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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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DOE OSTI2017

Algebraic Multigrid Benchmark

AMG is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the BoomerAMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL and is very similar to the AMG2013 benchmark with additional optimizations. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem with a 27-point stencil, which can be scaled up and is designed to solve a very large problem. A second problem simulates a time dependent problem, in which successively various smnllcr systems are solved.

Yang, Ulrike↗
DOE OSTI2021

On ParELAG's Parallel Element-based Algebraic Multigrid and its MFEM Miniapps for H(curl) and H(div) Problems: a report including lowest and next to the lowest order numerical results

This paper presents the utilization of element-based algebraic multigrid (AMGe) hierarchies, implemented in the ParELAG (Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers) library, to produce multilevel preconditioners and solvers for H(curl) and H(div) formulations. This involves the construction of hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and AMS (Auxiliary-space Maxwell Solver) or ADS (Auxiliary-space Divergence Solver) on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Parallel Element-based Algebraic Multigrid for H (c url ) and H (div) Problems Using the ParELAG Library

This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the ParELAG (Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers) library, to produce multilevel preconditioners and solvers for H (c url ) and H (div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and AMS (Auxiliary-space Maxwell Solver) or ADS (Auxiliary-space Divergence Solver) on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Parallel performance of algebraic multigrid domain decomposition

Algebraic multigrid (AMG) is a widely used scalable solver and preconditioner for large-scale linear systems resulting from the discretization of a wide class of elliptic PDEs. While AMG has optimal computational complexity, the cost of communication has become a significant bottleneck that limits its scalability as processor counts continue to grow on modern machines. This article examines the design, implementation, and parallel performance of a novel algorithm, algebraic multigrid domain decomposition (AMG-DD), designed specifically to limit communication. The goal of AMG-DD is to provide a low-communication alternative to standard AMG V-cycles by trading some additional computational overhead for a significant reduction in communication cost. Numerical results show that AMG-DD achieves superior accuracy per communication cost compared with AMG, and speedup over AMG is demonstrated on a large GPU cluster.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Manycore Parallel Computing for a Hybridizable Discontinuous Galerkin Nested Multigrid Method

In this work, we present a parallel computing strategy for a hybridizable discontinuous Galerkin (HDG) nested geometric multigrid (GMG) solver. Parallel GMG solvers require a combination of coarse-grain and fine-grain parallelism to improve time-to-solution performance. In this work we focus on fine-grain parallelism. We use Intel's second generation Xeon Phi (Knights Landing) manycore processor. The GMG method achieves ideal convergence rates of 0.2 or less, for high polynomial orders. A matrix free (assembly free) technique is exploited to save considerable memory usage and increase arithmetic intensity. HDG enables static condensation, and due to the discontinuous nature of the discretization, we developed a matrix vector multiply routine that does not require any costly synchronizations or barriers. Our algorithm is able to attain 80% of peak bandwidth performance for higher order polynomials. This is possible due to the data locality inherent in the HDG method. Very high performance is realized for high order schemes, due to good arithmetic intensity, which declines as the order is reduced.

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 OSTI2017

Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT)

Here we develop a two-grid convergence theory for the parallel-in-time scheme known as multigrid reduction in time (MGRIT), as it is implemented in the open-source package [XBraid: Parallel Multigrid in Time, http://llnl.gov/casc/xbraid]. MGRIT is a scalable and multilevel approach to parallel-in-time simulations that nonintrusively uses existing time-stepping schemes, and in a specific two-level setting it is equivalent to the widely known parareal algorithm. The goal of this paper is twofold. First, we present a two-level MGRIT convergence analysis for linear problems where the spatial discretization matrix can be diagonalized, and then apply this analysis to our two basic model problems, the heat equation and the advection equation. One important assumption is that the coarse and fine time-grid propagators can be diagaonalized by the same set of eigenvectors, which is often the case when the same spatial discretization operator is used on the coarse and fine time grids. In many cases, the MGRIT algorithm is guaranteed to converge, and we demonstrate numerically that the theoretically predicted convergence rates are sharp in practice for our model problems. Second, we explore how the convergence of MGRIT compares to the stability of the chosen time-stepping scheme. In particular, we demonstrate that a stable time-stepping scheme does not necessarily imply convergence of MGRIT, although MGRIT with FCF-relaxation always converges for the diffusion dominated problems considered here.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Optimal Polynomial Smoothers and One‐Sided V‐Cycles for Poisson Problems

The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier‐Stokes equations needs robust preconditioning strategies. One such strategy is multigrid. To realize the potential of multigrid methods, effective smoothing strategies are needed. Chebyshev polynomial smoothers, in conjunction with pointwise Jacobi or additive Schwarz methods (ASMs), prove to be an effective smoother. Other polynomial smoothers, however, may provide superior convergence to the multigrid preconditioner. The authors compare the standard Chebyshev polynomial smoothers to both the novel fourth‐kind Chebyshev polynomial smoothers proposed by Lottes as well as smoothers based on the polynomial of best uniform approximation to as proposed by Kraus, Vassilevski, and Zikatanov. At the cost of symmetry, further improvements may be made. For example, a order polynomial smoother on both sides of the V‐cycle may be substituted with an order polynomial smoother on one side at no additional cost. The choice of omitting the postsmoother in favor of higher‐order polynomial presmoothing is advantageous in cases where the multigrid approximation property constant is large. The authors consider a 2D model problem based on finite differences to motivate the choice of polynomial smoother, order, and whether to apply postsmoothing for the target application of high‐order ‐geometric multigrid methods for GPU architectures. Results from both domains demonstrate the substantial improvement of these approaches over the standard Chebyshev polynomial smoother with a symmetric V‐cycle.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Parallel three-dimensional simulations of quasi-static elastoplastic solids

Hypo-elastoplasticity is a flexible framework for modeling the mechanics of many hard materials under small elastic deformation and large plastic deformation. Under typical loading rates, most laboratory tests of these materials happen in the quasi-static limit, but there are few existing numerical methods tailor-made for this physical regime. Here, we extend to three dimensions a recent projection method for simulating quasi-static hypo-elastoplastic materials. The method is based on a mathematical correspondence to the incompressible Navier–Stokes equations, where the projection method of Chorin (1968) is an established numerical technique. We develop and utilize a three-dimensional parallel geometric multigrid solver employed to solve a linear system for the quasi-static projection. Our method is tested through simulation of three-dimensional shear band nucleation and growth, a precursor to failure in many materials. As an example system, we employ a physical model of a bulk metallic glass based on the shear transformation zone theory, but the method can be applied to any elastoplasticity model. We consider several examples of three-dimensional shear banding, and examine shear band formation in physically realistic materials with heterogeneous initial conditions under both simple shear deformation and boundary conditions inspired by friction welding.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A non-intrusive parallel-in-time adjoint solver with the XBraid library

In this paper, an adjoint solver for the multigrid-in-time software library XBraid is presented. XBraid provides a non-intrusive approach for simulating unsteady dynamics on multiple processors while parallelizing not only in space but also in the time domain (XBraid: Parallel multigrid in time, http://llnl.gov/casc/xbraid). It applies an iterative multigrid reduction in time algorithm to existing spatially parallel classical time propagators and computes the unsteady solution parallel in time. Techniques from Automatic Differentiation are used to develop a consistent discrete adjoint solver which provides sensitivity information of output quantities with respect to design parameter changes. The adjoint code runs backwards through the primal XBraid actions and accumulates gradient information parallel in time. It is highly non-intrusive as existing adjoint time propagators can easily be integrated through the adjoint interface. The adjoint code is validated on advection-dominated flow with periodic upstream boundary condition. We report it provides similar strong scaling results as the primal XBraid solver and offers great potential for speeding up the overall computational costs for sensitivity analysis using multiple processors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Reaching bandwidth saturation using transparent injection parallelization

Although logically available, applications may not exploit enough instantaneous communication concurrency to maximize network utilization on HPC systems. This is exacerbated in hybrid programming models that combine single program multiple data with OpenMP or CUDA. We present the design of a “multi-threaded” runtime able to transparently increase the instantaneous network concurrency and to provide near saturation bandwidth, independent of the application configuration and dynamic behavior. The runtime offloads communication requests from application level tasks to multiple communication servers. The servers use system specific performance models to attain network saturation. Our techniques alleviate the need for spatial and temporal application level message concurrency optimizations. Experimental results show improved message throughput and bandwidth by as much as 150% for 4 KB messages on InfiniBand and by as much as 120% for 4 KB messages on Cray Aries. For more complex operations such as all-to-all collectives, we observe as much as 30% speedup. This translates into 23% speedup on 12,288 cores for a NAS FT implemented using FFTW. We observe as much as 76% speedup on 1500 cores for an already optimized UPC+OpenMP geometric multigrid application using hybrid parallelism. For the geometric multigrid GPU implementation, we observe as much as 44% speedup on 512 GPUs.

Chaimov, Nicholas↗
DOE OSTI2017

Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study

The need for parallelism in the time dimension is being driven by changes in computer architectures, where performance increases are now provided through greater concurrency, not faster clock speeds. This creates a bottleneck for sequential time marching schemes because they lack parallelism in the time dimension. Multigrid reduction in time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficiency is defined as achieving similar performance when compared to an equivalent linear problem. The benchmark nonlinear problem is the p-Laplacian, where p=4 corresponds to a well-known nonlinear diffusion equation and p=2 corresponds to the standard linear diffusion operator, our benchmark linear problem. The key difficulty encountered is that the nonlinear time-step solver becomes progressively more expensive on coarser time levels as the time-step size increases. To overcome such difficulties, multigrid research has historically targeted an accumulated body of experience regarding how to choose an appropriate solver for a specific problem type. To that end, this paper develops a library of MGRIT optimizations and modifications, most important an alternate initial guess for the nonlinear time-step solver and delayed spatial coarsening, that will allow many nonlinear parabolic problems to be solved with parallel scaling behavior comparable to the corresponding linear problem.

parareal↗
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↗
DOE OSTI2023

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 parallelin-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 implemented in a high-performance but general way in XBraid and explored for more complicated problems. We will additionally research, as time allows, improvements to these techniques, as well as multigrid relaxation techniques based on Least Squares Shadowing (LSS by Wang) and a nonintrusive block tridiagonal solver based on MGRIT, called TriMGRIT.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Scaling Structured Multigrid to 500K+ Cores Through Coarse-Grid Redistribution

The effcient solution of sparse, linear systems resulting from the discretization of partial differential equations is crucial to the performance of many physics-based simulations. The algorithmic optimality of multilevel approaches for common discretizations makes them a good candidate for an efficient parallel solver. Yet, modern architectures for high-performance computing systems continue to challenge the parallel scalability of multilevel solvers. While algebraic multigrid methods are robust for solving a variety of problems, the increasing importance of data locality and cost of data movement in modern architectures motivates the need to carefully exploit structure in the problem. Robust logically structured variational multigrid methods, such as Black Box Multigrid (BoxMG), maintain structure throughout the multigrid hierarchy. This avoids indirection and increased coarse- grid communication costs typical in parallel algebraic multigrid. Nevertheless, the parallel scalability of structured multigrid is challenged by coarse-grid problems where the overhead in communication dominates computation. In this paper, an algorithm is introduced for redistributing coarse-grid problems through incremental agglomeration. Guided by a predictive performance model, this algorithm provides robust redistribution decisions for structured multilevel solvers. A two-dimensional diffusion problem is used to demonstrate the significant gain in performance of this algorithm over the previous approach that used agglomeration to one processor. In addition, the parallel scalability of this approach

97 MATHEMATICS AND COMPUTING↗