[Illinois] Data-Centric Models for Multilevel Algorithms
07 Feb 2016 | Online Presentations | Contributor(s): Samuel Guiterrez
Today, computational scientists must contend with a diverse set of supercomputer architectures that are capable of exposing unprecedented levels of parallelism and complexity. Effectively placing, moving, and operating on data residing in complex distributed memory hierarchies is quickly becoming an intractable problem for developers of parallel scientific software. Because of this, there is an increased desire to explore data-centric task-based programming models because of their ability to...
[Illinois] New FOSLS Formulation of Nonlinear Stokes Flow for Glaciers
07 Feb 2016 | Online Presentations | Contributor(s): Jeffrey Allen
This talk describes two First-order System Least-squares (FOSLS) formulations of the nonlinear Stokes flow used to model glaciers and ice sheets. The first is a Stress formulation and the second a Stress-Vorticity formulation. Both use fluidity, which is the reciprocal of viscosity and avoid the difficulties of infinite viscosity. Coercivity and continuity in appropriate Sobolev norms will be discussed. A Nested Iteration (NI), Newton-FOSLS-AMG approach is employed, in which the majority of...
[Illinois] Non-Blocking Conjugate Gradient Methods for Extreme Scale Computing
07 Feb 2016 | Online Presentations | Contributor(s): Paul Eller
Many scientific and engineering applications use Krylov subspace methods to solve large systems of linear equations. For extreme scale parallel computing systems, the dot products in these methods (implemented using allreduce operations in MPI) can limit performance because they are a synchronization point or barrier. Therefore we seek to develop Krylov subspace methods that avoid blocking allreduce operations and provide greater parallel efficiency.
We present a rearranged preconditioned...
[Illinois] Reducing Communication Costs for Sparse Matrix Multiplication within Algebraic Multigrid
07 Feb 2016 | Online Presentations | Contributor(s): Thomas Benson
Recent years have seen renewed interest in the numerical solution of the Stokes Equations. Of particular interest is the use of inf-sup stable pairs of finite elements for which weak enforcement of the incompressibility condition implies strong enforcement as well, such as with BDMelements. While there have been recent developments in preconditioning methods for the linear systems arising from this discretization, they are nonstandard preconditioning approaches. In this talk, we explore...
[Illinois] Range Decomposition: A Low Communication Algorithm for Solving PDEs on Massively Parallel Machines
07 Feb 2016 | Online Presentations | Contributor(s): Tom Manteuffel
The Range Decomposition (RD) algorithm uses nested iteration and adaptive mesh refinement locally before performing a global communication step. Only several such steps are observed to be necessary before reaching a solution within a small multiple of discretization error. The target application is peta- and exascale machines where traditional parallel numerical PDE communication patterns stifle scalability. The RD algorithm uses a partition of unity to equally distribute the error, and...
[Illinois] Multigrid Methods Conference
04 Feb 2016 | Workshops
Optimization and Inverse Problems
Data Mining, Large Graphs, and Markov Chains
Nonsymmetric and Indefinite Problems
Hybrid Direct-Iterative Linear Solvers
Parallel Multigrid on Multicore Systems and Heterogeneous Architectures
Time Parallel Methods
Iterative Methods in Applications (e.g., Electromagnetics, Energy, Environmental, MHD, Neutronics, Transport/Reaction)
[Illinois] A Massively Parallel Semicoarsening Multigrid for 3D Reservoir Simulation on Multi-core and Multi-GPU Architectures
04 Feb 2016 | Online Presentations | Contributor(s): Abdulrahman Manea
In this work, we have designed and implemented a massively parallel version of the Semicoarsening Black Box Multigrid Solver , which is capable of handling highly heterogeneous and anisotropic 3D reservoirs, on a parallel architecture with multiple GPU’s. For comparison purposes, the same algorithm was also implemented on a shared-memory multi-core parallel architecture using OpenMP. The parallel implementation exploits the parallelism in every module of the original Multigrid...
[Illinois] On the Design of a Finite Element Multigrid Solver for Mimetic Finite Difference Schemes
04 Feb 2016 | Online Presentations | Contributor(s): Carmen Rodrigo
The focus of this work is to study the relation between mimetic finite difference schemes on triangular grids and some finite element methods for two model problems based on curl-rot and grad-div operators. With this purpose, modified Nédélec and Raviart-Thomas finite element methods are derived respectively. This connection allows us to design an efficient multigrid method for the curl-rot problem, by considering canonical inter-grid transfer operators arising from the finite...
[Illinois] Least-Squares Finite Element Method and Nested Iteration for Electromagnetic Two-Fluid Plasma Models
04 Feb 2016 | Online Presentations | Contributor(s): Christopher Leibs
Efforts are currently being directed towards a fully implicit, electromagnetic, JFNK-based solver, motivating the necessity of developing a fluid-based, electromagnetic, preconditioning strategy . The two-fluid plasma (TFP) model is an ideal approximation to the kinetic Jacobian. The TFP model couples both an ion and an electron fluid with Maxwell's equations. The fluid equations consist of the conservation of momentum and number density. A Darwin approximation of Maxwell is used to...
[Illinois] A Multigrid Method for the Self-Adjoint Angular Flux Form of the Radiation-Transport Equation Based on Cellwise Block Jacobi Iteration
04 Feb 2016 | Online Presentations | Contributor(s): Jeffrey Densmore
Cellwise block Jacobi iteration is a technique for radiation-transport calculations in which the angular flux for all directions is solved for simultaneously within a spatial cell with the angular flux in neighboring cells held fixed. Each step of the iteration then involves the inversion of a small to moderate-sized matrix for every cell. The resulting arithmetic intensity may make cellwise block Jacobi iteration suitable for advanced, heterogeneous computing architectures. However, the...
[Illinois] Understanding the Propagation of Silent Data Corruption in Algebraic Multigrid
04 Feb 2016 | Online Presentations | Contributor(s): Jon Calhoun
Sparse linear solvers from a fundamental kernel in high performance computing (HPC). Exascale systems are expected to be more complex than systems of today being composed of thousands of heterogeneous processing elements that operate at near-threshold-voltage to meet power constraints. The combination of near near-threshold-voltage and number of processing elements required to reach exascale increases the rate of silent data corruption (SDC). With the rate of SDC expected to be higher,...
[Illinois] A Performance Comparison of Algebraic Multigrid Preconditioners on GPUs and MIC
04 Feb 2016 | Online Presentations | Contributor(s): Karl Rupp
Algebraic multigrid (AMG) preconditioners for accelerators such as graphics processing units (GPUs) and Intel's many-integrated core (MIC) architecture typically require a careful, problem-dependent trade-off between efficient hardware use, robustness, and convergence rate in order to minimize time-to-solution. Several variants of AMG with fine-grained parallelism have been proposed recently, but a comparison across different hardware architectures is difficult since the proposed...
[Illinois] Monolithic Multigrid Methods for Coupled Multi-Physics Problems
04 Feb 2016 | Online Presentations | Contributor(s): Scott Maclachlan
While block-diagonal and approximate block-factorization preconditioners are often considered for coupled problems, monolithic approaches can offer improved performance, particularly when the coupling between equations is strong. In this talk, we discuss the extension of Braess-Sarazin relaxation techniques to the linear systems resulting from the linearization and finite-element discretization of such coupled systems. Defining an easy-to-invert approximation to the (1,1) block of the system...
[Illinois] Application of Multigrid Techniques to Magnetic and Electromagnetic Systems
04 Feb 2016 | Online Presentations | Contributor(s): Benjamin Cowan
We discuss the use of multigrid techniques for several novel systems related to electromagnetics. One of these is the magnetostatic problem, in which systems can involve highly anisotropic and nonlinear materials. We describe the linear problems arising in several variations of this problem, including fully static, hysteretic, and eddy-current. We show results of tests of several AMG methods for solving these systems in 2D and 3D. We then discuss the challenges in solving the nonlinear...
[Illinois] Geometric Multigrid for MHD Simulations with Nedelec Finite Elements on Tetrahedral Grids
04 Feb 2016 | Online Presentations | Contributor(s): Chris Hansen
The Magneto-HydroDynamic (MHD) model is used extensively to simulate macroscopic plasma dynamics in Magnetic Confinement Fusion (MCF) devices. In these simulations, the span of time scales from fast wave dynamics to the desired evolution of equilibrium due to transport processes is large, resulting in stiff linear systems for implicit time advance. Many existing codes leverage toroidal symmetry, a common feature of many MCF devices, to develop efficient preconditioners for the linearized...
[Illinois] Parallel Multigrid Preconditioner Based on Automatic 3D Tetradedric Meshes
04 Feb 2016 | Online Presentations | Contributor(s): Frederic Vi
Multigrid methods are efficient for solving large sparse linear systems. Geometric (GMG) and Algebraic Multigrid (AMG) have both their own benefits and limitations. Combining the simplicity of AMG with the efficiency of GMG lead us to the development of an Hybrid Multigrid preconditionner. From an initial fine mesh we first build coarser unnested meshes and then deduce interpolation and restriction matrices based on interpolation of a mesh into a coarser one. We finally use Galerkin relation...
[Illinois] HPGMG: Benchmarking Computers Using Multigrid
04 Feb 2016 | Online Presentations | Contributor(s): Jed Brown
HPGMG (https://hpgmg.org) is a geometric multigrid benchmark designed to measure the performance and versatility of computers. For a benchmark to be representative of applications, good performance on the benchmark should be sufficient to ensure good performance on most important applications and only those system features necessary for some important applications should be stressed by the benchmark. Moreover, the specification should be scale-free and the method should solve a...
[Illinois] A Scalable Algorithm for Inverse Medium Problems with Multiple Sources
04 Feb 2016 | Online Presentations | Contributor(s): Keith Kelly
We consider the problem of acoustic scattering as described by the free-space, time-harmonic scalar wave equation given by
along with radiation boundary conditions. Here, is a point in , is the source term, and is the wavenumber. Our formulation is based on potential theory. First we write as , where is a constant value, is a known function of space, and is the unknown medium perturbation. Then,...
[Illinois] Support Graph Smoothing Techniques
04 Feb 2016 | Online Presentations | Contributor(s): Alyson Fox
Many tasks in large-scale network analysis and simulation require efficient approximation of the solution to the linear system $ Lx=b$, where $ L$ is a graph Laplacian. However, due to the large size and complexity of scale-free graphs, standard iterative methods do not perform optimally. The use of support graph techniques for preconditioning graph Laplacian systems has been studied, but efficiently finding optimal preconditioners is challenging for general scale-free graphs. An attractive...
[Illinois] Task-Graph and Functional Programming Models: The New Paradigm
04 Feb 2016 | Online Presentations | Contributor(s): Ben Bergen
The Message Passing Interface (MPI) is an example of a distributed-memory communication model that has served us well through the CISC processor era. However, because of MPI's low-level interface, which requires the user to manage raw memory buffers, and its bulk-synchronous communication model, MPI will have great difficulty in scaling to exascale systems and beyond. Additionally, the MPI model cannot be easily extended to include the fault tolerance and resilience features that will be...
[Illinois] A Fast Multigrid Approach for Solving the Helmholtz Equation with a Point Source
04 Feb 2016 | Online Presentations | Contributor(s): Eran Treister
Solving the discretized Helmholtz equations with high wave numbers in large dimensions is a challenging task. However, in many scenarios, the solution of these equations is required for a point source. In this case, the problem can be be reformulated and split into two parts: one in a solution of the eikonal equation for the travel time, and the second is a solution of a complex-valued advection-diffusion-reaction (ADR) equation for the amplitude. The eikonal equation for the travel time can...
[Illinois] Compatible Relaxation Based Geometric-Algebraic Multigrid
04 Feb 2016 | Online Presentations | Contributor(s): Fei Cao
We develop compatible relaxation algorithms for smoothed aggregation-based multigrid coarsening. In the proposed method, we use the geometry of the given discrete problem on the finest level to coarsen the system together with compatible relaxation to from the sparsity structure of the interpolation operator and then apply energy minimization techniques to compute its entries. Of particular interest in this work is the idea to use a geometric coarsening algorithm based on a new more sharp...
[Illinois] Hub Snub: Removing Vertices with High Degree from Coarse-grid Correction
04 Feb 2016 | Online Presentations | Contributor(s): Geoffry Sanders
Network scientists often employ numerical solutions to linear systems as subroutines of data mining algorithms. Due to the ill-conditioned nature of the systems, obtaining solutions with standard iterative methods is often prohibitively costly; current research aims to automatically construct preconditioners that are optimally efficient and scalable for a very broad class of graph associated matrices and network topologies. Frequently, networks encountered have skewed degree distribution and...
[Illinois] On the Preconditioning of a High-Order RDG-based All-Speed Navier-Stokes Solver
04 Feb 2016 | Online Presentations | Contributor(s): Brian Weston
We investigate the preconditioning of an all-speed Navier-Stokes solver, based on the orthogonal-basis Reconstructed Discontinuous Galerkin (RDG) space discretization, and integrated using a high-order fully-implicit time discretization method. The work is motivated by applications in Additive Manufacturing (AM), requiring simulations of laser-induced powder melting with formation and subsequent solidification of liquid metal pools, with numerous numerically challenging issues. These include...
[Illinois] Is the Ideal Approximation Operator Always "Ideal" for a Particular C/F Splitting?
04 Feb 2016 | Online Presentations | Contributor(s): Erin Molloy
Given a coarse grid, the ideal prolongation operator is defined by , where the weight matrix, , interpolates a set of fine grid variable (-points) from a set of coarse grid variable (-points), and the identity matrix, , represents the injection of -points to and from the coarse grid (Falgout and Vassilevski, 2004). In this talk, we consider , constructed from both traditional splittings and splittings corresponding to aggregates, for...