Abstract : Multigrid methods are an optimal computational complexity linear solver and preconditioner that is often utilized to solve large-scale problems. The purpose of this minisymposium is to bring together researchers working on high-performance multigrid solvers. The presentations include topics on both performance aspects, advanced architectures, and applications.
Wayne Mitchell (Lawrence Livermore National Laboratory)
Abstract : A crucial concept for algbraic multigrid (AMG) coarsening and interpolation is that of strength of connection (SoC) between degrees of freedom. The classical SoC measure is based on the relative sizes of entries in each row of the matrix and relies on heuristics that assume an M-matrix structure. This simple measure is cheap to evaluate and successful for many problems, but also relies on a user-defined strength threshold, which may need to be tuned for specific problems. In addition, the classical SoC measure can have difficulty identifying the proper strong connections for operators that are not M-matrices, particularly when there are both positive and negative off-diagonal entries in the same row. In this work, we examine low-cost techniques for building an auxiliary strength matrix that shares important properties with the original matrix operator while also being more amenable to the classical SoC measure. Applying classical SoC to this auxiliary strength matrix improves robustness of the SoC measure for a wider class of problems and across a wider range of strength thresholds.
[03867] Performance improvements of algebraic multigrid algorithms on modern system architectures
Format : Talk at Waseda University
Author(s) :
Luc Berger-Vergiat (Sandia National Laboratories)
Jonathan Hu (Sandia National Laboratories)
Christian Glusa (Sandia National Laboratories)
Chris Siefert (Sandia National Laboratories)
Abstract : Multigrid methods are an important class of linear solvers and preconditioners for their high scalability on large computing systems. Implementing these algorithms on GPU based platforms remains a challenging task. In this talk we will discuss the current state of the MueLu package of Trilinos which provides algebraic multigrid methods on CPUs and GPUs and present results gathered on recent architectures.
[05263] Monolithic Multigrid and Block Preconditioning for Magnetic Confinement Fusion Relevant Resistive MHD Simulations
Format : Talk at Waseda University
Author(s) :
Peter Ohm (RIKEN Center for Computational Science)
John Shadid (Sandia National Laboratories)
Jesus Bonilla (Los Alamos National Lab)
Edward Phillips (Sandia National Laboratories)
Raymond Tuminaro (Sandia National Laboratories)
Jonathan Hu (Sandia National Laboratories)
Xian-Zhu Tang (Los Alamos National Lab)
Michael Crockatt (Sandia National Laboratories)
Abstract : A base-level mathematical basis for the continuum fluid modeling of dissipative plasma system is the resistive magnetohydrodynamic model. This model requires the solution of the governing partial differential equations (PDEs) describing conservation of mass, momentum, and thermal energy, along with various reduced forms of Maxwell’s equations for the electromagnetic fields. The resulting systems are characterized by strong nonlinear and nonsymmetric coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that these interactions produce. These characteristics make scalable and efficient iterative solution, of the resulting poorly-conditioned discrete systems, extremely difficult.
In this talk we utilize Drekar, a multi-physics simulation code built on top of the Trilinos framework, for the simulation of various resistive MHD problems. We consider the use of block preconditioners as well as monolithic multigrid for solving coupled physics block systems.
[03865] Combined On/Off Node Performance Model for SPMV in Multigrid
Format : Talk at Waseda University
Author(s) :
Chris Siefert (Sandia National Laboratories)
Abstract : We propose combining traditional on-node memory-bandwidth based performance models (i.e., roofline) with inter-node performance models (i.e., ping-pong) to build a combined performance model to predict the performance of sparse matrix-vector products in the context of the Trilinos/MueLu algebraic multigrid software. We demonstrate the combined model on both CPU and GPU platforms and compare against actual Trilinos sparse matrix-vector product (SPMV) performance using Trilinos/MueLu.
[03275] Block-structured and hierarchical hybrid grid matrix-free multigrid solvers for CFD applications at scale
Format : Talk at Waseda University
Author(s) :
Harald Koestler (Friedrich-Alexander Universität Erlangen-Nürnberg)
Abstract : In this talk an overview of applications and performance for matrix-free multigrid solvers implemented in the HPC software frameworks HyTeG and ExaStencils is provided. This includes scaling and efficiency results on CPU and GPU clusters and sample applications from geophysics, charged particle and ocean simulations.
[05232] Recent Advances in Linear Solvers for Ice Sheet Modeling
Format : Talk at Waseda University
Author(s) :
Jonathan Hu (Sandia National Laboratories)
Jerry Watkins (Sandia National Laboratories)
Max Carlson (Sandia National Laboratories)
Mauro Perego (Sandia National Laboratories)
Kim Liegeois (Sandia National Laboratories)
Oscar Antepara (Lawrence Berkeley National Laboratory)
Samuel Williams (Lawrence Berkeley National Laboratories)
Abstract : We present recent ongoing work performed under the SciDAC FAnSSIE project to improve linear solver performance within land-ice simulations on GPU-based architectures. We'll review the current algorithmic approach that relies on specialized semicoarsening algebraic multigrid, challenges in adapting this approach to GPU architectures, and progress in improving performance and scaling. Finally, we'll present numerical results from the Albany simulation code on the HPE Cray Ex supercomputer Perlmutter.
[05118] A Matrix-Free Approach for Algebraic Multigrid for High-Order Systems
Format : Talk at Waseda University
Author(s) :
Graham Harper (Sandia National Laboratories)
Abstract : We present a matrix-free approach for algebraic multigrid (AMG) for high-order systems. In particular, we focus on the case where an application may have a mix of domains with geometric structure and regions without geometric structure. We mix geometric multigrid (GMG) and AMG, but we approach the overall problem from a matrix-free AMG perspective on the finer levels. We present numerical results using Trilinos and MueLu to verify our methods.
[03319] Mixed formulations and monolithic multigrid methods for smectic-A liquid crystals
Format : Online Talk on Zoom
Author(s) :
Abdalaziz Hamdan (Imperial College London)
Patrick Farrell (University of Oxford)
Scott MacLachlan (Memorial University of Newfoundland)
Abstract : Xia et al. recently proposed a new continuum model for smectic A liquid crystals. Here, we present a mixed finite-element formulation of that model and discuss the construction of solvers for the resulting nonlinear systems. We consider Newton-Krylov-Multigrid approaches, using Newton’s method to linearize and develop monolithic geometric multigrid preconditioners for the resulting saddle-point systems. We demonstrate this is an effective solver strategy when using a coupled “star” relaxation scheme and nested iteration.
