Massively parallel finite difference elasticity using block-structured adaptive mesh refinement with a geometric multigrid solver
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Runnels, Brandon (ORCID:0000000330435227)↗
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Cold Plasma Measurements
We have continued the simulation campaign in support of our ongoing magnetospheric cold plasma research project. This project aims to develop the next-generation particle instruments to measure the properties of the cold particle populations in the Earth’s magnetosphere. For this purpose, simulations have been performed with a Particle-In-Cell (PIC) code called the Curvilinear PIC (CPIC). The code is formulated in curvilinear geometry and couples the standard PIC algorithm with algorithms for the generation and adaptation of the underlaying computational mesh. It conforms to complex objects like spacecraft and it can place more grid points in regions where higher resolution is needed. The code also features a scalable solver based on the multigrid algorithm and it is fully parallelized via domain decomposition and MPI.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
TorchBraid: High-Performance Layer-Parallel Training of Deep Neural Networks with MPI and GPU Acceleration
TorchBraid is a high-performance implementation of layer-parallel training for deep neural networks (DNNs) supporting MPI-based parallelism and GPU acceleration. Layer-parallel training has been developed to overcome the serialization inherent in forward and backward propagation of DNNs that limits utilization of computational resources in the strong scaling limit. To achieve this, TorchBraid integrates the PyTorch neural network framework with the state-of-the-art XBraid time-parallel library. Furthermore, this article presents the use and performance of TorchBraid, in addition to solutions for overcoming the algorithmic challenges inherent in combining automatic differentiation with layer-parallel. Results are presented with and without GPU acceleration for the Tiny ImageNet and MNIST image classification data sets, as well as recurrent neural networks. Overall, TorchBraid enables fast training of DNNs, both in a strong and weak scaling context. In addition to the TorchBraid software, several new advances in applying layer-parallel algorithms are detailed. Integration of layer-parallel with data-parallel algorithms is presented for the first time, showing the computational advantages of the combination. Standard deep learning techniques, like batch-normalization, are developed for layer-parallel training. Finally, a new approach combining layer-parallel with spatial coarsening in order to accelerate training for 3D image classification shows roughly a 10× speedup over serial execution.
Layer-parallel↗
Asynchronous Truncated Multigrid-Reduction-in-Time
In this paper, we present the new “asynchronous truncated multigrid-reduction-in-time” (AT-MGRIT) algorithm for introducing time parallelism to the solution of discretized time-dependent problems. The new algorithm is based on the multigrid-reduction-in-time (MGRIT) approach, which, in certain settings, is equivalent to another common multilevel parallel-in-time method, Parareal. In contrast to Parareal and MGRIT that both consider a global temporal grid over the entire time interval on the coarsest level, the AT-MGRIT algorithm uses truncated local time grids on the coarsest level, each grid covering certain temporal subintervals. Further, these local grids can be solved completely in an independent way from each other, which reduces the sequential part of the algorithm and, thus, increases parallelism in the method. Here, we study the effect of using truncated local coarse grids on the convergence of the algorithm, both theoretically and numerically, and show, using challenging nonlinear problems, that the new algorithm consistently outperforms classical Parareal/MGRIT in terms of time to solution.
97 MATHEMATICS AND COMPUTING↗
Adaptive Mesh Refinement for Parallel in Time Methods
The project applied the multigrid-reduction-in-time (MGRIT) algorithm to an existing sub-cycled adaptive mesh refinement (AMR) code to investigate the performance of flows dominated by inertial physics. Previous work demonstrated good performance from MGRIT+AMR applied to flows dominated by diffusive physics. Consistent with previous experience, inertial physics negatively affected convergence rates and performance. Efforts to circumvent this issue by appealing to the physics of turbulence were investigated. It has been demonstrated that scales can be effectively transferred between multigrid levels for a turbulent flow resulting in a) partial convergence observed and b) nearly identical results to sequential time-stepping. Performance improvements have not yet been demonstrated - attempts at coarsening the grid on coarser MG levels compromises the solution quality and leads to divergence. This report summarize the accomplishments for the time-frame from 10/5/2020 to 12/31/2020 with an informal no-cost-extension to 05/20/2021
97 MATHEMATICS AND COMPUTING↗
A New Semistructured Algebraic Multigrid Method
Multigrid methods are well suited to large massively parallel computer architectures because they are mathematically optimal and display good parallelization properties. Since current architecture trends are favoring regular compute patterns to achieve high performance, the ability to express structure has become much more important. The hypre software library provides high-performance multigrid preconditioners and solvers through conceptual interfaces, including a semistructured interface that describes matrices primarily in terms of stencils and logically structured grids. This paper presents a new semistructured algebraic multigrid (SSAMG) method built on this interface. The numerical convergence and performance of a CPU implementation of this method are evaluated for a set of semistructured problems. In conclusion, SSAMG achieves significantly better setup times than hypre’s unstructured AMG solvers and comparable convergence. In addition, the new method is capable of solving more complex problems than hypre’s structured solvers.
97 MATHEMATICS AND COMPUTING↗
Reducing communication in algebraic multigrid with multi-step node aware communication
Algebraic multigrid (AMG) is often viewed as a scalable [Formula: see text] solver for sparse linear systems. Yet, AMG lacks parallel scalability due to increasingly large costs associated with communication, both in the initial construction of a multigrid hierarchy and in the iterative solve phase. This work introduces a parallel implementation of AMG that reduces the cost of communication, yielding improved parallel scalability. It is common in Message Passing Interface (MPI), particularly in the MPI-everywhere approach, to arrange inter-process communication, so that communication is transported regardless of the location of the send and receive processes. Performance tests show notable differences in the cost of intra- and internode communication, motivating a restructuring of communication. In this case, the communication schedule takes advantage of the less costly intra-node communication, reducing both the number and the size of internode messages. Node-centric communication extends to the range of components in both the setup and solve phase of AMG, yielding an increase in the weak and strong scaling of the entire method.
Computer Science↗
A Parallel Solver for Graph Laplacians
Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.
Konolige, Tristan↗
Non-Intrusive Parallel-in-Time Solvers for Partial Differential Equations (Final Report)
Many time-dependent problems and simulations are often modeled using Partial Differential Equations. Traditional modeling approaches that use sequential time-stepping are reaching a bottleneck in optimizing efficiency. The Center of Applied Science and Computing at Lawrence Livermore National Laboratory extensively works on parallelizing these algorithms to leverage the increasing computational power from the growing number of processors in computer hardware. In particular, they aim to design non-intrusive algorithms that can generalize to a variety of problems and sizes without requiring additional information from or modifications on the original problems. Multigrid Reduction in Time (MGRIT) is a parallel-in-time algorithm that is designed to be non-intrusive. This project focuses on increasing the efficiency of MGRIT by approximating the coarse-grid operator using machine learning approaches as a means to find the most non-intrusive, or general, solution.
97 MATHEMATICS AND COMPUTING↗
Multigrid reduction in time with Richardson extrapolation
The advent of exascale computing will leave many users with access to more computational resources than they can simultaneously use, e.g., billion-way parallelism. In particular, this is true for time-dependent simulations that limit parallelism to the spatial domain. One method to add parallelism in time to existing simulation codes and thus take advantage of ever larger compute resources is Multigrid Reduction in Time (MGRIT). The goal is to achieve a smaller time-to-solution through parallelism in time. In this paper, MGRIT is enhanced with Richardson extrapolation in a cost-efficient way to produce a parallel-in-time method with improved accuracy. Overall, this leads to a large improvement in the accuracy per computational cost of MGRIT.
97 MATHEMATICS AND COMPUTING↗
Fast Multigrid Reduction-in-Time for Advection via Modified Semi-Lagrangian Coarse-Grid Operators
Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs) but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. Here, the key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-Lagrangian-based coarse-grid operator that leads to fast and scalable multilevel time integration of linear advection PDEs. The coarse-grid operator is composed of a semi-Lagrangian discretization followed by a correction term, with the correction designed so that the leading-order truncation error of the composite operator is approximately equal to that of the ideal coarse-grid operator. Parallel results show substantial speed-ups over sequential time integration for variable-wave-speed advection problems in one and two spatial dimensions, and using high-order discretizations up to order five. The proposed approach establishes the first practical method that provides small and scalable MGRIT iteration counts for advection problems.
97 MATHEMATICS AND COMPUTING↗
Convergence of the multigrid reduction in time algorithm for the linear elasticity equations
This paper presents some recent advances for parallel-in-time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel-in-time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel-in-time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF-relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel-in-time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. Finally, the paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.
97 MATHEMATICS AND COMPUTING↗
Parallel-in-Time Solution of Hyperbolic PDE Systems via Characteristic-Variable Block Preconditioning
We consider the parallel-in-time solution of both linear and nonlinear hyperbolic partial differential equation (PDE) systems in one spatial dimension. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that intervariable coupling between characteristic variables is weak, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small, while that between the original variables is not. For an ℓ-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of ℓ scalar linear(ized)-advection-like problems, each associated with a different characteristic wave-speed in the underlying linear(ized) PDE. Furthermore, we approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions. For many test problems, the solver converges in just a handful of iterations and with mesh-independent convergence rates.
97 MATHEMATICS AND COMPUTING↗
Array-based, parallel hierarchical mesh refinement algorithms for unstructured meshes
In this paper, we describe an array-based hierarchical mesh refinement capability through uniform refinement of unstructured meshes for efficient solution of PDE's using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial coarse mesh that can be used for a variety of purposes such as in multigrid solvers/preconditioners, to do solution convergence and verification studies and to improve overall parallel efficiency by decreasing I/O bandwidth requirements (by loading smaller meshes and in memory refinement). We also describe a high-order boundary reconstruction capability that can be used to project the new points after refinement using high-order approximations instead of linear projection in order to minimize and provide more control on geometrical errors introduced by curved boundaries.The capability is developed under the parallel unstructured mesh framework "Mesh Oriented dAtaBase" (MOAB Tautges et al. (2004)). We describe the underlying data structures and algorithms to generate such hierarchies in parallel and present numerical results for computational efficiency and effect on mesh quality. Furthermore, we also present results to demonstrate the applicability of the developed capability to study convergence properties of different point projection schemes for various mesh hierarchies and to a multigrid finite-element solver for elliptic problems.
97 MATHEMATICS AND COMPUTING↗
Simulations of plasmas and fluids using anti-symmetric models
ALMA (Anti-symmetric, Large-Moment, Accelerated) is a fast, flexible, and scalable toolkit designed to solve hyperbolic conservation law systems in hybrid supercomputers. Here this manuscript describes the theoretical background and implementation of ALMA, which uses the anti-symmetric formulation of fluids to obtain simple, robust, and easily paralellizable code. Practical GPU acceleration is realized on entire applications with an overall gain factor of 2 to 4. ALMA also provides a parallel, GPU accelerated sparse solver based on geometric multigrid, capable of diagonalizing linear systems with 239 unknowns. Here we demonstrate ALMA's scaling and performance in petascale supercomputers and use standard fluid models to verify the overall approach with canonical benchmark problems.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A PETSc Parallel-in-Time Solver based on MGRIT algorithm
We address the development of a modular implementation of the MGRIT (MultiGrid-In-Time) algorithm to solve linear and nonlinear systems that arise from the discretization of evolutionary models with a parallel-in-time approach in the context of the PETSc (the Portable, Extensible Toolkit for Scientific computing) library. Our aim is to give the opportunity of predicting the performance gain achievable when using the MGRIT approach instead of the Time Stepping integrator (TS). To this end, we analyze the performance parameters of the algorithm that provide a-priori the best number of processing elements and grid levels to use to address the scaling of MGRIT, regarded as a parallel iterative algorithm proceeding along the time dimension.
Mele, Valeria↗
Multigrid methods with space–time concurrency
Here, we consider the comparison of multigrid methods for parabolic partial differential equations that allow space–time concurrency. With current trends in computer architectures leading towards systems with more, but not faster, processors, space–time concurrency is crucial for speeding up time-integration simulations. In contrast, traditional time-integration techniques impose serious limitations on parallel performance due to the sequential nature of the time-stepping approach, allowing spatial concurrency only. This paper considers the three basic options of multigrid algorithms on space–time grids that allow parallelism in space and time: coarsening in space and time, semicoarsening in the spatial dimensions, and semicoarsening in the temporal dimension. We develop parallel software and performance models to study the three methods at scales of up to 16K cores and introduce an extension of one of them for handling multistep time integration. We then discuss advantages and disadvantages of the different approaches and their benefit compared to traditional space-parallel algorithms with sequential time stepping on modern architectures.
97 MATHEMATICS AND COMPUTING↗
Applying Time-Parallelization to Turbulent Flows
Parallelization of the temporal domain is explored for the solution of turbulent flows. Multigrid reduction-in-time (MGRIT) is used to advance the large-scale fluid dynamics in time sequentially on the coarsest space-time grid but propagate the information in time parallel on all other levels. The goal of this process is to accurately and efficiently resolve the coarse-scale turbulence structure and use that to drive the fine-scales of the turbulent flow. The extra forcing from nonlinear multigrid facilitates the coupling and interaction between fine and coarse scales, through which the multiscale nonlinear physics is properly captured. Adaptive mesh refinement is employed to finely resolve only the regions with strong gradients, which provides further computational efficiency. The underlying computational fluid dynamics solver is a fourth-order finite-volume scheme with the standard 4-stage Runge-Kutta method. An advanced approach is devised and implemented to enable MGRIT to solve highly turbulent flows successfully. Furthermore, the method is applied to solve a Taylor-Green vortex problem and a doubleshear-layer turbulent mixing flow. Results are promising, validating that MGRIT with the filtering approach has the potential to efficiently solve general turbulent flows.
Computational Fluid Dynamics↗