Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods

The advent of parallel computers has led to the development of new solution algorithms for time-dependent partial differential equations. Two recently developed methods, multigrid waveform relaxation and time-parallel multigrid, have been designed to solve parabolic partial differential equations on many time-levels simultaneously. This paper compares the convergence properties of these methods, based on the results of an exponential Fourier mode analysis for a model problem.     

Theoretical analysis of some spectral multigrid methods

We develop a theoretical analysis of a multigrid algorithm applied to spectral element discretization of linear elliptic problems. For a 1-D problem with non-constant coefficients we prove essentially the independence of the two-level convergence factor with respect to both the degree of the polynomial approximation and the number of spectral elements. We also sketch some ideas for the analysis of the 2-D case when only one spectral element is involved.     

Contour Matching Using Wavelet Transform and Multigrid Methods

In this paper,wavelet transform and multigrid method are combined to make the method more practical.It is known that Gaussian filtering causes shrinkage of data.To overcome this disadvantage,Gaussian filtering is replaced with wavelet transform.This method introduces no curve shrinkage.Then,the linearized form of objective equation is proposed.This makes contour matching easier to implement.Finally,the multigrid method is used to speed up the convergence.     

Algebraic spectral multigrid methods

For the Helmholtz equation a spectral discretization with a symmetric and sparse matrix is presented. Certain algebraic spectral multigrid methods can be efficiently used for solving the linear systems.     

Scalable implementation of multigrid methods using partial semi-aggregation of coarse grids

The Journal of Supercomputing - Multigrid methods are efficient and fast algorithms for solving elliptic equations. However, they suffer from the degradation of parallel efficiency on coarser...     

A calculation procedure for three-dimensional steady recirculating flows using multigrid methods

An efficient finite difference calculation procedure for three-dimensional recirculating flows is presented. The algorithm is based on a coupled solution of the three-dimensional momentum and continuity equations in primitive variables by the multigrid technique. A symmetrical coupled Gauss-Seidel technique is used for iterations and is observed to provide good rates of smoothing. Calculations have been made of the fluid motion in a three-dimensional cubic cavity with a moving top wall. The efficiency of the method is demonstrated by performing calculations at different Reynolds numbers with finite difference grids as large as 66 × 66 × 66 nodes. The CPU times and storage requirements for these calculations are observed to be very modest. The algorithm has the potential to be the basis for an efficient general-purpose calculation procedure for practical fluid flows.     

Convergence analysis of geometrical multigrid methods for solving data-sparse boundary element equations

The convergence analysis of multigrid methods for boundary element equations arising from negative-order pseudo-differential operators is quite different from the usual finite element multigrid analysis for elliptic partial differential equations. In this paper, we study the convergence of geometrical multigrid methods for solving large-scale, data-sparse boundary element equations. In particular, we investigate multigrid methods for \(\mathcal{H}\)-matrices arising from the adaptive cross approximation to the single layer potential operator.     

Interior point multigrid methods for topology optimization

In this paper, a new multigrid interior point approach to topology optimization problems in the context of the homogenization method is presented. The key observation is that nonlinear interior point methods lead to linear-quadratic subproblems with structures that can be favourably exploited within multigrid methods. Primal as well as primal-dual formulations are discussed. The multigrid approach is based on the transformed smoother paradigm. Numerical results for an example problem are presented. Received February 15, 1999     

Convergence analysis of multigrid methods with collective point smoothers for optimal control problems

In this paper we consider multigrid methods for solving saddle point problems. The choice of an appropriate smoothing strategy is a key issue in this case. Here we focus on the widely used class of collective point smoothers. These methods are constructed by a point-wise grouping of the unknowns leading to, e.g., collective Richardson, Jacobi or Gauss-Seidel relaxation methods. Their smoothing properties are well-understood for scalar problems in the symmetric and positive definite case. In this work the analysis of these methods is extended to a special class of saddle point problems, namely to the optimality system of optimal control problems. For elliptic distributed control problems we show that the convergence rates of multigrid methods with collective point smoothers are bounded independent of the grid size and the regularization (or cost) parameter.     

Algebraic multigrid methods based on element preconditioning

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the “region of robustness“ between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy.

For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem.

The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.     

On monotone including nonlinear multigrid methods and applications

In recent years, some work has been devoted to construct multigrid methods for solving nonlinear systems which compute monotone including convergent sequences of sub- and supersolutions. Mainly with regard to the numerical solution of quasilinear partial differential equations we improve and generalize some of the existing results. In this paper we show that the monotone enclosure of the multigrid method follows from the monotone enclosure of the smoother. The theoretical results are confirmed by examples of realistic problems.     

Coupled algebraic multigrid methods for the Oseen problem

We provide a concept combining techniques known from geometric multigrid methods for saddle point problems (such as smoothing iterations of Braess- or Vanka-type) and from algebraic multigrid (AMG) methods for scalar problems (such as the construction of coarse levels) to a coupled algebraic multigrid solver. Coupled here is meant in contrast to methods, where pressure and velocity equations are iteratively decoupled (pressure correction methods) and standard AMG is used for the solution of the resulting scalar problems. To prove the efficiency of our solver experimentally, it is applied to finite element discretizations of real life industrial problems.     

Nonlinear multigrid methods for total variation image denoising

The classical image denoising technique introduced by Rudin, Osher, and Fatemi [17] a decade ago, leads to solve a constrained minimization problem for the total variation (TV) of the image. The formal first variation of the minimization problem is a nonlinear and highly anisotropic boundary value problem. In this paper, a computational PDE method based on a nonlinear multigrid scheme for restoring noisy images is suggested. Here, we examine different discretizations for the Euler–Lagrange equation as well as different smoothers within the multigrid scheme. Then we describe the iterative total variation regularization scheme, which starts with an isotropic (smooth) problem and leads to smooth edges in the image. Within the iteration the problem becomes more and more anisotropic and converges to an image with sharp edges. Finally, we present some experimental results for synthetic and real images.     

On an energy minimizing basis for algebraic multigrid methods

This paper is devoted to the study of an energy minimizing basis first introduced in Wan, Chan and Smith (2000) for algebraic multigrid methods. The basis will be first obtained in an explicit and compact form in terms of certain local and global operators. The basis functions are then proved to be locally harmonic functions on each coarse grid element. Using these new results, it is illustrated that this basis can be numerically obtained in an optimal fashion. In addition to the intended application for algebraic multigrid method, the energy minimizing basis may also be applied for numerical homogenization.     

Image analysis using mathematical morphology

For the purposes of object or defect identification required in industrial vision applications, the operations of mathematical morphology are more useful than the convolution operations employed in signal processing because the morphological operators relate directly to shape. The tutorial provided in this paper reviews both binary morphology and gray scale morphology, covering the operations of dilation, erosion, opening, and closing and their relations. Examples are given for each morphological concept and explanations are given for many of their interrelationships.     

基于图像复杂度的隐写方法研究*

为了提供较大的隐写容量和保持良好的载密图像质量,依据人眼对纹理、边界和黑暗区域变化敏感性弱的视觉特点,结合小波变换提出了一种图像复杂度描述方法,将图像小块分为纹理、边界、黑暗和平滑四个不同类别,利用模函数设计隐写算法,在不同区域嵌入不同量的信息。实验结果表明,新的复杂度描述方法能准确区分不同类型的小块,隐写算法在提高嵌入容量的同时保持了较好的视觉质量。     

Image analysis using hahn moments

This paper shows how Hahn moments provide a unified understanding of the recently introduced Chebyshev and Krawtchouk moments. The two latter moments can be obtained as particular cases of Hahn moments with the appropriate parameter settings, and this fact implies that Hahn moments encompass all their properties. The aim of this paper is twofold: 1) To show how Hahn moments, as a generalization of Chebyshev and Krawtchouk moments, can be used for global and local feature extraction, and 2) to show how Hahn moments can be incorporated into the framework of normalized convolution to analyze local structures of irregularly sampled signals.     

Nested multigrid methods for time-periodic,parabolic optimal control problems

We present a nested multigrid method to optimize time-periodic, parabolic, partial differential equations (PDE). We consider a quadratic tracking objective with a linear parabolic PDE constraint. The first order optimality conditions, given by a coupled system of boundary value problems can be rewritten as an Fredholm integral equation of the second kind, which is solved by a multigrid of the second kind. The evaluation of the integral operator consists of solving sequentially a boundary value problem for respectively the state and the adjoints. Both problems are solved efficiently by a time-periodic space-time multigrid method.     

Radiosity and relaxation methods

To date, there has been some confusion in the computer graphics community about how the progressive radiosity (PR) method relates to standard numerical methods for solving linear systems of equations. We show that PR is actually equivalent to the combination of two numerical analysis techniques known as Southwell relaxation and Jacobi iteration. A new overshooting method similar to over relaxation can accelerate the convergence of the iterative radiosity methods