel computations that performs almost as well as the conventional approach by separating the order of computation from the details of the decomposition of vectors across the processes.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1992

Volume 2: Explicit, multistage upwind schemes for Euler and Navier-Stokes equations

The objective of this study was to develop a high-resolution-explicit-multi-block numerical algorithm, suitable for efficient computation of the three-dimensional, time-dependent Euler and Navier-Stokes equations. The resulting algorithm has employed a finite volume approach, using monotonic upstream schemes for conservation laws (MUSCL)-type differencing to obtain state variables at cell interface. Variable interpolations were written in the k-scheme formulation. Inviscid fluxes were calculated via Roe's flux-difference splitting, and van Leer's flux-vector splitting techniques, which are considered state of the art. The viscous terms were discretized using a second-order, central-difference operator. Two classes of explicit time integration has been investigated for solving the compressible inviscid/viscous flow problems--two-state predictor-corrector schemes, and multistage time-stepping schemes. The coefficients of the multistage time-stepping schemes have been modified successfully to achieve better performance with upwind differencing. A technique was developed to optimize the coefficients for good high-frequency damping at relatively high CFL numbers. Local time-stepping, implicit residual smoothing, and multigrid procedure were added to the explicit time stepping scheme to accelerate convergence to steady-state. The developed algorithm was implemented successfully in a multi-block code, which provides complete topological and geometric flexibility. The only requirement is C degree continuity of the grid across the block interface. The algorithm has been validated on a diverse set of three-dimensional test cases of increasing complexity. The cases studied were: (1) supersonic corner flow; (2) supersonic plume flow; (3) laminar and turbulent flow over a flat plate; (4) transonic flow over an ONERA M6 wing; and (5) unsteady flow of a compressible jet impinging on a ground plane (with and without cross flow). The emphasis of the test cases was validation of code, and assessment of performance, as well as demonstration of flexibility.

Elmiligui, Alaa↗

DOE OSTI2016

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method

Here, we propose two multilevel spectral techniques for constructing coarse discretization spaces for saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of fine-grid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the inf-sup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coarse discretization (referred to as operator complexity) deteriorates when we introduce additional functions in the coarse space. We also provide an efficient solver for the coarse (upscaled) saddle-point system by employing hybridization, which leads to a symmetric positive definite (s.p.d.) reduced system for the Lagrange multipliers, and to solve the latter s.p.d. system, we use our previously developed spectral AMGe solver. Numerical experiments, in both two and three dimensions, are provided to illustrate the efficiency of the proposed upscaling technique.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Layer-Parallel Training of Deep Residual Neural Networks

Residual neural networks (ResNets) are a promising class of deep neural networks that have shown excellent performance for a number of learning tasks, e.g., image classification and recognition. Mathematically, ResNet architectures can be interpreted as forward Euler discretizations of a nonlinear initial value problem whose time-dependent control variables represent the weights of the neural network. Hence, training a ResNet can be cast as an optimal control problem of the associated dynamical system. For similar time-dependent optimal control problems arising in engineering applications, parallel-in-time methods have shown notable improvements in scalability. This paper demonstrates the use of those techniques for efficient and effective training of ResNets. The proposed algorithms replace the classical (sequential) forward and backward propagation through the network layers with a parallel nonlinear multigrid iteration applied to the layer domain. This adds a new dimension of parallelism across layers that is attractive when training very deep networks. From this basic idea, we derive multiple layer-parallel methods. The most efficient version employs a simultaneous optimization approach where updates to the network parameters are based on inexact gradient information in order to speed up the training process. Finally, using numerical examples from supervised classification, we demonstrate that the new approach achieves a training performance similar to that of traditional methods, but enables layer-parallelism and thus provides speedup over layer-serial methods through greater concurrency.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Optimizing multigrid reduction-in-time and Parareal coarse-grid operators for linear advection

Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs) in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. Therefore, one of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge–Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2002

Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.

Jothiprasad, Giridhar↗

NASA NTRS1997

Multilevel methods for elliptic problems on unstructured grids

Multilevel methods on unstructured grids for elliptic problems are reviewed. The advantages of these techniques are the flexible approximation of the boundaries of complicated physical domains and the ability to adapt the grid to the resolution of fine scaled structures. Multilevel methods, which include multigrid methods and domain decomposition methods, depend on the correct splitting of appropriate finite element spaces. The standard splittings used in the structured grid case cannot be directly extended to unstructured grids due to their requirement for a hierarchical grid structure. Issues related to the application of multilevel methods to unstructured grids are discussed, including how the coarse spaces and transfer operators are defined and how different types of boundary conditions are treated. An obvious way to generate a coarse mesh is to regrid the physical domain several times. Several alternatives are proposed and discussed: node nested coarse spaces, agglomerated coarse spaces and algebraically generated coarse spaces.

Chan, Tony F.↗

NASA NTRS1990

An investigation of design optimization using a 2-D viscous flow code with multigrid

Computational fluid dynamics (CFD) codes have advanced to the point where they are effective analytical tools for solving flow fields around complex geometries. There is also a need for their use as a design tool to find optimum aerodynamic shapes. In the area of design, however, a difficulty arises due to the large amount of computer resources required by these codes. It is desired to streamline the design process so that a large number of design options and constraints can be investigated without overloading the system. There are several techniques which have been proposed to help streamline the design process. The feasibility of one of these techniques is investigated. The technique under consideration is the interaction of the geometry change with the flow calculation. The problem of finding the value of camber which maximizes the ratio of lift over drag for a particular airfoil is considered. In order to test out this technique, a particular optimization problem was tried. A NACA 0012 airfoil was considered at free stream Mach number of 0.5 with a zero angle of attack. Camber was added to the mean line of the airfoil. The goal was to find the value of camber for which the ratio of lift over drag is a maximum. The flow code used was FLOMGE which is a two dimensional viscous flow solver which uses multigrid to speed up convergence. A hyperbolic grid generation program was used to construct the grid for each value of camber.

Doria, Michael L.↗