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DOE OSTI2016

A Full Multigrid Method for the Self-Adjoint Angular Flux Equation Based on Cellwise Block Jacobi Iteration

Previous work by Densmore et al. has demonstrated that multigrid methods for the self-adjoint angular flux (SAAF) equation with cellwise block Jacobi smoothing can be extremely effective at solving for the angular flux in one dimension. The extension of that work to two dimensions demonstrated that while the smoother works well, the use of a direct coarse-grid approximation can severely limit the rate of convergence for optically thin cells. In that paper, this convergence limit was evaded through the use of a partially Galerkin coarse-grid approximation. However, relying on a Galerkin coarse-grid approximation is not ideal as constructing the coarse-grid discretization in this approach is complicated. The motivation for this summary is to find a method that can work without the need for a Galerkin coarse-grid approximation. The Full Multigrid (FMG) method is well known as a method that can theoretically solve a problem to its discretization accuracy with optimal linear computational complexity, that is in just O(N) operations for N unknowns. Indeed, we develop an FMG method for the 2-D SAAF equation that yields discretization accuracy and does not require the use of a Galerkin coarse-grid approximation. Until recently, aside from proofs about the order of the error for FMG, little work had been done in quantifying when and how FMG can obtain discretization accuracy. However, recent work by Rodrigo et al. has introduced a Fourier analysis technique called worst-case right-hand side analysis used to calculate the FMG accuracy measure. The FMG accuracy measure is the upper bound on the constant of proportionality between the error of the FMG solution and the discretization error. In this summary, we develop and implement an FMG method for the 2-D Cartesian SAAF equations. We then perform worst-case right-hand side analysis and give numerical examples that demonstrate under what circumstances the method can solve the 2-D SAAF equations to discretization accuracy, for optically thin cells, without the need for a Galerkin coarse-grid approximation. (authors)

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Multilevel well modeling in aggregation-based nonlinear multigrid for multiphase flow in porous media

A full approximation scheme (FAS) nonlinear multigrid solver for two-phase flow and transport problems driven by wells with multiple perforations is developed here. It is an extension to our previous work on FAS solvers for diffusion and transport problems. The solver is applicable to discrete problems defined on unstructured grids as the coarsening algorithm is aggregation-based and algebraic. To construct coarse basis that can better capture the radial flow near wells, coarse grids in which perforated well cells are not near the coarse-element interface are desired. This is achieved by an aggregation algorithm proposed in this paper that makes use of the location of well cells in the cell-connectivity graph. Numerical examples in which the FAS solver is compared against Newton's method on benchmark problems are given. In particular, for a refined version of the SAIGUP model, the FAS solver is at least 35% faster than Newton's method for time steps with a CFL number greater than 10.

58 GEOSCIENCES↗
DOE OSTI2014

Non-Galerkin Coarse Grids for Algebraic Multigrid

Algebraic multigrid (AMG) is a popular and effective solver for systems of linear equations that arise from discretized partial differential equations. And while AMG has been effectively implemented on large scale parallel machines, challenges remain, especially when moving to exascale. Particularly, stencil sizes (the number of nonzeros in a row) tend to increase further down in the coarse grid hierarchy, and this growth leads to more communication. Therefore, as problem size increases and the number of levels in the hierarchy grows, the overall efficiency of the parallel AMG method decreases, sometimes dramatically. This growth in stencil size is due to the standard Galerkin coarse grid operator, $P^T A P$, where $P$ is the prolongation (i.e., interpolation) operator. For example, the coarse grid stencil size for a simple three-dimensional (3D) seven-point finite differencing approximation to diffusion can increase into the thousands on present day machines, causing an associated increase in communication costs. We therefore consider algebraically truncating coarse grid stencils to obtain a non-Galerkin coarse grid. First, the sparsity pattern of the non-Galerkin coarse grid is determined by employing a heuristic minimal “safe” pattern together with strength-of-connection ideas. Second, the nonzero entries are determined by collapsing the stencils in the Galerkin operator using traditional AMG techniques. The result is a reduction in coarse grid stencil size, overall operator complexity, and parallel AMG solve phase times.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

An algebraic multigrid method for Q 2 - Q 1 mixed discretizations of the Navier-Stokes equations: AMG for Q 2 - Q 1 discretization

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily colocated at mesh points. Specifically, we investigate a Q 2 -Q 1 mixed finite element discretization of the incompressible Navier–Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees of freedom (DOFs) are defined at spatial locations where there are no corresponding pressure DOFs. Thus, AMG approaches leveraging this colocated structure are not applicable. This article instead proposes an automatic AMG coarsening that mimics certain pressure/velocity DOF relationships of the Q 2 -Q 1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity DOFs resembles that on the finest grid. To define coefficients within the intergrid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier–Stokes problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

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↗
DOE OSTI2021

Optimizing multigrid reduction-in-time and Parareal coarse-grid operators for linear advection

Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs) in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. Therefore, one of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge–Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Overlapping Schwarz Methods Are Not Anisotropy‐Robust Multigrid Smoothers

We analyze overlapping multiplicative Schwarz methods as smoothers in the geometric multigrid solution of two-dimensional anisotropic diffusion problems. For diffusion equations, it is well known that the smoothing properties of point-wise smoothers, such as Gauss Seidel, rapidly deteriorate as the strength of anisotropy increases. On the other hand, global smoothers based on line smoothing are known to generally provide good smoothing for diffusion problems, independent of the anisotropy strength. Here, a natural question is whether global methods are really necessary to achieve good smoothing in such problems, or whether it can be obtained with locally overlapping block smoothers using sufficiently large blocks and overlap. Through local Fourier analysis and careful numerical experimentation, we show that global methods are indeed necessary to achieve anisotropy-robust smoothing. Specifically, for any fixed block size bounded sufficiently far away from the global domain size, we find that the smoothing properties of overlapping multiplicative Schwarz rapidly deteriorate with increasing anisotropy, irrespective of the amount of overlap between blocks. Moreover, our results indicate that anisotropy-robust smoothing requires blocks of diameter 𝒪⁡(𝜖 −1/2 ) for anisotropy ratio 𝜖 ∈(0,1] .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Comparative Performance Analysis of Coarse Solvers for Algebraic Multigrid on Multicore and Manycore Architectures

In this paper, we study the performance of a two-level algebraic-multigrid algorithm, with a focus on the impact of the coarse-grid solver on performance. We consider two algorithms for solving the coarse-space systems: the preconditioned conjugate gradient method and a new robust HSS-embedded low-rank sparse-factorization algorithm. Our test data comes from the SPE Comparative Solution Project for oil-reservoir simulations. We contrast the performance of our code on one 12-core socket of a Cray XC30 machine with performance on a 60-core Intel Xeon Phi coprocessor. To obtain top performance, we optimized the code to take full advantage of fine-grained parallelism and made it thread-friendly for high thread count. We also developed a bounds-and-bottlenecks performance model of the solver which we used to guide us through the optimization effort, and also carried out performance tuning in the solver’s large parameter space. Finally, as a result, significant speedups were obtained on both machines.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Efficient Multigrid Reduction-in-Time for Method-of-Lines Discretizations of Linear Advection

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. In this report we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multigrid strategies. For both approaches, we start from an integral formulation of the continuous time dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multi-grid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies are compared for varying orders of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences in execution times

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

High resolution viscous fingering simulation in miscible displacement using a p-adaptive discontinuous Galerkin method with algebraic multigrid preconditioner

High resolution simulation of viscous fingering can offer an accurate and detailed prediction for subsurface engineering processes involving fingering phenomena. The fully implicit discontinuous Galerkin (DG) method has been shown to be an accurate and stable method to model viscous fingering with high Peclet number and mobility ratio. In this study, we present two techniques to speedup large scale simulations of this kind. The first technique relies on a simple p-adaptive scheme in which high order basis functions are employed only in elements near the finger fronts where the concentration has a sharp change. As a result, the number of degrees of freedom is significantly reduced and the simulation yields almost identical results to the more expensive simulation with uniform high order elements throughout the mesh. The second technique for speedup involves improving the solver efficiency. We present an algebraic multigrid (AMG) preconditioner which allows the DG matrix to leverage the robust AMG preconditioner designed for the continuous Galerkin (CG) finite element method. The resulting preconditioner works effectively for fixed order DG as well as p-adaptive DG problems. Finally, with the improvements provided by the p-adaptivity and AMG preconditioning, we can perform high resolution three-dimensional viscous fingering simulations required for miscible displacement with high Peclet number and mobility ratio in greater detail than before for well injection problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Agglomeration-based geometric multigrid solvers for compact discontinuous Galerkin discretizations on unstructured meshes

Here, we present a geometric multigrid solver for the Compact Discontinuous Galerkin method through building a hierarchy of coarser meshes using a simple agglomeration method which handles arbitrary element shapes and dimensions. The method is easily extendable to other discontinuous Galerkin discretizations, including the Local DG method and the Interior Penalty method. We demonstrate excellent solver performance for Poisson's equation, provided a flux formulation is used for the operator coarsening and a suitable switch function chosen for the numerical fluxes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Compiler-based code generation and autotuning for geometric multigrid on GPU-accelerated supercomputers

GPUs, with their high bandwidths and computational capabilities are an increasingly popular target for scientific computing. Unfortunately, to date, harnessing the power of the GPU has required use of a GPU-specific programming model like CUDA, OpenCL, or OpenACC. Thus, in order to deliver portability across CPU-based and GPU-accelerated supercomputers, programmers are forced to write and maintain two versions of their applications or frameworks. In this paper, we explore the use of a compiler-based autotuning framework based on CUDA-CHiLL to deliver not only portability, but also performance portability across CPU- and GPU-accelerated platforms for the geometric multigrid linear solvers found in many scientific applications. We also show that with autotuning we can attain near Roofline (a performance bound for a computation and target architecture) performance across the key operations in the miniGMG benchmark for both CPU- and GPU-based architectures as well as for a multiple stencil discretizations and smoothers. We show that our technology is readily interoperable with MPI resulting in performance at scale equal to that obtained via hand-optimized MPI+CUDA implementation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Scalable line and plane relaxation in a parallel structured multigrid solver

The efficient solution of sparse, linear systems that arise through the discretization of partial differential equations remains a key challenge for a range of high performance scientific simulations. One approach for reducing data movement and improving performance is by exposing and exploiting structure in a problem through the use of robust structured multilevel solvers. By choosing coarsening that preserves the structure of the problem, these methods maintain efficient structured computation and communication throughout the multigrid hierarchy. However, when coarsening is not permitted to be dependent on the operator, anisotropy must be addressed by the smoother — producing error compatible for coarse-grid correction with structured coarsening. Here, the components required in a scalable parallel structured solver are described with a focus on memory and communication efficiency of robust smoothers. While the implementation of communication and memory reduction techniques in smoothers integrated in a complete 3D solver present a significant engineering challenge, a novel approach is proposed that addresses these challenges systematically through a change to the solver’s execution model. Enabled by user-level threading paired with a set of data and communication abstractions, this approach permits seamless aggregation of communication in plane smoothers — directly reusing code for a 2D distributed multilevel cycle. Results show an effective reduction in communication costs for coarse-grid problems, and result in a speedup of 8.7x in smoothing routines shown in Fig. 12 using this approach. This produces a significant improvement to strong scalability while maintaining favorable weak scaling behavior. Finally, a parallel scaling study using a series of refined meshes is included that demonstrates the effectiveness of this approach in an application of interest.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Numerical Multilevel Upscaling for Incompressible Flow in Reservoir Simulation: An Element-Based Algebraic Multigrid (AMGe) Approach

Here, we study the application of a finite element numerical upscaling technique to the incompressible two-phase porous media total velocity formulation. Specifically, an element-agglomeration-based algebraic multigrid (AMGe) technique with improved approximation properties is used, for the first time, to generate upscaled and accurate coarse systems for the reservoir simulation equations. The upscaling technique is applied to both the mixed system for velocity and pressure and to the hyperbolic transport equations, providing fully upscaled systems. By introducing additional degrees of freedom associated with nonplanar interfaces between agglomerates, the coarse velocity space has guaranteed approximation properties. The employed AMGe technique provides coarse spaces with desirable local mass conservation and stability properties analogous to the original pair of Raviart--Thomas and piecewise discontinuous polynomial spaces, resulting in strong mass conservation for the upscaled systems. Due to the guaranteed approximation properties and the generic nature of the AMGe method, recursive multilevel upscaling is automatically obtained. Furthermore, this technique works for both structured and unstructured meshes. Multiscale mixed finite elements exhibit accuracy for general unstructured meshes but do not in general lead to nested hierarchy of spaces. Multiscale multilevel mimetic finite differences generate nested spaces but lack the adaptivity of the flux representation on coarser levels that the proposed AMGe approach offers. Thus, the proposed approach can be seen as a rigorous bridge that merges the best properties of these two existing methods. The accuracy and stability of the studied multilevel AMGe upscaling technique is demonstrated on two challenging test cases.

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

Multigrid Optimization for Large-Scale Ptychographic Phase Retrieval

Ptychography is a popular imaging technique that combines diffractive imaging with scanning microscopy. The technique consists of a coherent beam that is scanned across an object in a series of overlapping positions, leading to reliable and improved reconstructions. Ptychographic microscopes allow for large fields to be imaged at high resolution at the cost of additional computational expense. Herein, we propose a multigrid-based optimization framework to reduce the computational burdens of large-scale ptychographic phase retrieval. Our proposed method exploits the inherent hierarchical structures in ptychography through tailored restriction and prolongation operators for the object and data domains. Our numerical results show that our proposed scheme accelerates the convergence of its underlying solver and outperforms the ptychographical iterative engine, a workhorse in the optics community.

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↗