Tzanio V. Kolev

Scaling Hypre’s Multigrid Solvers to 100,000 Cores

The hypre software library (http://www.llnl.gov/CASC/hypre/) is a collection of high performance preconditioners and solvers for large sparse linear systems of equations on massively parallel machines. This paper investigates the scaling properties of several of the popular multigrid solvers and system building interfaces in hypre on two modern parallel platforms. We present scaling results on over 100,000 cores and even solve a problem with over a trillion unknowns.

High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics

The numerical approximation of the Euler equations of gas dynamics in a movingLagrangian frame is at the heart of many multiphysics simulation algorithms. In this paper, we present a general framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements. This method is an extension of the approach outlined in [Dobrev et al., Internat. J. Numer. Methods Fluids, 65 (2010), pp. 1295--1310] and can be formulated for any finite dimensional approximation of the kinematic and thermodynamic fields, including generic finite elements on two- and three-dimensional meshes with triangular, quadrilateral, tetrahedral, or hexahedral zones. We discretize the kinematic variables of position and velocity using a continuous high-order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high-order parametric mapping from a standard reference element. This enables the use of curvilinear zone geometry, higher-ord...

Parallel Time Integration with Multigrid

We consider optimal-scaling multigrid solvers for the linear systems that arise from the discretization of problems with evolutionary behavior. Typically, solution algorithms for evolution equations are based on a time-marching approach, solving sequentially for one time step after the other. Parallelism in these traditional time-integration techniques is limited to spatial parallelism. However, current trends in computer architectures are leading toward systems with more, but not faster, processors. Therefore, faster compute speeds must come from greater parallelism. One approach to achieving parallelism in time is with multigrid, but extending classical multigrid methods for elliptic operators to this setting is not straightforward. In this paper, we present a nonintrusive, optimal-scaling time-parallel method based on multigrid reduction (MGR). We demonstrate optimality of our multigrid-reduction-in-time algorithm (MGRIT) for solving diffusion equations in two and three space dimensions in numerical ex...

A tensor artificial viscosity using a finite element approach

We derive a tensor artificial viscosity suitable for use in a 2D or 3D unstructured arbitrary Lagrangian-Eulerian (ALE) hydrodynamics code. This work is similar in nature to that of Campbell and Shashkov [1]; however, our approach is based on a finite element discretization that is fundamentally different from the mimetic finite difference framework. The finite element point of view leads to novel insights as well as improved numerical results. We begin with a generalized tensor version of the Von Neumann-Richtmyer artificial viscosity, then convert it to a variational formulation and apply a Galerkin discretization process using high order Gaussian quadrature to obtain a generalized nodal force term and corresponding zonal heating (or shock entropy) term. This technique is modular and is therefore suitable for coupling to a traditional staggered grid discretization of the momentum and energy conservation laws; however, we motivate the use of such finite element approaches for discretizing each term in the Euler equations. We review the key properties that any artificial viscosity must possess and use these to formulate specific constraints on the total artificial viscosity force term as well as the artificial viscosity coefficient. We also show, that under certain simplifying assumptions, the two-dimensional scheme from [1] can be viewed as an under-integrated version of our finite element method. This equivalence holds on general distorted quadrilateral grids. Finally, we present computational results on some standard shock hydro test problems, as well as some more challenging problems, indicating the advantages of the new approach with respect to symmetry preservation for shock wave propagation over general grids.

Multigrid Smoothers for Ultraparallel Computing

This paper investigates the properties of smoothers in the context of algebraic multigrid (AMG) running on parallel computers with potentially millions of processors. The development of multigrid smoothers in this case is challenging, because some of the best relaxation schemes, such as the Gauss-Seidel (GS) algorithm, are inherently sequential. Based on the sharp two-grid multigrid theory from [R. D. Falgout and P. S. Vassilevski, SIAM J. Numer. Anal., 42 (2004), pp. 1669-1693] and [R. D. Falgout, P. S. Vassilevski, and L. T. Zikatanov, Numer. Linear Algebra Appl., 12 (2005), pp. 471-494] we characterize the smoothing properties of a number of practical candidates for parallel smoothers, including several

Scaling Algebraic Multigrid Solvers: On the Road to Exascale

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A Step towards Energy Efficient Computing: Redesigning a Hydrodynamic Application on CPU-GPU

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The approximation of the Maxwell eigenvalue problem using a least-squares method

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Improving algebraic multigrid interpolation operators for linear elasticity problems

, polynomial, and hybrid schemes. We show, in particular, that the popular hybrid GS algorithm has multigrid smoothing properties which are independent of the number of processors in many practical applications, provided that the problem size per processor is large enough. This is encouraging news for the scalability of AMG on ultraparallel computers. We also introduce the more robust

H(curl) auxiliary mesh preconditioning

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