Scott P. MacLachlan

Adaptive Smoothed Aggregation (

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 problem being discretized may be unavailable to the solver. Algebraic multigrid (AMG) and multilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry. These methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error. For smoothed aggregation (SA) multigrid methods applied to discretizations of elliptic problems, these assumptions typically consist of knowledge of the near-kernel or near-nullspace of the weak form. This paper introduces an extension of the SA method in which good convergence properties are achieved in situations where explicit knowledge of the near-kernel components is unavailable. This extension is accomplished in an adaptive process that uses the method itself to determine near-kernel components and adjusts the coarsening processes accordingly.

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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 problem being discretized may be unavailable to the solver. Algebraic multigrid (AMG) and multilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry. These methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error. For smoothed aggregation (SA) methods applied to discretizations of elliptic problems, these assumptions typically consist of knowledge of the near-nullspace of the weak form. This paper introduces an extension of the SA method in which good convergence properties are achieved in situations where explicit knowledge of the near-nullspace components is unavailable. This extension is accomplished by using the method itself to determine near-nullspace components and adjusting the coarsening processes accordingly.

SA) 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...

Adaptive Smoothed Aggregation (

It is well known that two-level and multilevel preconditioned conjugate gradient (PCG) methods provide efficient techniques for solving large and sparse linear systems whose coefficient matrices are symmetric and positive definite. A two-level PCG method combines a traditional (one-level) preconditioner, such as incomplete Cholesky, with a projection-type preconditioner to get rid of the effect of both small and large eigenvalues of the coefficient matrix; multilevel approaches arise by recursively applying the two-level technique within the projection step. In the literature, various such preconditioners are known, coming from the fields of deflation, domain decomposition, and multigrid (MG). At first glance, these methods seem to be quite distinct; however, from an abstract point of view, they are closely related. The aim of this paper is to relate two-level PCG methods with symmetric two-grid (V(1,1)-cycle) preconditioners (derived from MG approaches), in their abstract form, to deflation methods and a two-level domain-decomposition approach inspired by the balancing Neumann-Neumann method. The MG-based preconditioner is often expected to be more effective than these other two-level preconditioners, but this is shown to be not always true. For common choices of the parameters, MG leads to larger error reductions in each iteration, but the work per iteration is more expensive, which makes this comparison unfair. We show that, for special choices of the underlying one-level preconditioners in the deflation or domain-decomposition methods, the work per iteration of these preconditioners is approximately the same as that for the MG preconditioner, and the convergence properties of the resulting two-level PCG methods will also be (approximately) the same. This means that, in this respect, the particular choice of the two-level preconditioner is less important than the choice of the parameters. Numerical experiments are presented to emphasize the theoretical results.

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In this paper, an iterative solution method for a fourth-order accurate discretization of the Helmholtz equation is presented. The method is a generalization of that presented in (SIAM J. Sci. Comput. 2006; 27:1471–1492), where multigrid was employed as a preconditioner for a Krylov subspace iterative method. The multigrid preconditioner is based on the solution of a second Helmholtz operator with a complex-valued shift. In particular, we compare preconditioners based on a point-wise Jacobi smoother with those using an ILU(0) smoother, we compare using the prolongation operator developed by de Zeeuw in (J. Comput. Appl. Math. 1990; 33:1–27) with interpolation operators based on algebraic multigrid principles, and we compare the performance of the Krylov subspace method Bi-conjugate gradient stabilized with the recently introduced induced dimension reduction method, IDR(s). These three improvements are combined to yield an efficient solver for heterogeneous problems. Copyright

SA)

We consider the numerical simulation of two-phase fluid flow, where bubbles or droplets of one phase move against a background of the other phase. Such flows are governed by the Navier-Stokes equations, the solution of which may be approximated using a pressure-correction approach. Within such an approach, the computational cost is often dominated by the solution of a linear system corresponding to a discrete Poisson equation with discontinuous coefficients. In this paper, we explore the efficient solution of these linear systems using robust multilevel solvers, such as deflated variants of the preconditioned conjugate gradient method, or robust multigrid techniques. We consider these families of methods in more detail and compare their performance in the simulation of bubbly flows. Some of these methods turn out to be very effective and reduce the amount of work to solve the pressure-correction system substantially, resulting in efficient calculations for two-phase flows on highly resolved grids.

Parallel Time Integration with Multigrid

In this paper, we consider the numerical solution of the Helmholtz equation, arising from the study of the wave equation in the frequency domain. The approach proposed here differs from those recently considered in the literature, in that it is based on a decomposition that is exact when considered analytically, so the only degradation in computational performance is due to discretization and roundoff errors. In particular, we make use of a multiplicative decomposition of the solution of the Helmholtz equation into an analytical plane wave and a multiplier, which is the solution of a complex-valued advection-diffusion-reaction equation. The use of fast multigrid methods for the solution of this equation is investigated. Numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies.

A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation

SUMMARY 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 difierential equations. It automatically constructs a sequence of increasingly smaller matrix problems that hopefully enables e‐cient resolution of all scales present in the solution. The methodology is based on measuring how a so-called algebraically smooth error value at one point depends on its value at another. Such a concept of strength of connection is well understood for operators whose principal part is an Mmatrix; however, the strength concept for more general matrices is not yet clearly understood, and this lack of knowledge limits the scope of AMG applicability. The purpose of this paper is to motivate a general deflnition of strength of connection, discuss its implementation, and present the results of initial numerical experiments. Copyright c ∞ 2005 John Wiley & Sons, Ltd.

A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

SUMMARY Since their popularization in the late 1970s and early 1980s, multigrid methods have been a central tool in the numerical solution of the linear and nonlinear systems that arise from the discretization of many PDEs. In this paper, we present a local Fourier analysis (LFA, or local mode analysis) framework for analyzing the complementarity between relaxation and coarse-grid correction within multigrid solvers for systems of PDEs. Important features of this analysis framework include the treatment of arbitrary finite-element approximation subspaces, leading to discretizations with staggered grids, and overlapping multiplicative Schwarz smoothers. The resulting tools are demonstrated for the Stokes, curl–curl, and grad–div equations. Copyright 2011 John Wiley & Sons, Ltd.

Fast and robust solvers for pressure-correction in bubbly flow problems

In the mathematical modeling of real-life applications, systems of equations with complex coefficients often arise. While many techniques of numerical linear algebra, e.g., Krylov-subspace methods, extend directly to the case of complex-valued matrices, some of the most effective preconditioning techniques and linear solvers are limited to the real-valued case. Here, we consider the extension of the popular algebraic multigrid method to such complex-valued systems. The choices for this generalization are motivated by classical multigrid considerations, evaluated with the tools of local Fourier analysis, and verified on a selection of problems related to real-life applications.