Steve F McCormick

• Ph. D.
• University of Colorado Boulder

This paper provides a rounding-error analysis for two-grid methods that use one relaxation step both before and after coarsening. The analysis is based on floating point arithmetic and focuses on a two-grid scheme that is perturbed on the coarse grid to allow for an approximate coarse-grid solve. Leveraging previously published results, this two-gr...

Block Floating Point (BFP) arithmetic is currently seeing a resurgence in interest because it requires less power, less chip area, and is less complicated to implement in hardware than standard floating point arithmetic. This paper explores the application of BFP to mixed- and progressive-precision multigrid methods, enabling the solution of linear...

Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff–Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both displacement and normal rotation boundary conditions must be applied. A popular alternative is to employ Nitsche’s method...

This paper builds on the algebraic theory in the companion paper [Algebraic Error Analysis for Mixed-Precision Multigrid Solvers] to obtain discretization-error-accurate solutions for linear elliptic partial differential equations (PDEs) by mixed-precision multigrid solvers. It is often assumed that the achievable accuracy is limited by discretizat...

This paper establishes the first theoretical framework for analyzing the rounding-error effects on multigrid methods using mixed-precision iterative-refinement solvers. While motivated by the sparse symmetric positive definite (SPD) matrix equations that arise from discretizing linear elliptic PDEs, the framework is purely algebraic such that it ap...

Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff-Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both functional and derivative boundary conditions must be applied. A popular alternative is to employ Nitsche's method to wea...

Algebraic muligrid (AMG) is a go-to solver for symmetric positive definite linear systems resulting from the discretization of elliptic PDEs, or the spatial discretization of parabolic PDEs. For diffusion-like problems, the time to solution typically scales linearly with the number of unknowns. However, convergence theory and most variations of AMG...

Algebraic multigrid (AMG) is a widely used scalable solver and preconditioner for large-scale linear systems resulting from the discretization of a wide class of elliptic PDEs. While AMG has optimal computational complexity, the cost of communication has become a significant bottleneck that limits its scalability as processor counts continue to gro...

This paper proposes a new, low-communication algorithm for solving PDEs on massively parallel computers. The range decomposition (RD) algorithm exposes coarse-grain parallelism by applying nested iteration and adaptive mesh refinement locally before performing a global communication step. Just a few such steps are observed to be sufficient to obtai...

Existing multigrid methods for cloth simulation are based on geometric multigrid. While good results have been reported, geometric methods are problematic for unstructured grids, widely varying material properties, and varying anisotropies, and they often have difficulty handling constraints arising from collisions. This paper applies the algebraic...

Many problems of interest in plasma modeling are subject to the ‘tyranny of scales’, specifically, problems that encompass physical processes that operate on timescales that are separated by many orders of magnitude. Investigating such problems therefore requires the use of implicit timeintegration schemes, which advance problem solutions on the ti...

In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel al...

Many problems of interest in plasma modelling are subject to the ‘tyranny of scales’, specifically, problems that encompass physical processes that operate on timescales that are separated by many orders of magnitude. Investigating such problems therefore requires the use of implicit time-integration schemes, which advance problem solutions on the...

We consider two algebraic multilevel solvers for the solution of discrete problems arising from PDEs with random inputs. Our focus is on problems with large jumps in material coefficients. The model problem considered is that of a diffusion problem with uncertainties in the diffusion coefficients and realization values differing dramatically betwee...

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This paper investigates the performance of a parallel Newton, first-order system least-squares (FOSLS) finite-element method with local adaptive refinement and algebraic multigrid (AMG) applied to incompressible, resistive magnetohydrodynamics. In particular, an island coalescence test problem is studied that models magnetic reconnection using a re...

This paper combines first-order system least squares (FOSLS) with first-order system LL* (FOSLL*) to create a Hybrid method. The FOSLS approach minimizes the error, e(h) = u(h) - u, over a finite element subspace, V-h, in the operator norm: min(uh is an element of Vh)parallel to L(u(h) - u)parallel to. The FOSLL* method looks for an approximation i...

This paper develops new adaptive mesh refinement strategies for first-order system least squares (FOSLS) in conjunction with algebraic multigrid (AMG) methods in the context of nested iteration (NI). The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all eleme...

Bootstrap algebraic multigrid (BAMG) is a multigrid‐based solver for matrix equations of the form Ax = b . Its aim is to automatically determine the interpolation weights used in algebraic multigrid by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0. This paper studies an i...

The biennial Copper Mountain Conference on Iterative Methods was held April 4-9, 2010. This meeting included more than 140 presentations covering many scientific computing areas, such as uncertainty quantification, optimization, Markov chains, saddle-point systems, inverse problems, direct factorizations, Krylov methods, algebraic multigrid, softwa...

In this paper, we propose new adaptive local refinement (ALR) strategies for first-order system least-squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration. The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over...

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...

This paper describes the use of an efficiency-based adaptive mesh refinement scheme, known as ACE, on a 2D reduced model of the incompressible, resistive magnetohydrodynamic (MHD) equations. A first-order system least squares (FOSLS) finite element formulation and algebraic multigrid (AMG) are used in the context of nested iteration. The FOSLS a po...

Bootstrap algebraic multigrid (BAMG) is a multigrid-based solver for matrix equations of the form Ax=b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid (AMG) by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax=0, and are then possibly...

The Dirac equation of quantum electrodynamics describes the interaction between electrons and photons. Large-scale numerical simulations of the theory require repeated solution of the two-dimensional Dirac equation, a system of two first-order partial differential equations coupled to a background U(1) gauge field. Traditional discretizations of th...

Adaptive local refinement (ALR) can substantially improve the performance of simulations that involve numerical solution of partial differential equations. In fact, local refinement capabilities are one of the attributes of first-order system least squares (FOSLS) in that it provides an inexpensive but effective a posteriori local error bound that...

The solution of the Navier–Stokes equations requires that data about the solution is available along the boundary. In some situations, such as particle imaging velocimetry, there is additional data available along a single plane within the domain, and there is a desire to also incorporate this data into the approximate solution of the Navier–Stokes...

Applying smoothed aggregation (SA) multigrid to solve a nonsymmetric linear system, Ax = b, is often impeded by the lack of a minimization principle that can be used as a basis for the coarse-grid correction process. This paper proposes a Petrov-Galerkin (PG) approach based on applying SA to either of two symmetric positive definite (SPD) matrices,...

This paper develops a nested iteration algorithm to solve time-dependent nonlinear systems of partial differential equations. For each time step, Newton's method is used to form approximate solutions from a sequence of nested spaces, where the resolution of the approximations increases as the algorithm progresses. Nested iteration results in most o...

Magnetohydrodynamics (MHD) is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles. Hence, the equations that describe the plasma form a nonlinear system that couples Navier–Stokes equations with Maxwell's equations. This paper shows that the first-order system least squares (FOSLS) finite element meth...

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...

An algebraic multigrid (AMG) method is presented for the calculation of the sta-tionary probability vector of an irreducible Markov chain. The method is based on standard AMG for nonsingular linear systems, but in a multiplicative, adaptive setting. A modified AMG interpo-lation formula is proposed that produces a nonnegative interpolation operator...

A smoothed aggregation multigrid method is presented for the numerical calcula-tion of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for Markov chains that have been proposed in t...

There are many applications of the least-squares finite element method for the numerical solution of partial differential equations because of a number of benefits that the least-squares method has. However, one of most well-known drawbacks of the least-squares finite element method is the lack of exact discrete mass conservation, in some contexts,...

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...

Consider the linear system Ax=b, where A is a large, sparse, real, symmetric, and positive-definite matrix and b is a known vector. Solving this system for unknown vector x using a smoothed aggregation (SA) multigrid algorithm requires a characterization of the algebraically smooth error, meaning error that is poorly attenuated by the algorithm's r...

Two efficiency-based grid refinement strategies are investigated for adaptive finite element solution of partial differential equations. In each refinement step, the elements are ordered in terms of decreasing local error, and the optimal fraction of elements to be refined is determined based on efficiency measures that take both error reduction an...

A multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described. The method is a special case of the adaptive smoothed aggregation and adaptive algebraic multigrid methods for sparse linear systems and is also closely related to certain extensively studied iterative aggrega...

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...

The focus of this paper is on incompressible flows in three dimensions modeled by least-squares finite element methods (LSFEM) and using a novel reformulation of the Navier–Stokes equations. LSFEM are attractive because the resulting discrete equations yield symmetric, positive definite systems of algebraic equations and the functional provides bot...

A spatial multigrid algorithm for isotropic neutron transport is presented in x-y geometry. The linear system is obtained using discrete ordinates in angle and corner balance nite dierencing in space. Spatial smoothing is accomplished by a four-color block Jacobi relaxation, where the diagonal blocks correspond to four cell blocks on the spatial gr...

In recent years, several extensions of the classical AMG method (see [2] and [10]) to handle more general finite element matrices have been proposed (see, e.g., [3], [9], and [7]). Other extensions are related to the so–called smoothed aggregation method; see e.g., [11] and the papers cited therein. For the most recent versions of both the AMG and...

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...

Multigrid methods are among the most important algorithms for computational scientists because they're the most efficient solvers for a wide range of problems. The modern era of multi-grid methods began more than 30 years ago with the publication of Achi Brandt's seminal papers. Although it was originally successful for solving elliptic partial dif...

With the ubiquity of large-scale computing resources has come significant attention to practical details of fast algorithms for the numerical solution of partial differential equations. Included in this group are the class of multigrid and algebraic multigrid algorithms that are effective solvers for many of the large matrix problems arising from t...

Following earlier work for Stokes equations, a least squares functional is developed for two- and three-dimensional Oseen equations. By introducing a velocity flux variable and associated curl and trace equations, ellipticity is established in an appropriate product norm. The form of Oseen equations examined here is obtained by linearizing the inco...

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...

The modeling of blood flow through a compliant vessel requires solving a system of coupled nonlinear partial differential equations (PDEs). Traditional methods for solving the system of PDEs do not scale optimally, i.e., doubling the discrete problem size results in a computational time increase of more than a factor of 2. However, the development...

Without Abstract

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 introduce a discretization technique, called the finite volume-element method (FVE), which combines finite volumes (cf. [2]) and finite elements (cf. [1]) into a general approach for problems posed in conservative form. We illustrate its use and analyze its accuracy by combining FVE with multigrid to solve incompressible potential...

We present a first-order system least-squares (FOSLS) method to approximate the solution to the equations of geometrically nonlinear elasticity in two dimensions. With assumptions of regularity on the problem, we show H 1 equivalence of the norm induced by the FOSLS functional in the case of pure displacement boundary conditions as well as local co...

The goal of this paper is to introduce a new multilevel solver for two-dimensional elliptic systems of nonlinear partial differential equations (PDEs), where the nonlinearity is of the type \(u \partial v\). The incompressible Navier--Stokes equations are an important representative of this class and are the target of this study. Using a first-orde...

In a companion paper (8), we propose a new multilevel solver for two-dimensional elliptic systems of partial difierential equations (PDEs) with nonlinearity of type u@v. The approach is based on a multilevel projection method (PML (9)) applied to a flrst-order system least-squares (FOSLS) functional that allows us to treat the nonlinearity directly...

Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete oper- ators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares s...

Efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multi-scale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple g...

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...

Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical probl...

Two related approaches for solving linear systems that arise from a higher-order finite element discretization of elliptic partial differential equations are described. The first approach explores direct application of an algebraic-based multigrid method (AMG) to iteratively solve the linear systems that result from higher-order discretizations. Wh...

First-order system least squares (FOSLS) is a recently developed methodology for solving partial differential equations. Among its advantages are that the finite element spaces are not restricted by the inf-sup condition imposed, for example, on mixed methods and that the least-squares functional itself serves as an appropriate error measure. This...

RESUMEN RESUMEN This paper develops new first - order system LL* (FOSLL*) formulations for scalar elliptic partial differential equations . It extends the work of [Z. Cal et al., SIAM J. Numer . Anal., 39 (2001), pp. 1418--1445], where the FOSLL* methodology was first introduced . One focus of that paper was to develop \ FL\ formulations that allo...

Least-squares finite element methods (LSFEMs) for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector, or the associated flux potential, explicitly as additional dependent variables. This reformulation highlights the smoothness of the flux vector for weak solutio...

The projection multilevel method can be an efficient solver for systems of nonlinear partial differential equations that, for certain classes of nonlinearities (including least-squares formulations of the Navier–Stokes equations), requires no linearization anywhere in the algorithm. This paper provides an abstract framework and establishes optimal...

First-order system least squares (FOSLS) is a methodology that offers an alternative to standard methods for solving partial differential equations. This paper studies the first-order system least-squares approach for scalar second-order elliptic boundary value problems with discontinuous coefficients. In a companion paper [M. Berndt, T. A. Manteuf...

Mathematical models for the mechanical coupling between a moving fluid and an elastic solid are inherently nonlinear because the shape of the Eulerian fluid domain is not known a priori – it is at least partially determined by the displacement of the elastic solid. In this paper, a first-order system least squares finite element formulation is used...

The standard multigrid algorithm is widely known to yield optimal convergence whenever all high-frequency error components correspond to large relative eigenvalues. This property guarantees that smoothers like Gauss–Seidel and Jacobi will significantly dampen all the high-frequency error components, and thus, produce a smooth error. This has been e...

Blood flow in large vessels is typically modeled using the Navier-Stokes equations for the fluid domain and elasticity equations for the vessel wall. As the wall deforms, additional complications are introduced because the shape of the fluid domain changes, necessitating the use of a re-mapping or re-griding process for the fluid region. Typically,...

Electrical impedance tomography (EIT) belongs to a family of imaging methods that employ boundary measurements to distinguish interior spatial variation of an electromagnetic parameter. The associated inverse problem is notoriously ill-posed, due to diffusive effects in the quasi-static regime, when electrical impedance reduces to its real part, re...

Accurate numerical modeling of complex physical, chemical, and biological systems requires numerical simulation capability over a large range of length scales, with the ability to cap- ture rapidly varying phenomena localized in space and/or time. Adaptive mesh refinement (AMR) is a numerical process for dynamically introducing local fine resolutio...

Least-squares finite element methods (LSFEMs) for scalar linear partial differential equations (PDEs) of hyperbolic type are studied. The space of admissible boundary data is identified precisely, and a trace theorem and a Poincare inequality are formulated. The PDE is restated as the minimization of a least-squares functional, and the well-posedne...

Substantial effort has been focused over the last two decades on developing multi- level iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical pro...

Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple...

Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical probl...

This paper presents numerical results for the asynchronous version of the fast adap-tive composite-grid algorithm (AFACx). These results confirm the level-independent convergence bounds established theoretically in a companion paper. These numerical results include the case of AFACx applied to first-order system least-squares finite element discret...

We study the discretization accuracy for first-order system least squares (FOSLS) applied to Poisson's equation as a model problem. The FOSLS formulation is based on an H 1 elliptic bilinear form F . Since the order of convergence of the discretization in the L 2 and H 1 norms depends on the regularity of F , we examine this property in detail...

We study the discretization accuracy for first-order system least squares (FOSLS) applied to Poisson's equation as a model problem. The FOSLS formulation is based on an H elliptic bilinear form F . Since the order of convergence of the discretization in the L and H norms depends on the regularity of F , we examine this property in detail. We then u...

A fully variational approach is developed for solving nonlinear elliptic equations that enables accurate discretization and fast solution methods. The equations are converted to a first-order system that is then linearized via Newton's method. First-order system least squares (FOSLS) is used to formulate and discretize the Newton step, and the resu...

A new fully variational approach is studied for elliptic grid generation (EGG). It is based on a general algorithm developed in a companion paper [A. L. Codd, T. A. Manteuffel, and S. F. McCormick, SIAM I Numer. Anal., 41 (2003), pp. 2197-2209] that involves using Newton's method to linearize an appropriate equivalent first-order system, first-orde...

We introduce spectral AMGe (ρAMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritz-type finite element discretizations of partial differential equations. The method requires access to the element stiffness matrices, which enables accurate approximation of algebraically "smooth" vectors (i.e., error co...

This paper develops two first-order system least-squares (FOSLS) approaches for the solution of the pure traction problem in planar linear elasticity. Both are two-stage algorithms that first solve for the gradients of displacement (which immediately yield deformation and stress), then for the displacement itself (if desired). One approach, which u...

This paper develops a least-squares approach to the solution of the incompressible Navier--Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier--Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares...

The L 2 --norm version of first--order system least squares (FOSLS) attempts to reformulate a given system of partial di#erential equations so that applying a least--squares principle yields a functional whose bilinear part is H 1 --elliptic. This means that the minimization process amounts to solving a weakly coupled system of elliptic scalar equa...

This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a least--squares problem for an equivalent first--order system. The main result is the proof of ellipticity, which is used in a companion paper to establish optimal convergence of multiplicative and addi...

Mathematical modeling of compliant blood vessels generally involves the Navier-Stokes equations on the evolving fluid domain and constitutive structural equations on the tissue domain. Coupling these systems while accounting for the changing shape of the fluid domain is a major challenge in numerical simulation. Many techniques have been developed...

A multigrid method for solving the 1-D slab-geometry SN equations with isotropic scattering and no absorption is presented. This scheme is highly compatible with massively parallel computer architectures and represents a first step towards similar multigrid methods for the SN equations in curvilinear and multi-dimensional geometries. Extensive theo...

This paper focuses on the multilevel projection method (PML) applied to numerical solution of the basic equations of fluid dynamics formulated as least-squares problems for first-order systems. The fundamental approach taken here is to pose the fluid flow equations in their first-order form, incorporate additional (usually redundant) equations wher...

This paper develops first-order system least-squares (FOSLS) functionals for solving the pure traction problem in three-dimensional linear elasticity. It is a direct extension of an earlier paper on planar elasticity [Z. Cai, T. A. Manteu#el, S. F. McCormick, and S. V. Parter, SIAM J. Numer. Anal., 35 (1998), pp. 320--335]. Two functionals are deve...