Coarse Grid Operators Research Articles

A Wavelet Multigrid Method Using Symmetric Biorthogonal Wavelets

In [1], the author introduced a wavelet multigrid method that used the wavelet transform to define the coarse grid, interpolation, and restriction operators for the multigrid method. In this paper, we modify the method by using symmetric biorthogonal wavelet transforms to define the requisite operators. Numerical examples are presented to demonstrate the effectiveness of the modified wavelet multigrid method for diffusion problems with highly oscillatory coefficients, as well as for advection-diffusion equations in which the advection is moderately dominant.

Compact Fourier Analysis for Multigrid Methods based on Block Symbols

The notion of compact Fourier analysis (CFA) is discussed. CFA allows description of multigrid (MG) in a nutshell and offers a clear overview on all MG components. The principal idea of CFA is to model the MG mechanisms by means of scalar generating functions and matrix functions (block symbols). The formalism of the CFA approach is presented by describing the symbols of the fine and coarse grid problems, the prolongation and restriction, the smoother, and the coarse grid correction, resp., smoothing corrections. CFA uses matrix functions and their features (e.g., product, inverse, adjugate, norm, spectral radius, eigenvectors, eigenvalues of multilevel $\omega$-circulant matrices), and scalar functions and their roots. This leads to an elementary description and allows for an easy analysis of MG algorithms. A first application is to utilize CFA for deriving MG as a direct solver, i.e., an MG cycle that will converge in just one iteration step. Necessary and sufficient conditions that have to be fulfilled by the MG components are given for obtaining MG functioning as a direct solver. Furthermore, new general and practical smoothers and transfer operators that lead to efficient MG methods are introduced. In addition, we study sparse approximations of the Galerkin coarse grid operator yielding efficient and practicable MG algorithms (approximately direct solvers). Numerical experiments demonstrate the theoretical results.

An Efficient Multigrid Method for Molecular Mechanics Modeling in Atomic Solids

AbstractWe propose a multigrid method to solve the molecular mechanics model (molecular dynamics at zero temperature). The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state as the initial guess of the molecular mechanics model. The efficiency of the algorithm is demonstrated by three examples with homogeneous deformation, namely, one dimensional chain under tensile deformation and aluminum under tension and shear deformations. The method exhibits linear-scaling computational complexity, and is insensitive to parameters arising from iterative solvers. In addition, we study two examples with inhomogeneous deformation: vacancy and nanoindentation of aluminum. The results are still satisfactory while the linear-scaling property is lost for the latter example.

Efficient simulation of surface tension-dominated flows through enhanced interface geometry interrogation

In this paper, three improvements for modelling surface tension-dominated interfacial flows using interface tracking-based solution algorithms are presented. We have developed an improved approach to curvature estimation for incorporation into modern mesh-based surface tension models such as the Continuum Surface Force (CSF) and Sharp Surface Force (SSF) models. The scheme involves generating samples of curvature estimates from the multitude of height functions that can be generated from VOF representations of interfaces, and applying quality statistics based on interface orientation and smoothness to choose optimal candidates from the samples. In this manner, the orientation-dependence of past schemes for height function-based curvature estimation is ameliorated, the use of compact stencils for efficient computation can be maintained, and robustness is enhanced even in the presence of noticeable subgrid-scale disturbances in the interface representation. For surface tension-dominated flows, the explicit capillary timestep restriction is relaxed through timescale-separated slope limiting that identifies spurious modes in curvature evolution and omits them from contributing to surface force computations, thus promoting efficiency in simulation through the use of less timesteps. Efficiency in flow simulation is further promoted by incorporating awareness of interface location into multigrid preconditioning for Krylov subspace-based solution of elliptic problems. This use of interface-cognizance in solving problems such as the Helmholtz equation and the Poisson equation enables multigrid-like convergence in discontinuous-coefficient elliptic problems without the expense of constructing the Galerkin coarse-grid operator. The key improvements in the surface tension modelling and the numerical linear algebra are also applicable to level-set-based interfacial flow simulation.

Multigrid solution of the optical flow system using a combined diffusion‐ and curvature‐based regularizer

AbstractOptical flow techniques are used to compute an approximate motion field in an image sequence. We apply a variational approach for the optical flow using a simple data term but introducing a combined diffusion‐ and curvature‐based regularizer. The same data term arises in image registration problems where a deformation field between two images is computed. For optical flow problems, usually a diffusion‐based regularizer should dominate, whereas for image registration a curvature‐based regularizer is more appropriate. The combined regularizer enables us to handle optical flow and image registration problems with the same solver and it improves the results of each of the two regularizers used on their own. We develop a geometric multigrid method for the solution of the resulting fourth‐order systems of partial differential equations associated with the variational approach for optical flow and image registration problems. The adequacy of using (collective) pointwise smoothers within the multigrid algorithm is demonstrated with the help of local Fourier analysis. Galerkin‐based coarse grid operators are applied for an efficient treatment of jumping coefficients. We show some multigrid convergence rates, timings and investigate the visual quality of the approximated motion or deformation field for synthetic and real‐world images. Copyright © 2008 John Wiley & Sons, Ltd.

A new wavelet multigrid method

The standard multigrid procedure performs poorly or may break down when used to solve certain problems, such as elliptic problems with discontinuous or highly oscillatory coefficients. The method discussed in this paper solves this problem by using a wavelet transform and Schur complements to obtain the necessary coarse grid, interpolation, and restriction operators. A factorized sparse approximate inverse is used to improve the efficiency of the resulting method. Numerical examples are presented to demonstrate the versatility of the method.

Numerical Study of Two-grid Preconditioners for 1-d Elliptic Problems with Highly Oscillating Discontinuous Coefficients

AbstractVarious advanced two-level iterative methods are studied numerically and compared with each other in conjunction with finite volume discretizations of symmetric 1-D elliptic problems with highly oscillatory discontinuous coefficients. Some of the methods considered rely on the homogenization approach for deriving the coarse grid operator. This approach is considered here as an alternative to the well-known Galerkin approach for deriving coarse grid operators. Different intergrid transfer operators are studied, primary consideration being given to the use of the so-called problemdependent prolongation. The two-grid methods considered are used as both solvers and preconditioners for the Conjugate Gradient method. The recent approaches, such as the hybrid domain decomposition method introduced by Vassilevski and the globallocal iterative procedure proposed by Durlofsky et al. are also discussed. A two-level method converging in one iteration in the case where the right-hand side is only a function of the coarse variable is introduced and discussed. Such a fast convergence for problems with discontinuous coefficients arbitrarily varying on the fine scale is achieved by a problem-dependent selection of the coarse grid combined with problem-dependent prolongation on a dual grid. The results of the numerical experiments are presented to illustrate the performance of the studied approaches.

3D optical flow computation using a parallel variational multigrid scheme with application to cardiac C-arm CT motion

Motivated by recent applications to 3D medical motion estimation, we consider the problem of 3D optical flow computation in real time. The 3D optical flow model is derived from a straightforward extension of the 2D Horn–Schunck model and discretized using standard finite differences. We compare memory costs and convergence rates of four numerical schemes: Gauss–Seidel and multigrid with three different strategies of coarse grid operators discretization: direct coarsening, lumping and Galerkin approaches. Experimental results to compute 3D motion from cardiac C-arm CT images demonstrate that our variational multi-grid based on Galerkin discretization outperforms significantly the Gauss–Seidel method. The parallel implementation of the proposed scheme using domain partitioning shows that the algorithm scales well up to 32 processors on a cluster of AMD Opteron CPUs which consists of four-way nodes connected by an Infiniband network.

Multigrid Methods for Block Toeplitz Matrices with Small Size Blocks

In this paper we discuss multigrid methods for ill-conditioned symmetric positive definite block Toeplitz matrices. Our block Toeplitz systems are general in the sense that the individual blocks are not necessarily Toeplitz, but we restrict our attention to blocks of small size. We investigate how transfer operators for prolongation and restriction have to be chosen such that our multigrid algorithms converge quickly. We point out why these transfer operators can be understood as block matrices as well and how they relate to the zeroes of the generating matrix function. We explain how our new algorithms can also be combined efficiently with the use of a natural coarse grid operator. We clearly identify a class of ill-conditioned block Toeplitz matrices for which our algorithmic ideas are suitable. In the final section we present an outlook to well-conditioned block Toeplitz systems and to problems of vector Laplace type. In the latter case the small size blocks can be interpreted as degrees of freedom associated with a node. A large number of numerical experiments throughout the article confirms convincingly that our multigrid solvers lead to optimal order convergence.

Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport

AbstractThis study investigates algebraic multilevel domain decomposition preconditioners of the Schwarz type for solving linear systems associated with Newton–Krylov methods. The key component of the preconditioner is a coarse approximation based on algebraic multigrid ideas to approximate the global behaviour of the linear system. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the non‐zero block structure of the Jacobian matrix. The scalability of the preconditioner is presented as well as comparisons with a two‐level Schwarz preconditioner using a geometric coarse grid operator. These comparisons are obtained on large‐scale distributed‐memory parallel machines for systems arising from incompressible flow and transport using a stabilized finite element formulation. The results demonstrate the influence of the smoothers and coarse level solvers for a set of 3D example problems. For preconditioners with more than one level, careful attention needs to be given to the balance of robustness and convergence rate for the smoothers and the cost of applying these methods. For properly chosen parameters, the two‐ and three‐level preconditioners are demonstrated to be scalable to 1024 processors. Copyright © 2006 John Wiley & Sons, Ltd.

On Multigrid for Overlapping Grids

The solution of elliptic partial differential equations on composite overlapping grids using multigrid is discussed. An approach is described that provides a fast and memory efficient scheme for the solution of boundary value problems in complex geometries. The key aspects of the new scheme are an automatic coarse grid generation algorithm, an adaptive smoothing technique for adjusting residuals on different component grids, and the use of local smoothing near interpolation boundaries. Other important features include optimizations for Cartesian component grids, the use of over-relaxed red-black smoothers, and the generation of coarse grid operators through Galerkin averaging. Numerical results in two and three dimensions show that very good multigrid convergence rates can be obtained for Dirichlet, Neumann, and mixed boundary conditions. A comparison to Krylov-based solvers shows that the multigrid solver can be much faster and require significantly less memory.

Convergence Analysis of a Multigrid Method for a Convection-Dominated Model Problem

The paper presents a convergence analysis of a multigrid solver for a system of linear algebraic equations resulting from the discretization of a convection-diffusion problem using a finite element method. We consider piecewise linear finite elements in combination with a streamline diffusion stabilization. We analyze a multigrid method that is based on canonical intergrid transfer operators, a "direct discretization" approach for the coarse-grid operators and a smoother of line-Jacobi type. A robust (diffusion and h-independent) bound for the contraction number of the two-grid method andthe multigrid W-cycle are proved for a special class of convection-diffusion problems, namely with Neumann conditions on the outflow boundary, Dirichlet conditions on the rest of the boundary, and a flow direction that is constant and aligned with gridlines. Our convergence analysis is based on modified smoothing and approximation properties. The arithmetic complexity of one multigrid iteration is optimal up to a logarithmic term.

A Phase Error Analysis of Multigrid Methods for Hyperbolic Equations

In this paper, we study the effects of the coarse grid correction process on multigrid convergence for hyperbolic problems in one and two dimensions. We approach this from the perspective of phase error, which allows us to exploit the hyperbolic nature of the underlying PDE. In particular, we consider three combinations of coarse grid operators and coarse grid solution approaches: (1) inexact coarse grid solve with direct discretization, (2) exact coarse grid solve with direct discretization, and (3) exact coarse grid solve with Galerkin coarse grid operator. For all these approaches, we show that the convergence behavior of multigrid can be precisely described by the phase error analysis ofthe coarse grid correction matrix, and we verify our results by numerical examples in one and two dimensions.

An algebraic 2‐level domain decomposition preconditioner with applications to the compressible Euler equations

AbstractA domain decomposition technique to solve large‐scale aerodynamics problems on unstructured grids is investigated. The linear system, arising when an implicit time‐advancing scheme is used, is preconditioned using a Schwarz‐based method. The key idea of the Schwarz preconditioner is to solve (approximately) a linear problem on each subdomain, then to exchange information only with neighbouring subdomains. Unfortunately, the performance of the Schwarz‐type preconditioner deteriorates as the number of processors grows. So, in this case, a key element for obtaining a scalable preconditioner is to provide a coarse level operator. Since many of the coarse operators proposed in literature are difficult to implement on unstructured 2D and 3D meshes, a purely algebraic procedure, that requires the entries of the matrix only, has been developed. This procedure may be seen as an Algebraic MultiGrid (AMG) method applied as a coarse grid correction operator. The key idea is to take advantage of the use of local data of domain decomposition preconditioners, and of the automatic coarsening procedures of AMG methods.Two possible schemes to introduce the coarse grid operator will be described. Both cases have been implemented and tested in a distributed parallel environment, using the MPI library. It will be shown that for suitable values of the rank of the coarse grid operator it is possible to obtain a considerable reduction in the number of iterations compared to the Schwarz preconditioner without coarse operator. Copyright © 2002 John Wiley & Sons, Ltd.

Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics

AbstractMultigrid has been a popular solver method for finite element and finite difference problems with regular grids for over 20 years. The application of multigrid to unstructured grid problems, in which it is often difficult or impossible for the application to provide coarse grids, is not as well understood. In particular, methods that are designed to require only data that are easily available in most finite element applications (i.e. fine grid data), constructing the grid transfer operators and coarse grid operators internally, are of practical interest. We investigate three unstructured multigrid methods that show promise for challenging problems in 3D elasticity: (1) non‐nested geometric multigrid, (2) smoothed aggregation, and (3) plain aggregation algebraic multigrid. This paper evaluates the effectiveness of these three methods on several unstructured grid problems in 3D elasticity with up to 76 million degrees of freedom. Published in 2002 by John Wiley & Sons, Ltd.

Efficient multigrid methods for solving upwind-biased discretizations of the convection equation

This paper studies a novel multigrid approach to the solution for a second-order upwind-biased discretization of the convection equation in two dimensions. This approach is based on semi-coarsening and well-balanced explicit correction terms, added to coarse-grid operators to maintain on coarse grids the same cross-characteristic interaction as on the target (fine) grid. Multicolor relaxation schemes are used on all the levels, allowing a very efficient parallel implementation. The results of the smoothing analysis and numerical tests can be summarized as follows: (1) The residual asymptotic convergence rate is about 3 per cycle, which far surpasses the theoretical limit (4/3) predicted for standard multigrid algorithms with full-coarsening. (2) The full multigrid (FMG) algorithm with two V(0,2) cycles on the target grid and just one V(0,2) cycle on all the coarse grids always provides an approximate solution with the algebraic error less than the discretization error. (3) A novel algorithm for deriving a discrete solution approximating the true continuous solution with a relative accuracy given in advance is developed. The computational complexity of this algorithm is (nearly) optimal (comparable with the complexity of the FMG algorithm applied to solve the problem on the optimally spaced target grid).

Coarse-Grid Correction for Nonelliptic and Singular Perturbation Problems

The suitability of coarse-grid correction operators for multigrid solution of discretized nonelliptic partial differential equations is studied. It is found that the requirements for the orders of intergrid transfers and the order of approximation of the coarse-grid operators differ in an important way from the well-known "rules" pertaining to elliptic operators. Both Galerkin coarsening and coarsening schemes based on the differential operator are examined, and new coarse-grid correction strategies are suggested in light of this analysis.

Analysis of matrix-dependent multigrid algorithms

Convergence theory for a multigrid method with matrix-dependent restriction, prolongation and coarse-grid operators is developed for a class of SPD problems. It motivates the construction of improved multigrid versions for diffusion problems with discontinuous coefficients. A computational two-level analysis method for a class of separable problems is also available. It motivates the design of matrix-dependent multigrid algorithms and, in particular, multiple coarse-grid correction algorithms for highly indefinite equations. Numerical experiments show the advantage of the present methods for several examples. © 1998 John Wiley & Sons, Ltd.

The Black Box Multigrid Numerical Homogenization Algorithm

In mathematical models of flow through porous media, the coefficients typically exhibit severe variations in two or more significantly different length scales. Consequently, the numerical treatment of these problems relies on ahomogenizationorupscalingprocedure to define an approximate coarse-scale problem that adequately captures the influence of the fine-scale structure. Inherent in such a procedure is a compromise between its computational cost and the accuracy of the resulting coarse-scale solution. Although techniques that balance the conflicting demands of accuracy and efficiency exist for a few specific classes of fine-scale structure (e.g., fine-scale periodic), this is not the case in general. In this paper we propose a new, efficient, numerical approach for thehomogenizationof the permeability in models of single-phase saturated flow. Our approach is motivated by the observation that multiple length scales are captured automatically by robust multilevel iterative solvers, such as Dendy'sblack box multigrid. In particular, the operator-induced variational coarsening in black box multigrid produces coarse-grid operators that capture the essential coarse-scale influence of the medium's fine-scale structure. We derive an explicit local, cell-based, approximate expression for the symmetric, 2×2 homogenized permeability tensor that is defined implicitly by the black box coarse-grid operator. The effectiveness of this black box multigrid numerical homogenization method is demonstrated through numerical examples.

Matrix-Dependent Multigrid Homogenization for Diffusion Problems

For problems with strongly varying or discontinuous diffusion coefficients we present a method to compute coarse-scale operators and to approximately determine the effective diffusion tensor on the coarse-scale level. The approach is based on techniques that are used in multigrid, such as matrix-dependent prolongations and the construction of coarse-grid operators by means of the Galerkin approximation. In numerical experiments we compare our multigrid-homogenization method with continuous homogenization, renormalization, and simple averaging approaches.