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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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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Evaluating asynchronous Schwarz solvers on GPUs

With the commencement of the exascale computing era, we realize that the majority of the leadership supercomputers are heterogeneous and massively parallel. Even a single node can contain multiple co-processors such as GPUs and multiple CPU cores. For example, ORNL’s Summit accumulates six NVIDIA Tesla V100 GPUs and 42 IBM Power9 cores on each node. Synchronizing across compute resources of multiple nodes can be prohibitively expensive. Hence, it is necessary to develop and study asynchronous algorithms that circumvent this issue of bulk-synchronous computing. In this study, we examine the asynchronous version of the abstract Restricted Additive Schwarz method as a solver. We do not explicitly synchronize, but allow the communication between the sub-domains to be completely asynchronous, thereby removing the bulk synchronous nature of the algorithm. We accomplish this by using the one-sided Remote Memory Access (RMA) functions of the MPI standard. We study the benefits of using such an asynchronous solver over its synchronous counterpart. We also study the communication patterns governed by the partitioning and the overlap between the sub-domains on the global solver. Finally, we show that this concept can render attractive performance benefits over the synchronous counterparts even for a well-balanced problem.

Nayak, Pratik↗
DOE OSTI2021

A Scalable Semi-Implicit Barotropic Mode Solver for the MPAS-Ocean

A scalable semi-implicit barotropic mode solver for the ocean component of the model for prediction across scales has been implemented as a competitor to an existing explicit-subcycling scheme to allow faster and more stable simulations while not sacrificing accuracy. The semi-implicit solver adopts the pipelined preconditioned bi-conjugate gradient stabilization algorithm as an iterative solver in conjunction with the restricted additive Schwarz preconditioner that accelerates the convergence rate of the iterative solver. The preconditioner is constructed from a linearized barotropic system that also reorders the system for optimal performance, while the semi-implicit solver deals with the fully nonlinear barotropic system that requires reassembly of the coefficient matrix for every time step. Several numerical experiments, from simple one-dimensional tests to three-dimensional real-world tests, demonstrate that the semi-implicit solver has almost the same accuracy and better parallel scalability compared with the existing scheme while allowing faster and more stable simulations. Furthermore, the semi-implicit solver accelerates the barotropic mode up to 2.9 times faster than the existing scheme on 16,320 processors, leading to an overall runtime speedup of 1.9.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Parallel Dynamics Simulation Using a Krylov-Schwarz Linear Solution Scheme

Fast dynamics simulation of large-scale power systems is a computational challenge because of the need to solve a large set of stiff, nonlinear differential-algebraic equations at every time step. The main bottleneck in dynamic simulations is the solution of a linear system during each nonlinear iteration of Newton’s method. In this paper, we present a parallel Krylov- Schwarz linear solution scheme that uses the Krylov subspacebased iterative linear solver GMRES with an overlapping restricted additive Schwarz preconditioner. As a result, performance tests of the proposed Krylov-Schwarz scheme for several large test cases ranging from 2,000 to 20,000 buses, including a real utility network, show good scalability on different computing architectures.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Accelerating Multivariate Functional Approximation Computation with Domain Decomposition Techniques⋆

Modeling large datasets through Multivariate Functional Approximations (MFA) provide an elegant way to handle many visualization and scientific analysis workflows. The process necessitates scalable data partitioning methods to compute MFA representations efficiently without compromising the accuracy or continuity of the reconstructed solution. We propose a domain -decomposed method for computing the MFA with B -spline bases, which reduces the total work per task and uses a restricted Additive Schwarz (RAS) method to converge the control point data degrees -of -freedom along subdomain boundaries. We provide an in-depth analysis of the parallel approach with domain decomposition solvers, aiming to minimize local subdomain error residuals and recover high -order continuity at subdomain interfaces with appropriate choices of knot overlaps. The communication cost, determined by the overlap regions in the RAS implementation, is optimized to recover the numerical error profile of the single subdomain case. Our proposed method stands in contrast to previous methods, which typically only recover either C 0 or at best C 1 continuity for arbitrary B -spline degree expansions, or those that require post -processing to blend discontinuities in the reconstructed data. We demonstrate the effectiveness of our approach using analytical and real -world datasets in 1D, 2D, and 3D through both strong and weak scaling studies. The performance results indicate that the overall cost of computing the approximation is directly proportional to the underlying nearest -neighbor communication implementation, and is only weakly dependent on the overlap region size that determines the size of the messages. This finding underscores the efficiency and scalability of our proposed method, making it a promising solution for handling large datasets in scientific workflows.

additive Schwarz solvers↗
DOE OSTI2013

HPCG Benchmark Technical Specification

The High Performance Conjugate Gradient (HPCG) benchmark [cite SNL, UTK reports] is a tool for ranking computer systems based on a simple additive Schwarz, symmetric Gauss-Seidel preconditioned conjugate gradient solver. HPCG is similar to the High Performance Linpack (HPL), or Top 500, benchmark [1] in its purpose, but HPCG is intended to better represent how today’s applications perform. In this paper we describe the technical details of HPCG: how it is designed and implemented, what code transformations are permitted and how to interpret and report results.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Ifpack2 User's Guide 1.0

This is the definitive user manual for the I FPACK 2 package in the Trilinos project. I FPACK 2 pro- vides implementations of iterative algorithms (e.g., Jacobi, SOR, additive Schwarz) and processor- based incomplete factorizations. I FPACK 2 is part of the Trilinos T PETRA solver stack, is templated on index, scalar, and node types, and leverages node-level parallelism indirectly through its use of T PETRA kernels. I FPACK 2 can be used to solve to matrix systems with greater than 2 billion rows (using 64-bit indices). Any options not documented in this manual should be considered strictly experimental .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods

In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on finite element method--spectral element method (FEM-SEM) equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order (p = 1) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, requiring specialized solvers to treat the anisotropy. We make use of an element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and Bassi and Rebay (BR2) discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Finally, numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

On performance of Krylov smoothing for fully-coupled AMG preconditioners for VMS resistive MHD

Here, this study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid (AMG) preconditioned Newton- Krylov solution approach applied to a fully-implicit variational multiscale (VMS) nite element (FE) resistive magnetohydrodynamics (MHD) formulation. In this context a Newton iteration is used for the nonlinear system and a Krylov (GMRES) method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the AMG preconditioner for the linear solutions and the performance of the smoothers play a critical role. Krylov smoothers are considered in an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition (DD) with incomplete LU factorization solves on each subdomain. Three time dependent resistive MHD test cases are considered to evaluate the method. The results demonstrate that the GMRES smoother can be faster due to a decrease in the preconditioner setup time and a reduction in outer GMRESR solver iterations, and requires less memory (typically 35% less memory for global GMRES smoother) than the DD ILU smoother.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

On the Degradation of the Effectiveness of Nonlinear Diffusion Acceleration with Parallel Block Jacobi Splitting

MOOSE's radiation transport module Rattlesnake comprises a first order S N solver that can be accelerated with a flexible Nonlinear Diffusion Acceleration. At each NDA iteration, the streaming and collision operator of the S{sub N} equations are (approximately) inverted using a transport sweep. When executed on multiple processors parallel block Jacobi (PBJ) splitting is utilized lagging the processor sub-domain's incoming angular fluxes making the implemented method asynchronous, i.e. the convergence properties of NDA become dependent on the number of concurrent tasks and details of the partitioning. Within this work, we study the NDA convergence properties for the two-dimensional C5G7 benchmark problem PBJ with p = 1, 2, 4, 8, 12, 16, 24, 32, 48, 60, 72, 96 and 144 concurrent tasks. Rosa studied the convergence properties of PBJ splitting methods with and without transport synthetic acceleration (TSA) in slab geometry and two-dimensional Cartesian geometry, respectively. Rosa's iteration scheme lags both the incoming angular fluxes and the scattering source making it very similar to the method described here. Rosa found by means of Fourier analysis that convergence of unaccelerated PBJ can degrade for optically thin subdomains, i.e. for a large number of processors given a fixed domain size. Depending on the choice of parameters application of TSA can lead to an effective algorithm. In contrast to Rosa's work, this work focuses on PBJ splitting for NDA accelerated solution of the multigroup eigenvalue problem. Later, Zerr and Anistratov used their integral transport matrix method in conjunction with PBJ. This iteration scheme is different as it does not lag the scattering source. Nevertheless, Fourier analysis with Diamond Difference spatial discretization showed that the method degrades with increasing the overall size of the spatial domain while keeping the sub-domain size constant. In the case of an infinite medium, the observed spectral radius becomes unity. These references establish the iterative performance of PBJ for one-group fixed source problems discretized on rectangular cells, but for the Rattlesnake first order NDA method, an emphasis lies on multigroup eigenvalue problems. In this setting, the additional complication arises that the low order equations are not exactly inverted, but only to some iterative criterion. Further, as the low order problem is solved using Jacobian-Free Newton Krylov methods, a suitable preconditioner for the GMRES iterations is required. As the effectiveness of the preconditioners may decrease when more concurrent processes are used, we expect the low order solve to impact the overall effectiveness of the NDA method in parallel execution as well. Therefore, this work focuses on the question of how detrimental the asynchronicity of the high and low order iterative solutions is for the effectiveness of NDA solutions of realistic, unstructured grid, multigroup eigenvalue problems. The iterative convergence properties of Rattlesnake's NDA method for multigroup eigenvalue problems are studied for varying number of concurrent processes. Rattlesnake's NDA method is asynchronous because the high order transport sweep uses PBJ splitting and because the low order solve's preconditioners, algebraic multigrid and additive Schwarz, are asynchronous. We found that up to 144 processes the number of Picard iterations increases by a factor of 2-3 affecting but not destroying NDA's parallel performance for the C5G7 benchmark problem, a representative LWR (light water reactor) configuration. The solution of the NDA low order eigenvalue problem posed greater problems than expected for the solvers currently available in Rattlesnake mainly because of problems to effectively precondition the modified low-order problem. As a matter of fact, the low order problem becomes an advection-diffusion problem. Non-robustness was observed for the AMG preconditioned coarse mesh acceleration for more than 60 concurrent processes, which was fixed by increasing the number of permissible V-cycles per AMG call. For fine mesh acceleration AMG proved to be ineffective and we resorted to ASM. For both AMG and ASM, increasing number of linear iterations per nonlinear are observed when increasing the number of processes, but for ASM this problem appears to be worse. The results from this work also indicate that initial development efforts should be diverted from implementation of a high-order parallel sweeping algorithm to the effective and robust iterative solution of the low order NDA multigroup eigenvalue problem. (authors)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Hierarchical Krylov and nested Krylov methods for extreme-scale computing

The solution of large, sparse linear systems is typically a dominant phase of computation for simulations based on partial differential equations, which are ubiquitous in scientific and engineering applications. While preconditioned Krylov methods are widely used and provide many advantages for solving sparse linear systems that do not have highly convergent, geometric multigrid solvers or specialized fast solvers, Krylov methods encounter well-known scaling difficulties for over 10,000 processor cores because each iteration requires at least one vector inner product, which in turn requires a global synchronization that scales poorly because of internode latency. To aid in overcoming these difficulties, we have developed hierarchical Krylov methods and nested Krylov methods in the PETSc library that reduce the number of global inner products required across the entire system (where they are expensive), though freely allow vector inner products across smaller subsets of the entire system (where they are inexpensive) or use inner iterations that do not invoke vector inner products at all. Nested Krylov methods are a generalization of inner-outer iterative methods with two or more layers. Hierarchical Krylov methods are a generalization of block Jacobi and overlapping additive Schwarz methods, where each block itself is solved by Krylov methods on smaller blocks. Conceptually, the hierarchy can continue recursively to an arbitrary number of levels of smaller and smaller blocks. As a specific case, we introduce the hierarchical FGMRES method, or h-FGMRES, and we demonstrate the impact of two-level h-FGMRES with a variable preconditioner on the PFLOTRAN subsurface flow application. We also demonstrate the impact of nested FGMRES, BiCGStab and Chebyshev methods. These hierarchical Krylov methods and nested Krylov methods significantly reduced overall PFLOTRAN simulation time on the Cray XK6 when using 10,000 through 224,000 cores through the combined effects of reduced global synchronization due to fewer global inner products and stronger inner hierarchical or nested preconditioners.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

FROSch Preconditioners for Land Ice Simulations of Greenland and Antarctica

Numerical simulations of Greenland and Antarctic ice sheets involve the solution of large-scale highly nonlinear systems of equations on complex shallow geometries. This work is concerned with the construction of Schwarz preconditioners for the solution of the associated tangent problems, which are challenging for solvers mainly because of the strong anisotropy of the meshes and wildly changing boundary conditions that can lead to poorly constrained problems on large portions of the domain. In this study, two-level generalized Dryja--Smith--Widlund (GDSW)--type Schwarz preconditioners are applied to different land ice problems, i.e., a velocity problem, a temperature problem, as well as the coupling of the former two problems. We employ the message passing interface (MPI)--parallel implementation of multilevel Schwarz preconditioners provided by the package FROSch (fast and robust Schwarz) from the Trilinos library. The strength of the proposed preconditioner is that it yields out-of-the-box scalable and robust preconditioners for the single physics problems. To the best of our knowledge, this is the first time two-level Schwarz preconditioners have been applied to the ice sheet problem and a scalable preconditioner has been used for the coupled problem. The preconditioner for the coupled problem differs from previous monolithic GDSW preconditioners in the sense that decoupled extension operators are used to compute the values in the interior of the subdomains. Additionally, several approaches for improving the performance, such as reuse strategies and shared memory OpenMP parallelization, are explored. In our numerical study we target both uniform meshes of varying resolution for the Antarctic ice sheet as well as nonuniform meshes for the Greenland ice sheet. We present several weak and strong scaling studies confirming the robustness of the approach and the parallel scalability of the FROSch implementation. Among the highlights of the numerical results are a weak scaling study for up to 32,K processor cores (8,K MPI ranks and 4 OpenMP threads) and 566,M degrees of freedom for the velocity problem as well as a strong scaling study for up to 4,K processor cores (and MPI ranks) and 68,M degrees of freedom for the coupled problem.

97 MATHEMATICS AND COMPUTING↗