James Brannick

Pennsylvania State University | Penn State · Department of Mathematics

Algebraic multigrid (AMG) methods are powerful solvers with linear or near-linear computational complexity for certain classes of linear systems, Ax=b. Broadening the scope of problems that AMG can effectively solve requires the development of improved interpolation operators. Such development is often based on AMG convergence theory. However, conv...

In this paper, two approaches for modeling three-component fluid flows using diffusive interface method are discussed. Thermodynamic consistency of the proposed models is preserved when using an energetic variational framework to derive the coupled systems of partial differential equations that comprise the resulting models. The issue of algebraic...

In this paper, we consider a classical form of optimal algebraic multigrid (AMG) interpolation that directly minimizes the two-grid convergence rate and compare it with the so-called ideal form that minimizes a certain weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid a...

The diffusive interface method is an approach for modeling interactions among complex substances. The main idea behind this method is to introduce phase field labeling functions in order to model the contact line by smooth change from one type of material to another. The approach has been widely used and successfully incorporated to numerous practi...

This paper introduces both a two-grid and a bootstrap multigrid finite element approximation to the Laplace-Beltrami (surface Laplacian) eigen-problem on a closed surface. The proposed multigrid method is suitable for solving problems with eigenvalues having large multiplicity, computing interior eigenvalues, and solving the shifted indefinite eige...

Preface: Special Issue – Weizmann Workshop 2013 - Volume 8 Issue 2 - Alfio Borzì, James Brannick, Francisco Gaspar, Irad Yavneh

Observations of several types of neutron stars indicate surface temperature inhomogeneities. In recent years magneto-thermal simulations have supported the idea that the magnetic field and anisotropic heat conduction play important roles in generating these inhomogeneities. Simulations rely on crustal microphysics input heretofore calculated at the...

The overlap operator is a lattice discretization of the Dirac operator of quantum chromodynamics, the fundamental physical theory of the strong interaction between the quarks. As opposed to other discretizations it preserves the important physical property of chiral symmetry, at the expense of requiring much more effort when solving systems with th...

In this paper we motivate, discuss the implementation and present the resulting numerics for a new definition of strength of connection which is based on the notion of algebraic distance. This algebraic distance measure, combined with compatible relaxation, is used to choose suitable coarse grids and accurate interpolation operators for algebraic m...

This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in...

In this paper, we introduce a diffuse interface model for describing the dynamics of mixtures involving multiple (two or more) phases. The coupled hydrodynamical system is derived through an energetic variational approach. The total energy of the system includes the kinetic energy and the mixing (interfacial) energies. The least action principle (o...

The computation of stationary distributions of Markov chains is an important task in the simulation of stochastic models. The linear systems arising in such applications involve non-symmetric M-matrices, making algebraic multigrid methods a natural choice for solving these systems. In this paper we investigate extensions and improvements of the boo...

We focus on the study of multigrid methods with aggressive coarsening and polynomial smoothers for the solution of the linear systems corresponding to finite difference/element discretizations of the Laplace equation. Using local Fourier analysis we determine automatically the optimal values for the parameters involved in defining the polynomial sm...

We develop an algebraic multigrid method for solving the non-Hermitian Wilson discretization of the 2-dimensional Dirac equation. The proposed approach uses a bootstrap setup algorithm based on a multigrid eigensolver. It computes test vectors which define the least squares interpolation operators by working mainly on coarse grids, leading to an ef...

This paper develops an algebraic multigrid preconditioner for the graph Laplacian. The proposed approach uses aggressive coarsening based on the aggregation framework in the setup phase and a polynomial smoother with sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration as a preconditioner to the flexible Conjugate Gradient...

The outermost several hundred meters of a neutron star crust is similar to a white dwarf interior, consisting of nuclei screened by a relativistic, degenerate electron gas. Free neutrons don't appear until a density of 4x10^11 g/cc. Below a depth of several tens of meters, corresponding to 10^6-10^8 g/cc, the nuclei are thought to crystallize. Unli...

We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hi...

We present an Algebraic Multigrid (AMG) method for graph Laplacian problems. The coarse graphs are constructed recursively by pair-wise aggregation, or matching as in [3] and we use an Algebraic Multilevel Iterations (AMLI) [1, 6] for the solution phase.

We construct and analyze a preconditioner of the linear elasticity system discretized by conforming linear finite elements in the framework of the auxiliary space method. The auxiliary space preconditioner is based on two auxiliary spaces corresponding to discretizations of the scalar Poisson equation by linear finite elements and the generalized f...

This paper presents estimates of the convergence rate and complexity of an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to the graph Laplacian. A bound is derived on the energy norm of the projection operator onto any piecewise constant vector space, which results in an estimate of the two-level conve...

This paper presents a strength of connection measure for algebraic multilevel algorithms for a class of linear systems corresponding to the graph Laplacian on a general graph. The coarsening in the multilevel algorithm is based on partitioning in subgraphs (using matching) of the underlying graph. Our idea is to define a local measure of the qualit...

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, fro...

Algebraic multigrid is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial differential equations. It automatically constructs a sequence of increasingly smaller matrix problems that enable efficient resolution of all scales present in the solution. O...

We develop an algebraic multigrid (AMG) setup scheme based on the bootstrap framework for multiscale scientific computation. Our approach uses a weighted least squares definition of interpolation, based on a set of test vectors that are computed by a bootstrap setup cycle and then improved by a multigrid eigensolver and a local residual-based adapt...

This work concerns the development of an algebraic multilevel method for computing state vectors of Markov chains. We present an efficient bootstrap algebraic multigrid (AMG) method for this task. In our proposed approach, we employ a multilevel eigensolver, with interpolation built using ideas based on compatible relaxation, algebraic distances, a...

We present an adaptive multigrid Dirac solver developed for Wilson clover fermions which offers order-of-magnitude reductions in solution time compared to conventional Krylov solvers. The solver incorporates even-odd preconditioning and mixed precision to solve the Dirac equation to double precision accuracy and shows only a mild increase in time t...

We present an adaptive multigrid solver for application to the non-Hermitian Wilson-Dirac system of QCD. The key components leading to the success of our proposed algorithm are the use of an adaptive projection onto coarse grids that preserves the near null space of the system matrix together with a simplified form of the correction based on the so...

We construct and analyze a preconditioner of the linear elastiity system discretized by conforming linear finite elements in the framework of the auxiliary space method. The auxiliary space preconditioner is based on discretization of a scalar elliptic equation with Generalized Finite Element Method. Comment: 14 pages, 1 figure, submitted NLAA Marc...

We introduce a coarsening algorithm for algebraic multigrid (AMG) based on the concept of compatible relaxation (CR). The algorithm is significantly different from standard methods, most notably because it does not rely on any notion of strength of connection. We study its behavior on a number of model problems, and evaluate the performance of an A...

Classical multigrid solution of linear systems with matrices that have highly variable entries and are nearly singular is made difficult by the compounding difficulties introduced by these two model features. Efficient multigrid solution of nearly singular matrices is known to be possible, provided the so-called Brandt-McCormick (or eigenvector app...

A significant amount of the computational time in large Monte Carlo simulations of lattice field theory is spent inverting the discrete Dirac operator. Unfortunately, traditional covariant finite difference discretizations of the Dirac operator present serious challenges for standard iterative methods. For interesting physical parameters, the discr...

We report on the first successful QCD multigrid algorithm which demonstrates constant convergence rates independent of quark mass and lattice volume for the Wilson Dirac operator. The new ingredient is the adaptive method for constructing the near null space on which the coarse grid multigrid Dirac operator acts. In addition we speculate on future...

We present promising initial results of our adaptive multigrid solver developed for application directly to the non-Hermitian Wilson-Dirac system in 4 dimensions, as opposed to the solver developed in [1] for the corresponding normal equations. The key behind the success of this algorithm is the use of an adaptive projection onto coarse grids that...

This paper analyzes a multigrid (MG) V-cycle scheme for solving the discretized 2D Poisson equation with corner singularities. Using weighted Sobolev spaces Kma(Ω) and a space decomposition based on elliptic projections, we prove that the MG V-cycle with standard smoothers (Richardson, weighted Jacobi, Gauss–Seidel, etc.) and piecewise linear inter...

We present a new multigrid solver that is suitable for the Dirac operator in the presence of disordered gauge fields. The key behind the success of the algorithm is an adaptive projection onto the coarse grids that preserves the near null space. The resulting algorithm has weak dependence on the gauge coupling and exhibits very little critical slow...

A significant amount of the computational time in large Monte Carlo simulations of lattice quantum chromodynamics (QCD) is spent inverting the discrete Dirac operator. Unfortunately, traditional covariant finite difference discretizations of the Dirac operator present serious challenges for standard iterative methods. For interesting physical param...

We present a new multigrid solver that is suitable for the Dirac operator in the presence of disordered gauge fields. The key behind the success of the algorithm is an adaptive projection onto the coarse grids that preserves the near null space. The resulting algorithm has weak dependence on the gauge coupling and exhibits mild critical slowing dow...

The linear systems arising in lattice quantum chromodynamics (QCD) pose significant challenges for traditional iterative solvers. The Dirac operator associated with these systems is nearly singular, indicating the need for efficient preconditioners. Multilevel preconditioners cannot, however, be easily constructed for these systems becasue the Dira...

This paper presents an adaptive algebraic multigrid setup algorithm for positive definite linear systems arising from discretizations of elliptic partial differential equations. The proposed method uses compatible relaxation to select the set of coarse variables. The nonzero supports for the coarse-space basis are determined by approximation of the...

Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements that scale linearly with the degrees of freedom. Unfortunately, they have been historically limited by their applicability to elliptic-type...

Algebraic multigrid (AMG) is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial differential equations. It automatically constructs a sequence of increasingly smaller matrix problems that hopefully enables efficient resolution of all scales present i...

In this paper, we highlight new multigrid solver advances in the Terascale Optimal PDE Simulations (TOPS) project in the Scientific Discovery Through Advanced Computing (SciDAC) program. We discuss two new algebraic multigrid (AMG) developments in TOPS: the adaptive smoothed aggregation method (αSA) and a coarse-grid selection algorithm based on co...