Steplength optimization and linear multigrid methods

Summary Recently steplength parameters have been used in linear multigrid methods. In this paper we give a theoretical analysis of the effects of steplength optimization in a rather general framework which covers two different implementations of steplength optimization in standard multigrid methods.     

Artificial damping in multigrid methods

When the solution and problem coefficients are highly oscillatory, the computed solution may not show characteristics of the original physical problem unless the numerical mesh is sufficiently fine. In the case, the coarse grid problem of a multigrid (MG) algorithm must be still huge and poorly-conditioned, and therefore, it is hard to solve by either a direct method or an iterative scheme. This article suggests a MG algorithm for such problems in which the coarse grid problem is slightly modified by an artificial damping (compressibility) term. It has been numerically observed that the artificial damping, even if slight, makes the coarse grid problem much easier to solve, without deteriorating the overall convergence rate of the MG method. For most problems, 2–6 times speed up have been observed.     

Nonnested multigrid methods for linear problems

This article presents an application of nonnested and unstructured multigrid methods to linear elastic problems. A variational formulation for transfer operators and some multigrid strategies, including adaptive algorithms, are presented. Expressions for the performance evaluation of multigrid strategies and its comparison with direct and preconditioned conjugate gradient algorithms are also presented. A C++ implementation of the multigrid algorithms and the quadtree and octree data structures for transfer operators are discussed. Some two‐ and three‐dimensional elasticity examples are analyzed. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:313–331, 2001     

Using block smoothers in multigrid methods

We present a general setting for the analysis of blocked smoothers using the LFA. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Grid transfer operators for multigrid methods

Local Fourier analysis (LFA) is a classical tool for proving convergence theorems for multigrid methods (MGMs). Analogously, the symbols of the involved matrices are studied to prove convergence results for MGMs for Toeplitz matrices. We show that in the case of elliptic partial differential equations (PDEs) with constant coefficients, the two different approaches lead to an equivalent optimality condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA which allows to deal not only with differential problems but also, e.g., with integral problems. A class of grid transfer operators related to the B-spline's refinement equation is discussed as well. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relaxation schemes for spectral multigrid methods

The effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined. Iterative methods are an attractive alternative to direct methods because Fourier transform techniques enable the discrete matrix-vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations. Preconditioning is found to be essential for acceptable rates of convergence. Preconditioners based on second-order finite difference methods are used. A comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary. The investigations show that iterations based on incomplete LU decompositions provide the most efficient methods for solving these algebraic systems.     

Cascadic multigrid methods for parabolic problems

In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.     

Defect-correction multigrid methods for nonlinear problems

Auzinger and Stetter [1] combine a multigrid method with defect-correction iteration and derive a composite iterative procedure which they call the DCMG (defect-correction multigrid) cycle. Using a high-order discrete operator in the coarsegrid correction and a lower-order operator in relaxation, the DCMG cycle achieves the higher-order approximation [4]. In an analogous way, DCMG can be used to solve nonlinear PDEs by using the nonlinear operator in correction and a related linear operator in relaxation. We prove convergence of such a DCMG scheme and give an error estimation.     

Algebraic multigrid methods for Laplacians of graphs

Classical algebraic multigrid theory relies on the fact that the system matrix is positive definite. We extend this theory to cover the positive semidefinite case as well, by formulating semiconvergence results for these singular systems. For the class of irreducible diagonal dominant singular M-matrices we show that the requirements of the developed theory hold and that the coarse level systems are still of the same class, if the C/F-splitting is good enough. An important example for matrices that are irreducible diagonal dominant M-matrices are Laplacians of graphs. Recent shape optimizing methods for graph partitioning require to solve singular linear systems involving these Laplacians. We present convergence results as well as experimental results for numerous graphs arising from finite element discretizations with up to 10⁶ vertices.     

A note on terminology in multigrid methods

We compare terminology used in the literature on multigrid methods for compressible computational fluid dynamics to that used in linear multigrid theory. Several popular iterative and direct smoothers are presented side-by-side using the same terminology. We argue for greater analysis of these methods in order to place them into a more rigorous framework and to identify the most promising candidates for future development. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

V-cycle multigrid methods for Wilson nonconforming element

Here two types of optimal V-cycle multigrid algorithms are presented for Wilson nonconforming finite element.     

Homogeneous multigrid for embedded discontinuous Galerkin methods

BIT Numerical Mathematics - We formulate a multigrid method for an embedded discontinuous Galerkin (EDG) discretization scheme for Poisson’s equation. This multigrid method is homogeneous in...     

A generalized predictive analysis tool for multigrid methods

Multigrid and related multilevel methods are the approaches of choice for solving linear systems that result from discretization of a wide class of PDEs. A large gap, however, exists between the theoretical analysis of these algorithms and their actual performance. This paper focuses on the extension of the well‐known local mode (often local Fourier) analysis approach to a wider class of problems. The semi‐algebraic mode analysis (SAMA) proposed here couples standard local Fourier analysis approaches with algebraic computation to enable analysis of a wider class of problems, including those with strong advective character. The predictive nature of SAMA is demonstrated by applying it to the parabolic diffusion equation in one and two space dimensions, elliptic diffusion in layered media, as well as a two‐dimensional convection‐diffusion problem. These examples show that accounting for boundary conditions and heterogeneity enables accurate predictions of the short‐term and asymptotic convergence behavior for multigrid and related multilevel methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Monotone multigrid methods for elliptic variational inequalities II

Summary. We derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements. The coarse grid corrections are computed from certain obstacle problems. The actual constraints are fixed by the preceding nonlinear fine grid smoothing. This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency. We give estimates for the asymptotic convergence rates. The numerical results indicate a significant improvement as compared with previous multigrid approaches. Received March 26, 1994 / Revised version received September 22, 1994     

Convergence analysis of multigrid methods with residual scaling techniques

In this paper, multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered. The idea of perturbed two-grid methods proposed in [7] is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors. We will show that if the convergence factors of the two-grid methods are uniformly bounded by σ (σ<0.5), then the convergence factors of the W-cycle multigrid methods are uniformly bounded by σ/(1−σ), whether the residuals are scaled at some or all levels. This result extends Notay’s Theorem 3.1 in [7] to more general cases. The result also confirms the viewpoint that the W-cycle multigrid method will converge sufficiently well as long as the convergence factor of the two-grid method is small enough. In the case where the convergence factor of the two-grid method is not small enough, by appropriate choice of the cycle index γ, we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than σ/(1−σ). Numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor. The convergence rates of the two-grid methods and the multigrid methods show that the W-cycle multigrid methods perform better if the convergence rate of the two-grid method becomes smaller. These numerical experiments support the proposed theoretical results in this paper.     

V-cycle convergence of some multigrid methods for ill-posed problems

For ill-posed linear operator equations we consider some V-cycle multigrid approaches, that, in the framework of Bramble, Pasciak, Wang, and Xu (1991), we prove to yield level independent contraction factor estimates. Consequently, we can incorporate these multigrid operators in a full multigrid method, that, together with a discrepancy principle, is shown to act as an iterative regularization method for the underlying infinite-dimensional ill-posed problem. Numerical experiments illustrate the theoretical results.

     

Convergence of nonconforming multigrid methods without full elliptic regularity

We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound () for the contraction number of the -cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable -cycle algorithm is an optimal preconditioner.

     

A generalized BPX multigrid framework covering nonnested V-cycle methods

More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or noninherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and it produces a variable V-cycle, or nonuniform convergence rate V-cycle methods, or other nonoptimal results in analysis thus far.

This paper completes a long-time effort in extending the BPX multigrid framework so that it truly covers the nonnested V-cycle. We will apply the extended BPX framework to the analysis of many V-cycle nonnested multigrid methods. Some of them were proven previously only for two-level and W-cycle iterations. Some numerical results are presented to support the theoretical analysis of this paper.

     

Estimates for multigrid methods based on red-black Gauss-Seidel smoothings

Summary TheMGR[v] algorithms of Ries, Trottenberg and Winter, the Algorithms 2.1 and 6.1 of Braess and the Algorithm 4.1 of Verfürth are all multigrid algorithms for the solution of the discrete Poisson equation (with Dirichlet boundary conditions) based on red-black Gauss-Seidel smoothing. Both Braess and Verfürth give explicit numerical upper bounds on the rate of convergence of their methods in convex polygonal domains. In this work we reconsider these problems and obtain improved estimates for theh–2h Algorithm 4.1 as well asW-cycle estimates for both schemes in non-convex polygonal domains. The proofs do not depend on the strengthened Cauchy inequality.Sponsored by the Air Force Office of Scientific Research under Contract No. AFOSR-86-0163     

Some numerical experiments with multigrid methods on Shishkin meshes

Piecewise uniform meshes introduced by Shishkin, are a very useful tool to construct robust and efficient numerical methods to approximate the solution of singularly perturbed problems. For small values of the diffusion coefficient, the step size ratios, in this kind of grids, can be very large. In this case, standard multigrid methods are not convergent. To avoid this troublesome, in this paper we propose a modified multigrid algorithm, which works fine on Shishkin meshes. We show some numerical experiments confirming that the proposed multigrid method is convergent, and it has similar properties that standard multigrid for classical elliptic problems.