Analysis of a Class of Parallel Multigrid Smoothers

This paper proposes and analyzes a class of multigrid smoothers called the parallel multiplicative (PM) smoother by subspace decomposition techniques. It shows that the well known additive and multiplicative smoothers and the JSOR smoother are special cases of the PM smoother, and their smoothing properties can be obtained directly from the PM analysis. Moreover, numerical results are presented in this paper to show that the JSOR smoother is more robust and effective than the damped Jacobi smoother on current MIMD parallel computers. AMS subject classification (2000) 65N55, 65Y05.Received May 2004. Revised September 2004. Communicated by Per Lötstedt.Dexuan Xie: This work was partially supported by the National Science Foundation through grant DMS-0241236.     

Mathematical Analysis of a Coarsening Model with Local Interactions

We consider particles on a one-dimensional lattice whose evolution is governed by nearest-neighbor interactions where particles that have reached size zero are removed from the system. Concentrating on configurations with infinitely many particles, we prove existence of solutions under a reasonable density assumption on the initial data and show that the vanishing of particles and the localized interactions can lead to non-uniqueness. Moreover, we provide a rigorous upper coarsening estimate and discuss generic statistical properties as well as some non-generic behavior of the evolution by means of heuristic arguments and numerical observations.     

Fourier Analysis of Multigrid for a Model Two-Dimensional Convection-Diffusion Equation

We present a Fourier analysis of multigrid for the two-dimensional discrete convection-diffusion equation. For constant coefficient problems with grid-aligned flow and semi-periodic boundary conditions, we show that the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4×4 blocks. This enables a trivial computation of the norm of the iteration matrix demonstrating rapid convergence in the case of both small and large mesh Peclet numbers, where the streamline-diffusion discretisation is used in the latter case. We also demonstrate that these results are strongly correlated with the properties of the iteration matrix arising from Dirichlet boundary conditions. AMS subject classification (2000) 65F10, 65N22, 65N30, 65N55     

Special Solutions to a Nonlinear Coarsening Model with Local Interactions

Journal of Nonlinear Science - We consider a class of mass transfer models on a one-dimensional lattice with nearest-neighbour interactions. The evolution is given by the backward parabolic...     

Nonlinear Approximation with Local Fourier Bases

It is shown that local Fourier bases are unconditional bases for the modulation spaces on R, including the Bessel potential spaces and the Segal algebra S ₀ . As a consequence, the abstract function spaces, that are defined by the approximation properties with respect to a local Fourier basis, are precisely the modulation space s. April 22, 1998. Date accepted: May 18, 1999.     

Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

In this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.     

Local Polynomial Fitting with Long-Memory, Short-Memory and Antipersistent Errors

Local polynomial smoothing for the trend function and its derivatives in nonparametric regression with long-memory, short-memory and antipersistent errors is considered. We show that in the case of antipersistence, the convergence rate of a nonparametric regression estimator is faster than for uncorrelated or short-range dependent errors. Moreover, it is shown that unified asymptotic formulas for the optimal bandwidth and the MSE hold for all of the three dependence structures. Also, results on estimation at the boundary are included. A bandwidth selector for nonparametric regression with different types of dependent errors is proposed. Its asymptotic property is investigated. The practical performance of the proposal is illustrated by simulated and real data examples.     

Multigrid Methods for Elliptic Optimal Control Problems with Pointwise State Constraints

An elliptic optimal control problem with constraints on the state variable is considered. The Lavrentiev-type regularization is used to treat the constraints on the state variable. To solve the problem numerically, the multigrid for optimization (MGOPT) technique and the collective smoothing multigrid (CSMG) are implemented. Numerical results are reported to illustrate and compare the efficiency of both multigrid strategies.     

Standard and Economical Cascadic Multigrid Methods for the Mortar Finite Element Methods

In this paper, standard and economical cascadic multigrid methods are con-sidered for solving the algebraic systems resulting from the mortar finite element meth-ods. Both cascadic multigrid methods do not need full elliptic regularity, so they can be used to tackle more general elliptic problems. Numerical experiments are reported to support our theory.     

Local Fourier Analysis for Edge-Based Discretizations on Triangular Grids

In this paper, we present a local Fourier analysis framework for analyzing the different components within multigrid solvers for edge-based discretizations on triangular grids. The different stencils associated with edges of different orientation in a triangular mesh make this analysis special. The resulting tool is demonstrated for the vector Laplace problem discretized by mimetic finite difference schemes. Results from the local Fourier analysis, as well as experimentally obtained results, are presented to validate the proposed analysis.     

Optimal Multigrid Methods with New Transfer Operators Based on Finite Difference Approximations

We constructed new interpolation operator in multigrid methods, which is efficient to transfer residual error from coarse grid to fine grid. This operator used idea of solving local residual equation using the standard stencil and the skewed stencil of the centered difference approximation to the Laplacian operator. We also compared our new multigrid methods with traditional multigrid methods, and found that new method is optimal.     

Multigrid and MGR[$v$] Methods for Diffusion Equations

The MGR[$v$] Algorithm of Ries, Trottenberg and Winter with v=0 and the Algorithm 2.1 of Braess are essentially the same multigrid algorithm for the discrete poisson equation. In this report we consider the extension to the general diffusion equation. In particular, we indicate the proof of the basic result $ρ≤\frac{1}{2}(1+Kh)$, thus extending the results of Braess and Ries Trottenberg and Winter. In addition to this theoretical result we present computational results which indicate that other constant coefficient estimates carry over to this case.     

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.     

The Convergence of Multigrid Methods for Nonsymmetric Elliptic Variational Inequalities

This paper is concerned with the convergence of multigrid methods (MGM) on nonsymmetric elliptic variational inequalities. On the basis of Wang and Zeng's work (1988), we develop the convergence results of the smoothing operator (i.e. PJOR and PSOR). We also extend the multigrid method of J.Mandel (1984) to nonsymmetric variational inequalities and obtain the convergence of MGM for these problems.     

Local properties of factored Fourier series

By using a convex sequence Bor [H. Bor, A study on local properties of factored Fourier series, Nonlinear Anal. 57 (2004) 191-197] has obtained a result dealing with local properties of factored Fourier series for summability. In this paper, we have generalized that result for summability. Some new results have also been obtained.     

Variography for Model Selection in Local Polynomial Regression with Spatial Data

In this work, we apply variographic techniques from spatial statistics to the problem of model selection in local polynomial regression with multivariate data. These techniques permit selection of the kernel and smoothing matrix with less computational load and interpretation of the regularity of the regression function in different directions. Moreover, they may represent the only feasible alternative for problems of a certain dimensionality.     

Universal Consistency of Local Polynomial Kernel Regression Estimates

Regression function estimation from independent and identically distributed data is considered. The L ₂ error with integration with respect to the design measure is used as an error criterion. It is shown that suitably defined local polynomial kernel estimates are weakly and strongly universally consistent, i.e., it is shown that the L ₂ errors of these estimates converge to zero almost surely and in L ₁ for all distributions.     

Local Hardy spaces and summability of Fourier transforms

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the amalgam space to W(L_p,ℓ_∞). This implies the almost everywhere convergence of the θ-means for all fW(L₁,ℓ_∞)L₁.