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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

An aggregation-based nonlinear multigrid solver for two-phase flow and transport in porous media

A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. We find that the resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newton's method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newton's method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.

58 GEOSCIENCES↗
DOE OSTI2014

s-Step Krylov Subspace Methods as Bottom Solvers for Geometric Multigrid

Geometric multigrid solvers within adaptive mesh refinement (AMR) applications often reach a point where further coarsening of the grid becomes impractical as individual sub domain sizes approach unity. At this point the most common solution is to use a bottom solver, such as BiCGStab, to reduce the residual by a fixed factor at the coarsest level. Each iteration of BiCGStab requires multiple global reductions (MPI collectives). As the number of BiCGStab iterations required for convergence grows with problem size, and the time for each collective operation increases with machine scale, bottom solves in large-scale applications can constitute a significant fraction of the overall multigrid solve time. In this paper, we implement, evaluate, and optimize a communication-avoiding s-step formulation of BiCGStab (CABiCGStab for short) as a high-performance, distributed-memory bottom solver for geometric multigrid solvers. This is the first time s-step Krylov subspace methods have been leveraged to improve multigrid bottom solver performance. We use a synthetic benchmark for detailed analysis and integrate the best implementation into BoxLib in order to evaluate the benefit of a s-step Krylov subspace method on the multigrid solves found in the applications LMC and Nyx on up to 32,768 cores on the Cray XE6 at NERSC. Overall, we see bottom solver improvements of up to 4.2x on synthetic problems and up to 2.7x in real applications. This results in as much as a 1.5x improvement in solver performance in real applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2012

Implementation and Optimization of miniGMG - a Compact Geometric Multigrid Benchmark

Multigrid methods are widely used to accelerate the convergence of iterative solvers for linear systems used in a number of different application areas. In this report, we describe miniGMG, our compact geometric multigrid benchmark designed to proxy the multigrid solves found in AMR applications. We explore optimization techniques for geometric multigrid on existing and emerging multicore systems including the Opteron-based Cray XE6, Intel Sandy Bridge and Nehalem-based Infiniband clusters, as well as manycore-based architectures including NVIDIA's Fermi and Kepler GPUs and Intel's Knights Corner (KNC) co-processor. This report examines a variety of novel techniques including communication-aggregation, threaded wavefront-based DRAM communication-avoiding, dynamic threading decisions, SIMDization, and fusion of operators. We quantify performance through each phase of the V-cycle for both single-node and distributed-memory experiments and provide detailed analysis for each class of optimization. Results show our optimizations yield significant speedups across a variety of subdomain sizes while simultaneously demonstrating the potential of multi- and manycore processors to dramatically accelerate single-node performance. However, our analysis also indicates that improvements in networks and communication will be essential to reap the potential of manycore processors in large-scale multigrid calculations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Toward textbook multigrid efficiency for fully implicit resistive magnetohydrodynamics

Multigrid methods can solve some classes of elliptic and parabolic equations to accuracy below the truncation error with a work-cost equivalent to a few residual calculations – so-called “textbook” multigrid efficiency. We investigate methods to solve the system of equations that arise in time dependent magnetohydrodynamics (MHD) simulations with textbook multigrid efficiency. We apply multigrid techniques such as geometric interpolation, full approximate storage, Gauss-Seidel smoothers, and defect correction for fully implicit, nonlinear, second-order finite volume discretizations of MHD. We apply these methods to a standard resistive MHD benchmark problem, the GEM reconnection problem, and add a strong magnetic guide field, which is a critical characteristic of magnetically confined fusion plasmas. We show that our multigrid methods can achieve near textbook efficiency on fully implicit resistive MHD simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Algebraic multigrid for directed graph Laplacian linear systems (NS‐LAMG)

Summary We propose nonsymmetric lean algebraic multigrid (NS‐LAMG), a new algebraic multigrid algorithm for directed graph Laplacian systems that combines ideas from undirected graph Laplacian multigrid solvers and multigrid algorithms for Markov chain stationary distribution systems. Low‐degree elimination, proposed in LAMG for undirected graphs, is generalized to directed graphs and is a key component of NS‐LAMG. In the setup phase, we propose a simple stationary‐aggregation multigrid algorithms for Markov chain stationary distribution systems solver enhanced by low‐degree elimination to find the right null‐space vector that is used for the intergrid transfer operators. Numerical results show that low‐degree elimination improves performance and that NS‐LAMG outperforms generalized minimal residual method with restart and stable bi‐conjugate gradient method for real‐world, directed graph Laplacian linear systems.

Fox, Alyson↗
DOE OSTI2019

Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods

An efficient hp-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to essentially optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on two-dimensional and three-dimensional uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multiphase elliptic interface problems with complex geometry. Extensions, e.g., to non-LDG discretizations and fully unstructured meshes, are briefly discussed.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Multigrid Reduction in Time for Chaotic and Hyperbolic Problems (Final Report)

The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations (PDEs), which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic and chaotic problems of interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic and chaotic problems on future architectures. The chosen methodology for solving these problems parallel-in-time is multigrid, because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. Multigrid is already commonly used in many DOE simulations for scalably and optimally solving space-only PDE problems. The areas of hyperbolic and chaotic problems are chosen because of their relevance to problems of programmatic interest to DOE. However, these problems are also well-known to be difficult for parallel-in-time methods, with the most common method, parareal, diverging in many cases. The current state of-the-art for parallel-in-time at LLNL is the multigrid reduction in time (MGRIT) XBraid package, which also struggles for such problems, while still showing some improvement over parareal. In summary, new methods are needed for an efficient parallel-in-time scheme for hyperbolic and chaotic problems, and this work shall research promising new multigrid methods in this area. In particular, this work shall continue researching the directions from the current collaboration with Dr. Falgout, which are laid out in the work Toward Parallel in Time for Chaotic Dynamical Systems and showed the first known results of a parallel-in-time speedup for a chaotic problem. This work outlines two key improvements to XBraid for chaotic problems, the so-called “theta” and “delta-correction” methods. Here, these two improvements will be further researched and improved (including with a new relaxation method inspired by on Least Squares Shadowing (LSS)) and explored for more complicated problems.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1981

Multigrid solvers on parallel computers

Massively parallel computers, as considered in this investigation, are not yet available. However, a large-scale parallel computer cannot usefully be designed before the hypothetical algorithms which will employ it are studied. Most of the studies of parallel partial differential equations (PDE) solvers are based on solution techniques much slower (on sequential machines) than multigrid methods. Multigrid methods are highly parallelizable. Each of their processes can simultaneously be performed at all grid points. The present investigation is concerned with a preliminary exploration of the potential of multigrid, or, more generally, Multi-Level Adaptive Techniques (MLAT) on computers with many processors. Basic processes are considered, taking into account coarse-grid approximation, relaxation, coarse-grid corrections, full multigrid algorithms, nonlinear problems and eigenvalue problems, fine-to-coarse correction, and chains of problems. Details of parallel multigrid processing are also examined.

Brandt, A.↗
NASA NTRS1986

Multigrid methods in structural mechanics

Although the application of multigrid methods to the equations of elasticity has been suggested, few such applications have been reported in the literature. In the present work, multigrid techniques are applied to the finite element analysis of a simply supported Bernoulli-Euler beam, and various aspects of the multigrid algorithm are studied and explained in detail. In this study, six grid levels were used to model half the beam. With linear prolongation and sequential ordering, the multigrid algorithm yielded results which were of machine accuracy with work equivalent to 200 standard Gauss-Seidel iterations on the fine grid. Also with linear prolongation and sequential ordering, the V(1,n) cycle with n greater than 2 yielded better convergence rates than the V(n,1) cycle. The restriction and prolongation operators were derived based on energy principles. Conserving energy during the inter-grid transfers required that the prolongation operator be the transpose of the restriction operator, and led to improved convergence rates. With energy-conserving prolongation and sequential ordering, the multigrid algorithm yielded results of machine accuracy with a work equivalent to 45 Gauss-Seidel iterations on the fine grid. The red-black ordering of relaxations yielded solutions of machine accuracy in a single V(1,1) cycle, which required work equivalent to about 4 iterations on the finest grid level.

Raju, I. S.↗
NASA NTRS1991

The improved robustness of multigrid elliptic solvers based on multiple semicoarsened grids

Multigrid convergence rates degenerate on problems with stretched grids or anisotropic operators, unless one uses line or plane relaxation. For 3-D problems, only plane relaxation suffices, in general. While line and plane relaxation algorithms are efficient on sequential machines, they are quite awkward and inefficient on parallel machines. A new multigrid algorithm is presented based on the use of multiple coarse grids, that eliminates the need for line or plane relaxation in anisotropic problems. This algorithm was developed and the standard multigrid theory was extended to establish rapid convergence for this class of algorithms. The new algorithm uses only point relaxation, allowing easy and efficient parallel implementation, yet achieves robustness and convergence rates comparable to line and plane relaxation multigrid algorithms. The algorithm described is a variant of Mulder's multigrid algorithm for hyperbolic problems. The latter uses multiple coarse grids to achieve robustness, but is unsuitable for elliptic problems, since its V-cycle convergence rate goes to one as the number of levels increases. The new algorithm combines the contributions from the multiple coarse grid via a local switch, based on the strength of the discrete operator in each coordinate direction.

Naik, Naomi H.↗
NASA NTRS1989

A highly parallel multigrid-like method for the solution of the Euler equations

We consider a highly parallel multigrid-like method for the solution of the two dimensional steady Euler equations. The new method, introduced as filtering multigrid, is similar to a standard multigrid scheme in that convergence on the finest grid is accelerated by iterations on coarser grids. In the filtering method, however, additional fine grid subproblems are processed concurrently with coarse grid computations to further accelerate convergence. These additional problems are obtained by splitting the residual into a smooth and an oscillatory component. The smooth component is then used to form a coarse grid problem (similar to standard multigrid) while the oscillatory component is used for a fine grid subproblem. The primary advantage in the filtering approach is that fewer iterations are required and that most of the additional work per iteration can be performed in parallel with the standard coarse grid computations. We generalize the filtering algorithm to a version suitable for nonlinear problems. We emphasize that this generalization is conceptually straight-forward and relatively easy to implement. In particular, no explicit linearization (e.g., formation of Jacobians) needs to be performed (similar to the FAS multigrid approach). We illustrate the nonlinear version by applying it to the Euler equations, and presenting numerical results. Finally, a performance evaluation is made based on execution time models and convergence information obtained from numerical experiments.

Tuminaro, Ray S.↗
NASA NTRS1993

Mapping robust parallel multigrid algorithms to scalable memory architectures

The convergence rate of standard multigrid algorithms degenerates on problems with stretched grids or anisotropic operators. The usual cure for this is the use of line or plane relaxation. However, multigrid algorithms based on line and plane relaxation have limited and awkward parallelism and are quite difficult to map effectively to highly parallel architectures. Newer multigrid algorithms that overcome anisotropy through the use of multiple coarse grids rather than line relaxation are better suited to massively parallel architectures because they require only simple point-relaxation smoothers. The parallel implementation of a V-cycle multiple semi-coarsened grid (MSG) algorithm or distributed-memory architectures such as the Intel iPSC/860 and Paragon computers is addressed. The MSG algorithms provide two levels of parallelism: parallelism within the relaxation or interpolation on each grid and across the grids on each multigrid level. Both levels of parallelism must be exploited to map these algorithms effectively to parallel architectures. A mapping of an MSG algorithm to distributed-memory architectures that demonstrate how both levels of parallelism can be exploited is described. The results is a robust and effective multigrid algorithm for distributed-memory machines.

Overman, Andrea↗
NASA NTRS1993

Multigrid solution of the Navier-Stokes equations on highly stretched grids with defect correction

Relaxation-based multigrid solvers for the steady incompressible Navier-Stokes equations are examined to determine their computational speed and robustness. Four relaxation methods with a common discretization have been used as smoothers in a single tailored multigrid procedure. The equations are discretized on a staggered grid with first order upwind used for convection in the relaxation process on all grids and defect correction to second order central on the fine grid introduced once per multigrid cycle. A fixed W(1,1) cycle with full weighting of residuals is used in the FAS multigrid process. The resulting solvers have been applied to three 2D flow problems, over a range of Reynolds numbers, on both uniform and highly stretched grids. In all cases the L(sub 2) norm of the velocity changes is reduced to 10(exp -6) in a few 10's of fine grid sweeps. The results from this study are used to draw conclusions on the strengths and weaknesses of the individual relaxation schemes as well as those of the overall multigrid procedure when used as a solver on highly stretched grids.

Sockol, Peter M.↗
NASA NTRS1993

Mapping robust parallel multigrid algorithms to scalable memory architectures

The convergence rate of standard multigrid algorithms degenerates on problems with stretched grids or anisotropic operators. The usual cure for this is the use of line or plane relaxation. However, multigrid algorithms based on line and plane relaxation have limited and awkward parallelism and are quite difficult to map effectively to highly parallel architectures. Newer multigrid algorithms that overcome anisotropy through the use of multiple coarse grids rather than relaxation are better suited to massively parallel architectures because they require only simple point-relaxation smoothers. In this paper, we look at the parallel implementation of a V-cycle multiple semicoarsened grid (MSG) algorithm on distributed-memory architectures such as the Intel iPSC/860 and Paragon computers. The MSG algorithms provide two levels of parallelism: parallelism within the relaxation or interpolation on each grid and across the grids on each multigrid level. Both levels of parallelism must be exploited to map these algorithms effectively to parallel architectures. This paper describes a mapping of an MSG algorithm to distributed-memory architectures that demonstrates how both levels of parallelism can be exploited. The result is a robust and effective multigrid algorithm for distributed-memory machines.

Overman, Andrea↗
NASA NTRS1994

Agglomeration multigrid for the three-dimensional Euler equations

A multigrid procedure that makes use of coarse grids generated by the agglomeration of control volumes is advocated as a practical approach for solving the three dimensional Euler equations on unstructured grids about complex configurations. It is shown that the agglomeration procedure can be tailored to achieve certain coarse grid properties such as the sizes of the coarse grids and aspect ratios of the coarse grid cells. The agglomeration is done as a preprocessing step and runs in linear time. The implications for multigrid of using arbitrary polyhedral coarse grids are discussed. The agglomeration multigrid technique compares very favorably with existing multigrid procedures both in terms of convergence rates and elapsed times. The main advantage of the present approach is the ease with which coarse grids of any desired degree of coarseness may be generated in three dimensions, without being constrained by considerations of geometry. Inviscid flows over a variety of complex configurations are computed using the agglomeration multigrid strategy.

Venkatakrishnan, V.↗
NASA NTRS1995

Multigrid techniques for unstructured meshes

An overview of current multigrid techniques for unstructured meshes is given. The basic principles of the multigrid approach are first outlined. Application of these principles to unstructured mesh problems is then described, illustrating various different approaches, and giving examples of practical applications. Advanced multigrid topics, such as the use of algebraic multigrid methods, and the combination of multigrid techniques with adaptive meshing strategies are dealt with in subsequent sections. These represent current areas of research, and the unresolved issues are discussed. The presentation is organized in an educational manner, for readers familiar with computational fluid dynamics, wishing to learn more about current unstructured mesh techniques.

Mavriplis, D. J.↗
NASA NTRS1994

New multigrid approach for three-dimensional unstructured, adaptive grids

A new multigrid method with adaptive unstructured grids is presented. The three-dimensional Euler equations are solved on tetrahedral grids that are adaptively refined or coarsened locally. The multigrid method is employed to propagate the fine grid corrections more rapidly by redistributing the changes-in-time of the solution from the fine grid to the coarser grids to accelerate convergence. A new approach is employed that uses the parent cells of the fine grid cells in an adapted mesh to generate successively coaser levels of multigrid. This obviates the need for the generation of a sequence of independent, nonoverlapping grids as well as the relatively complicated operations that need to be performed to interpolate the solution and the residuals between the independent grids. The solver is an explicit, vertex-based, finite volume scheme that employs edge-based data structures and operations. Spatial discretization is of central-differencing type combined with a special upwind-like smoothing operators. Application cases include adaptive solutions obtained with multigrid acceleration for supersonic and subsonic flow over a bump in a channel, as well as transonic flow around the ONERA M6 wing. Two levels of multigrid resulted in reduction in the number of iterations by a factor of 5.

Parthasarathy, Vijayan↗
NASA NTRS1999

Solving Upwind-Biased Discretizations: Multigrid Solver Using Semicoarsening - 2

This paper studies a novel multigrid approach to the solution for a second order upwind biased discretization of the convection equation in two dimensions. This approach is based on semi-coarsening and well balanced explicit correction terms added to coarse-grid operators to maintain on coarse-grid the same cross-characteristic interaction as on the target (fine) grid. Colored relaxation schemes are used on all the levels allowing a very efficient parallel implementation. The results of the numerical tests can be summarized as follows: 1) The residual asymptotic convergence rate of the proposed V(0, 2) multigrid cycle is about 3 per cycle. This convergence rate far surpasses the theoretical limit (4/3) predicted for standard multigrid algorithms using full coarsening. The reported efficiency does not deteriorate with increasing the cycle, depth (number of levels) and/or refining the target-grid mesh spacing. 2) The full multi-grid algorithm (FMG) with two V(0, 2) cycles on the target grid and just one V(0, 2) cycle on all the coarse grids always provides an approximate solution with the algebraic error less than the discretization error. Estimates of the total work in the FMG algorithm are ranged between 18 and 30 minimal work units (depending on the target (discretizatioin). Thus, the overall efficiency of the FMG solver closely approaches (if does not achieve) the goal of the textbook multigrid efficiency. 3) A novel approach to deriving a discrete solution approximating the true continuous solution with a relative accuracy given in advance is developed. An adaptive multigrid algorithm (AMA) using comparison of the solutions on two successive target grids to estimate the accuracy of the current target-grid solution is defined. A desired relative accuracy is accepted as an input parameter. The final target grid on which this accuracy can be achieved is chosen automatically in the solution process. the actual relative accuracy of the discrete solution approximation obtained by AMA is always better than the required accuracy; the computational complexity of the AMA algorithm is (nearly) optimal (comparable with the complexity of the FMG algorithm applied to solve the problem on the optimally spaced target grid).

Diskin, Boris↗