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At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

A scalable, fully implicit algorithm for the reduced two-field low-β extended MHD model

Here, we demonstrate a scalable fully implicit algorithm for the two-field low-β extended MHD model. This reduced model describes plasma behavior in the presence of strong guide fields, and is of significant practical impact both in nature and in laboratory plasmas. The model displays strong hyperbolic behavior, as manifested by the presence of fast dispersive waves, which make a fully implicit treatment very challenging. In this study, we employ a Jacobian-free Newton–Krylov nonlinear solver, for which we propose a physics-based preconditioner that renders the linearized set of equations suitable for inversion with multigrid methods. As a result, the algorithm is shown to scale both algorithmically (i.e., the iteration count is insensitive to grid refinement and timestep size) and in parallel in a weak-scaling sense, with the wall-clock time scaling weakly with the number of cores for up to 4096 cores. For a 4096 × 4096 mesh, we demonstrate a wall-clock-time speedup of ~6700 with respect to explicit algorithms. The model is validated linearly (against linear theory predictions) and nonlinearly (against fully kinetic simulations), demonstrating excellent agreement.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Algebraic Multigrid Solvers for Coupled PDE Systems (Final Report – B619169)

The Pennsylvania State University (“Subcontractor”) worked on the design of multigrid solvers for coupled systems of partial differential equations arising in numerical modeling of various applications, with a main emphasis on the design of new optimal algebraic multigrid interpolation. Generally, the aim of this work was to develop geometric and algebraic multilevel solvers that are robust and lend themselves to efficient implementation on massively parallel heterogeneous computers. The research in these areas built on previous works, focusing on the following topics: (1) design and analysis of algebraic coarsening algorithms for coupled PDE systems including Stokes equation, Maxwell equation and linear elasticity; (2) development of non-Galerkin coarsening techniques for the Wilson Dirac system; and (3) the use of this same Wilson MG solver for preconditioning the Overlap and Domain Wall formulations of the Dirac equation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A scalable multidimensional fully implicit solver for Hall magnetohydrodynamics

We propose an optimally performant fully implicit algorithm for the Hall magnetohydrodynamics (HMHD) equations based on multigrid-preconditioned Jacobian-free Newton-Krylov methods. HMHD is a challenging system to solve numerically because it supports stiff fast dispersive waves. The preconditioner is formulated using an operator-split approximate block factorization (Schur complement), informed by physics insight. We use a vector-potential formulation (instead of a magnetic field one) to allow a clean segregation of the problematic $\nabla$ x $\nabla$ x operator in the electron Ohm's law subsystem. This segregation allows the formulation of an effective damped block-Jacobi smoother for multigrid. We demonstrate by analysis that our proposed block-Jacobi iteration is convergent and has the smoothing property. The resulting HMHD solver is verified linearly with wave propagation examples, and nonlinearly with the GEM challenge reconnection problem by comparison against another HMHD code. We demonstrate the excellent algorithmic and parallel performance of the algorithm up to 16384 MPI tasks in two dimensions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
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↗
DOE OSTI2020

Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

This study presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time-stepping operator, analyzed directly and bounded in the norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time-stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge--Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge--Kutta schemes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Efficient solvers for hybridized three-field mixed finite element coupled poromechanics

We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures.

42 ENGINEERING↗
DOE OSTI2022

Generalized Bootstrap AMG and AIR-AMG for coupled PDE systems with a focus on spacetime discretizatoins

The Pennsylvania State University (“Subcontractor”) has worked on the design of new algebraic, parallel, multilevel methods that obtain the full space and time solution of systems of PDEs. In particular, the PI and his collaborators explored semi-intrusive approaches based on algebraic multigrid (AMG). The focus of the research has been on the development of these techniques for the Euler equations in 1d and 2d. The overall research focused on the development of adaptive AIR (approximate ideal restriction) AMG solvers for these problems. The PI also explored the use of smoothed aggregation and root-node energy-based AMG solvers for these problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

A divide-conquer-recombine algorithmic paradigm for large spatiotemporal quantum molecular dynamics simulations

We introduce an extension of the divide-and-conquer (DC) algorithmic paradigm called divide-conquer-recombine (DCR) to perform large quantum molecular dynamics (QMD) simulations on massively parallel supercomputers, in which interatomic forces are computed quantum mechanically in the framework of density functional theory (DFT). In DCR, the DC phase constructs globally informed, overlapping local-domain solutions, which in the recombine phase are synthesized into a global solution encompassing large spatiotemporal scales. For the DC phase, we design a lean divide-and-conquer (LDC) DFT algorithm, which significantly reduces the prefactor of the O(N) computational cost for N electrons by applying a density-adaptive boundary condition at the peripheries of the DC domains. Our globally scalable and locally efficient solver is based on a hybrid real-reciprocal space approach that combines: (1) a highly scalable real-space multigrid to represent the global charge density; and (2) a numerically efficient plane-wave basis for local electronic wave functions and charge density within each domain. Hybrid space-band decomposition is used to implement the LDC-DFT algorithm on parallel computers. A benchmark test on an IBM Blue Gene/Q computer exhibits an isogranular parallel efficiency of 0.984 on 786 432 cores for a 50.3 × 10{sup 6}-atom SiC system. As a test of production runs, LDC-DFT-based QMD simulation involving 16 661 atoms is performed on the Blue Gene/Q to study on-demand production of hydrogen gas from water using LiAl alloy particles. As an example of the recombine phase, LDC-DFT electronic structures are used as a basis set to describe global photoexcitation dynamics with nonadiabatic QMD (NAQMD) and kinetic Monte Carlo (KMC) methods. The NAQMD simulations are based on the linear response time-dependent density functional theory to describe electronic excited states and a surface-hopping approach to describe transitions between the excited states. A series of techniques are employed for efficiently calculating the long-range exact exchange correction and excited-state forces. The NAQMD trajectories are analyzed to extract the rates of various excitonic processes, which are then used in KMC simulation to study the dynamics of the global exciton flow network. This has allowed the study of large-scale photoexcitation dynamics in 6400-atom amorphous molecular solid, reaching the experimental time scales.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Scalable DPG multigrid solver for Helmholtz problems: A study on convergence

This paper presents a scalable multigrid preconditioner targeting large-scale systems arising from discontinuous Petrov–Galerkin (DPG) discretizations of high-frequency wave operators. This work is built on previously developed multigrid preconditioning techniques of Petrides and Demkowicz (Comput. Math. Appl. 87 (2021) pp. 12–26) and extends the convergence results from $\mathscr{O}$(10 7 ) degrees of freedom (DOFs) to $\mathscr{O}$(10 9 ) DOFs using a new scalable parallel MPI/OpenMP implementation. Novel contributions of this paper include an alternative definition of coarse-grid systems based on restriction of fine-grid operators, yielding superior convergence results. In the uniform refinement setting, a detailed convergence study is provided, demonstrating h and p robust convergence and linear scaling with respect to the wave frequency. Finally, the paper concludes with numerical results on hp -adaptive simulations including a large-scale seismic modeling benchmark problem with high material contrast.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Implementation of a Mesh refinement algorithm into the quasi-static PIC code QuickPIC

Plasma-based acceleration (PBA) has emerged as a promising candidate for the accelerator technology used to build a future linear collider and/or an advanced light source. In PBA, a trailing or witness particle beam is accelerated in the plasma wave wakefield (WF) created by a laser or particle beam driver. The WF is often nonlinear and involves the crossing of plasma particle trajectories in real space and thus particle-in-cell methods are used. The distance over which the drive beam evolves is several orders of magnitude larger than the wake wavelength. This large disparity in length scales is amenable to the quasi-static approach. Three-dimensional (3D), quasi-static (QS), particle-in-cell (PIC) codes, e.g., QuickPIC, have been shown to provide high fidelity simulation capability with 2-4 orders of magnitude speedup over 3D fully explicit PIC codes. In PBA, the witness beam needs to be matched to the focusing forces of the WF to reduce the emittance growth. In some linear collider designs, the matched spot size of the witness beam can be 2 to 3 orders of magnitude smaller than the spot size (and wavelength) of the wakefield. Such an additional disparity in length scales is ideal for mesh refinement where the WF within the witness beam is described on a finer mesh than the rest of the WF. A mesh refinement scheme is described that has been implemented into the 3D QS PIC code, QuickPIC. Very fine (high) resolution is used in a small spatial region that includes the witness beam and progressively coarser resolutions in the rest of the simulation domain. A fast multigrid Poisson solver has been implemented for the field solve on the refined meshes and a Fast Fourier Transform (FFT) based Poisson solver is used for the coarse mesh. The code has been parallelized with both MPI and OpenMP, and the parallel scalability has also been improved by using pipelining. A preliminary adaptive mesh refinement technique is described to optimize the computational time for simulations with an evolving witness beam size. Several test problems are used to verify that the mesh refinement algorithm provides accurate results. Additionally, the results are benchmarked against highly resolved simulations exhibiting near-azimuthal symmetry, performed using QPAD—a novel hybrid QS PIC code that uses a PIC description in the coordinates (r, ct – z) and a gridless description in the azimuthal angle, Φ.

Linear collider↗
DOE OSTI2018

A scalable geometric multigrid solver for nonsymmetric elliptic systems with application to variable-density flows

A geometric multigrid algorithm is introduced for solving nonsymmetric linear systems resulting from the discretization of the variable density Navier–Stokes equations on nonuniform structured rectilinear grids and high-Reynolds number flows. The restriction operation is defined such that the resulting system on the coarser grids is symmetric, thereby allowing for the use of efficient smoother algorithms. To achieve an optimal rate of convergence, the sequence of interpolation and restriction operations are determined through a dynamic procedure. A parallel partitioning strategy is introduced to minimize communication while maintaining the load balance between all processors. To test the proposed algorithm, we consider two cases: 1) homogeneous isotropic turbulence discretized on uniform grids and 2) turbulent duct flow discretized on stretched grids. Testing the algorithm on systems with up to a billion unknowns shows that the cost varies linearly with the number of unknowns. This O (N) behavior confirms the robustness of the proposed multigrid method regarding ill-conditioning of large systems characteristic of multiscale high-Reynolds number turbulent flows. The robustness of our method to density variations is established by considering cases where density varies sharply in space by a factor of up to 104, showing its applicability to two-phase flow problems. Strong and weak scalability studies are carried out, employing up to 30,000 processors, to examine the parallel performance of our implementation. Excellent scalability of our solver is shown for a granularity as low as 104 to 105 unknowns per processor. At its tested peak throughput, it solves approximately 4 billion unknowns per second employing over 16,000 processors with a parallel efficiency higher than 50%.

Computer Science↗
DOE OSTI2021

A Segregated Approach for Modeling the Electrochemistry in the 3-D Microstructure of Li-Ion Batteries and Its Acceleration Using Block Preconditioners

Abstract Battery performance is strongly correlated with electrode microstructure. Electrode materials for lithium-ion batteries have complex microstructure geometries that require millions of degrees of freedom to solve the electrochemical system at the microstructure scale. A fast-iterative solver with an appropriate preconditioner is then required to simulate large representative volume in a reasonable time. In this work, a finite element electrochemical model is developed to resolve the concentration and potential within the electrode active materials and the electrolyte domains at the microstructure scale, with an emphasis on numerical stability and scaling performances. The block Gauss-Seidel (BGS) numerical method is implemented because the system of equations within the electrodes is coupled only through the nonlinear Butler–Volmer equation, which governs the electrochemical reaction at the interface between the domains. The best solution strategy found in this work consists of splitting the system into two blocks—one for the concentration and one for the potential field—and then performing block generalized minimal residual preconditioned with algebraic multigrid, using the FEniCS and the Portable, Extensible Toolkit for Scientific Computation libraries. Significant improvements in terms of time to solution (six times faster) and memory usage (halving) are achieved compared with the MUltifrontal Massively Parallel sparse direct Solver. Additionally, BGS experiences decent strong parallel scaling within the electrode domains. Last, the system of equations is modified to specifically address numerical instability induced by electrolyte depletion, which is particularly valuable for simulating fast-charge scenarios relevant for automotive application.

25 ENERGY STORAGE↗
DOE OSTI2021

Tusas: A fully implicit parallel approach for coupled phase-field equations

In this study, we develop a fully-coupled, fully-implicit approach for phase-field modeling of solidification in metals and alloys. Predictive simulation of solidification in pure metals and metal alloys remains a significant challenge in the field of materials science, as microstructure formation during the solidification process plays a critical role in the properties and performance of the solid material. Our simulation approach consists of a finite element spatial discretization of the fully-coupled nonlinear system of partial differential equations at the microscale, which is treated implicitly in time with a preconditioned Jacobian-free Newton-Krylov method. The approach is algorithmically scalable as well as efficient due to an effective preconditioning strategy based on algebraic multigrid and block factorization. We implement this approach in the open-source Tusas framework, which is a general, flexible tool developed in C++ for solving coupled systems of nonlinear partial differential equations. The performance of our approach is analyzed in terms of algorithmic scalability and efficiency, while the computational performance of Tusas is presented in terms of parallel scalability and efficiency on emerging heterogeneous architectures. We demonstrate that modern algorithms, discretizations, and computational science, and heterogeneous hardware provide a robust route for predictive phase-field simulation of microstructure evolution during additive manufacturing.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Limitation of parallel flow in double diffusive convection: Two- and three-dimensional transitions in a horizontal porous domain

Two-dimensional (2D) and three-dimensional (3D) numerical simulations of double diffusion natural convection in an elongated enclosure filled with a binary fluid saturating a porous medium are carried out in the present work. The Boussinesq approximation is made in the formulation of the problem, and Neumann boundary conditions for temperature and concentration are adopted, respectively, on vertical and horizontal walls of the cavity. The used numerical method is based on the control volume approach, with the third order quadratic upstream interpolation scheme in approximating the advection terms. A semi implicit method algorithm is used to handle the velocity-pressure coupling. To avoid the excessively high computer time inherent to the solution of 3D natural convection problems, full approximation storage with full multigrid method is used to solve the problem. A wide range of the controlling parameters (Rayleigh-Darcy number Ra, lateral aspect ratio Ay, Lewis number Le, and the buoyancy ration N) is investigated. We clearly show that increasing the depth of the cavity (i.e., the lateral aspect ratio) has an important effect on the flow patterns. The 2D perfect parallel flows obtained for small lateral aspect ratio are drastically destabilized by increasing the cavity lateral dimension. This yields a 3D fluid motion with a much more complex flow pattern and the usually considered 2D parallel flow model cannot be applied.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Parallel-in-Time Integration for Nonlinear Hyperbolic Problems (Final Report)

The work for the subcontract was situated in the area of parallel-in-time integration for hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. Over the past two decades, much progress has been made in this area; however, this progress has largely been limited to diffusion-dominated PDEs, with some recent success in scalar linear hyperbolic PDEs. Given the ubiquity of numerical simulations of hyperbolic PDEs throughout the sciences, in particular, nonlinear hyperbolic systems, there is a strong need to develop efficient parallel-in-time techniques for hyperbolic problems beyond simple scalar and linear cases, which is the main topic of this subcontract. The main focus of the work was to further develop and perfect coarse-grid operators for the Multigrid Reduction-inTime (MGRIT) method applied to hyperbolic PDEs that were recently proposed in PhD thesis, based on a modified semi-Lagrangian approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Scalable multilevel Monte Carlo methods exploiting parallel redistribution on coarse levels

Here, we study an element agglomeration coarsening strategy that requires data redistribution at coarse levels when the number of coarse elements becomes smaller than the number of MPI processes used on the finest level. The overall procedure generates coarse elements (general unstructured unions of fine grid elements) within the framework of element-based algebraic multigrid methods (or AMGe) studied previously. The AMGe-generated coarse spaces have the ability to exhibit approximation properties of the same order as the fine-level spaces since by construction they contain the piecewise polynomials of the same order as on the fine level. These approximation properties are key for the successful use of AMGe in multilevel solvers for nonlinear partial differential equations as well as for multilevel Monte Carlo (MLMC) simulations. The ability to coarsen without being constrained by the number of MPI processes, as described in the present paper, allows to improve the scalability of these solvers as well as the overall MLMC method. The paper illustrates this latter fact with detailed scalability study of MLMC simulations applied to model Darcy equations with a stochastic log-normal permeability field.

AMGe↗
DOE OSTI2018

Deploy threading in Nalu solver stack

The goal of the ExaWind project is to enable predictive simulations of wind farms composed of many MW-scale turbines situated in complex terrain. Predictive simulations will require computational fluid dynamics (CFD) simulations for which the mesh resolves the geometry of the turbines, and captures the rotation and large deflections of blades. Whereas such simulations for a single turbine are arguably petascale class, multi-turbine wind farm simulations will require exascale-class resources. The primary code in the ExaWind project is Nalu, which is an unstructured-grid solver for the acousticallyincompressible Navier-Stokes equations, and mass continuity is maintained through pressure projection. The model consists of the mass-continuity Poisson-type equation for pressure and a momentum equation for the velocity. For such modeling approaches, simulation times are dominated by linear-system setup and solution for the continuity and momentum systems. For the ExaWind challenge problem, the moving meshes greatly affect overall solver costs as re-initialization of matrices and re-computation of preconditioners is required at every time step In this Milestone, we examine the effect of threading on the solver stack performance against flat-MPI results obtained from previous milestones using Haswell performance data full-turbine simulations. Whereas the momentum equations are solved only with the Trilinos solvers, we investigate two algebraic-multigrid preconditioners for the continuity equations: Trilinos/Muelu and HYPRE/BoomerAMG. These two packages embody smoothed-aggregation and classical Ruge-Stiiben AMG methods, respectively. In our FY18 Q2 report, we described our efforts to improve setup and solve of the continuity equations under flat-MPI parallelism. While significant improvement was demonstrated in the solve phase, setup times remained larger than expected. Starting with the optimized settings described in the Q2 report, we explore here simulation performance where OpenMP threading is employed in the solver stack. For Trilinos, threading is acheived through the Kokkos abstraction where, whereas HYPRE/BoomerAMG employs straight OpenMP. We examined results for our mid-resolution baseline turbine simulation configuration (229M DOF). Simulations on 2048 Haswell cores explored the effect of decreasing the number of MPI ranks while increasing the number of threads. Both HYPRE and Trilinos exhibited similar overal solution times, and both showed dramatic increases in simulation time in the shift from MPI ranks to OpenMP threads. This increase is attributed to the large amount of work per MPI rank starting at the single-thread configuration. Decreasing MPI ranks, while increasing threads, may be increasing simulation time due to thread synchronization and start-up overhead contributing to the latency and serial time in the model. These result showed that an MPI+OpenMP parallel decomposition will be more effective as the amount per MPI rank computation per MPI rank decreases and the communication latency increases. This idea was demonstrated in a strong scaling study of our low-resolution baseline model (29M DOF) with the Trilinos-HYPRE configuration. While MPI-only results showed scaling improvement out to about 1536 cores, engaging threading carried scaling improvements out to 4128 cores — roughly 7000 DOF per core. This is an important result as improved strong scaling is needed for simulations to be executed over sufficiently long simulated durations (i.e., for many timesteps). In addition to threading work described above, the team examined solver-performance improvements by exploring communication-overhead in the HYPRE-GMRES implementation through a communicationoptimal- GMRE algorithm (CO-GMRES), and offloading compute-intensive solver actions to GPUs. To those ends, a HYPRE mini-app was allow us to easily test different solver approaches and HYPRE parameter settings without running the entire Nalu code. With GPU acceleration on the Summitdev supercomputer, a 20x speedup was achieved for the overall preconditioner and solver execution time for the mini-app. A study on Haswell processors showed that CO-GMRES provides benefits as one increases MPI ranks.

17 WIND ENERGY↗
DOE OSTI2024

Parallel-in-Time Methods for Method-of-Lines Discretizations of Nonlinear Hyperbolic PDEs and Systems (Final Report)

The work for the subcontract is situated in the area of parallel-in-time integration for hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. The work in this subcontract builds on a variety of results that were obtained, as part of the work performed for Subcontract No. B648355, for the Multigrid Reduction-in-Time (MGRIT) method from [1] applied to hyperbolic PDEs. This subcontract extends these results further to more efficient methods and to the case of method-of-lines discretizations for nonlinear hyperbolic PDES and systems of PDEs. The following is a summary of the research performed and results achieved during milestone periods 1, 2 and 3 by the PI (Hans De Sterck) and Postdoctoral Research Associate (Oliver Krzysik), for required tasks 1-4 (as listed in the Statement of Work): Research over the previous year has been split into three main projects: (i) solution of acoustic equation system; (ii) solution of nonlinear scalar hyperbolic PDEs; (iii) solution of nonlinear hyperbolic systems of PDEs.

97 MATHEMATICS AND COMPUTING↗