A PROCESS FOR SOLVING A FEW EXTREME EIGENPAIRS OF LARGE SPARSE POSITIVE DEFINITE GENERALIZED EIGENVALUE PROBLEM

1. IntroductionWe are concerned in this work with finding a few extreme eigenvalues and theircorresponding eigenvectors of a generalized large scale eigenvalue problem in which thematrices are sparse and symmetric positive definite.Although finding a few extreme eigenpairs is of interest both in theory and practice,there are only few usable and efficient methods up to now. Reinsch and Baner ([12]),suggested a oR algorithm with Newton shift for the standard eigenproblem which included an ingen…     

Increasing efficiency of inverse iteration

Inverse iteration is simple but not very efficient method for computing few eigenvalues with minimal absolute values and corresponding eigenvectors of a symmetric matrix. The idea is to increase its efficiency by technique similar to multigrid methods used for solving linear systems. This approach is not new, but until now multigrid was mostly used for solving linear system which appear in Rayleigh quotient iteration, inverse iteration and related iterative methods. Instead of choosing appropriate coordinates (grids), our algorithm performs inverse iteration on a sequence of subspaces with decreasing dimensions (multispace). Block Lanczos method is used for the selection of a smaller subspace. This will produce a banded matrix, which makes inverse iteration even faster in the smaller dimensions.      

An inverse iteration method using multigrid for quantum chemistry

We present a method to compute the lowest eigenpairs of a generalized eigenvalue problem resulting from the discretization of a stationary Schrödinger equation by a fourth order finite difference scheme of Numerov type. We propose to use an inverse iteration method combined with a Rayleigh-Ritz procedure to correct several eigenvectors at the same time. The linear systems in the inverse iteration scheme are regularized by projections on lower dimensional spaces and approximately solved by a multigrid algorithm.We apply the method to the electronic structure calculation in quantum chemistry.     

An algebraic multigrid method with interpolation reproducing rigid body modes for semi-definite problems in two-dimensional linear elasticity

Based on the geometric grid information as geometric coordinates, an algebraic multigrid (AMG) method with the interpolation reproducing the rigid body modes (namely the kernel elements of semi-definite operator arising from linear elasticity) is constructed, and such method is applied to the linear elasticity problems with a traction free boundary condition and crystal problems with free boundary conditions as well. The results of various numerical experiments in two dimensions are presented. It is shown from the numerical results that the constructed AMG method is robust and efficient for such semi-definite problems, and the convergence is uniformly bounded away from one independent of the problem size. Furthermore, the AMG method proposed in this paper has better convergence rate than the commonly used AMG methods. Simultaneously, an AMG method that can preserve the quotient space, which means that if the exact solution of original problem belongs to the quotient space of discrete operator considered, then the numerical solution of AMG method is convergent in the same quotient space, is obtained using the technique of orthogonal decomposition.     

A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient

The discretization of eigenvalue problems for partial differential operators is a major source of matrix eigenvalue problems having very large dimensions, but only some of the smallest eigenvalues together with the eigenvectors are to be determined. Preconditioned inverse iteration (a “matrix-free” method) derives from the well-known inverse iteration procedure in such a way that the associated system of linear equations is solved approximately by using a (multigrid) preconditioner. A new convergence analysis for preconditioned inverse iteration is presented. The preconditioner is assumed to satisfy some bound for the spectral radius of the error propagation matrix resulting in a simple geometric setup. In this first part the case of poorest convergence depending on the choice of the preconditioner is analyzed. In the second part the dependence on all initial vectors having a fixed Rayleigh quotient is considered. The given theory provides sharp convergence estimates for the eigenvalue approximations showing that multigrid eigenvalue/vector computations can be done with comparable efficiency as known from multigrid methods for boundary value problems.     

An efficient algorithm for the Schrödinger–Poisson eigenvalue problem

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrödinger–Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrödinger eigenvalue problem, and use the block Lanczos method to compute the first k   eigenpairs of the Schrödinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrödinger eigenvalue problem are affected by the dopant in the Schrödinger–Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h³)$\mathrm{O}(h^3)$.     

Using cross-product matrices to compute the SVD

This paper concerns accurate computation of the singular value decomposition (SVD) of an matrix . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of and take the square roots of the large eigenvalues of it as the large singular values of ; (b) form the Rayleigh quotient of with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.Supported in part by the National Science Foundation of China (No. 10471074).     

On the Topology of Real Algebraic Plane Curves

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.     

New interpolation formulas of using geometric assumptions in the algebraic multigrid method

In this paper, new interpolation formulas for using geometric assumptions in the algebraic multigrid (AMG) method are reported. The theoretical and convergence analysis will be presented. The effectiveness and robustness of these interpolation formulas are demonstrated by numerical experiments. Not only is a rapid rate of convergence achieved, but the AMG algorithm used in conjunction with these formulas can also be used to solve various ill-conditioned systems of equations. The principal contribution of the present method is to extend the range of applications of the AMG method developed by Ruge and Stüben.     

A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.     

Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections

A framework is proposed for constructing algebraic multigrid transfer operators suitable for nonsymmetric positive definite linear systems. This framework follows a Schur complement perspective as this is suitable for both symmetric and nonsymmetric systems. In particular, a connection between algebraic multigrid and approximate block factorizations is explored. This connection demonstrates that the convergence rate of a two‐level model multigrid iteration is completely governed by how well the coarse discretization approximates a Schur complement operator. The new grid transfer algorithm is then based on computing a Schur complement but restricting the solution space of the corresponding grid transfers in a Galerkin‐style so that a far less expensive approximation is obtained. The final algorithm corresponds to a Richardson‐type iteration that is used to improve a simple initial prolongator or a simple initial restrictor. Numerical results are presented illustrating the performance of the resulting algebraic multigrid method on highly nonsymmetric systems. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on harmonic Ritz values and their reciprocals

This note summarizes an investigation of harmonic Ritz values to approximate the interior eigenvalues of a real symmetric matrix A while avoiding the explicit use of the inverse A^?1. We consider a bounded functional ψ that yields the reciprocals of the harmonic Ritz values of a symmetric matrix A. The crucial observation is that with an appropriate residual s, many results from Rayleigh quotient and Rayleigh–Ritz theory naturally extend. The same is true for the generalization to matrix pencils (A, B) when B is symmetric positive definite. These observations have an application in the computation of eigenvalues in the interior of the spectrum of a large sparse matrix. The minimum and maximum of ψ correspond to the eigenpairs just to the left and right of zero (or a chosen shift). As a spectral transformation, this distinguishes ψ from the original harmonic approach where an interior eigenvalue remains at the interior of the transformed spectrum. As a consequence, ψ is a very attractive vehicle for a matrix‐free, optimization‐based eigensolver. Instead of computing the smallest/largest eigenvalues by minimizing/maximizing the Rayleigh quotient, one can compute interior eigenvalues as the minimum/maximum of ψ. Copyright © 2009 John Wiley & Sons, Ltd.     

WEB-Spline–Based Multigrid Methods for the Stationary Stokes Problem

We discuss a multigrid technique in solving a large system of linear algebraic equations arising in the approximation of Stokes equations by a new strategy based on weighted extended B-spline (WEB-spline) methods. Three types of WEB-spline–based Stokes elements satisfying the inf-sup condition are considered. First for a linear-constant type of Stokes element, we give the detailed multigrid algorithm and its convergence proof. The convergence proof of the multigrid algorithm for a bubble-stabilized WEB-spline–based Stokes element is dealt with separately. Multigrid method in the case of bubble-condensed variational form is simplified using the techniques from the bubble-stabilized case.     

Differential Modular Forms on Shimura Curves, I

The quotient of a Shimura curve by the isogeny equivalence relation is not an object of algebraic geometry. The paper shows how this quotient space becomes a geometric object in a more general geometry obtained from 'usual algebraic geometry', by adjoining a new operation; this operation looks like a 'Fermat quotient' and should be viewed as an arithmetic analogue of usual derivations.     

An efficient hierarchical preconditioner for quadratic discretizations of finite element problems

Higher order finite element discretizations, although providing higher accuracy, are considered to be computationally expensive and of limited use for large‐scale problems. In this paper, we have developed an efficient iterative solver for solving large‐scale quadratic finite element problems. The proposed approach shares some common features with geometric multigrid methods but does not need structured grids to create the coarse problem. This leads to a robust method applicable to finite element problems discretized by unstructured meshes such as those from adaptive remeshing strategies. The method is based on specific properties of hierarchical quadratic bases. It can be combined with an algebraic multigrid (AMG) preconditioner or with other algebraic multilevel block factorizations. The algorithm can be accelerated by flexible Krylov subspace methods. We present some numerical results on the convection–diffusion and linear elasticity problems to illustrate the efficiency and the robustness of the presented algorithm. In these experiments, the performance of the proposed method is compared with that of an AMG preconditioner and other iterative solvers. Our approach requires less computing time and less memory storage. Copyright © 2010 John Wiley & Sons, Ltd.     

An algebraic multigrid method for high order time-discretizations of the div-grad and the curl-curl equations

We present an algebraic multigrid algorithm for fully coupled implicit Runge–Kutta and Boundary Value Method time-discretizations of the div-grad and curl-curl equations. The algorithm uses a blocksmoother and a multigrid hierarchy derived from the hierarchy built by any algebraic multigrid algorithm for the stationary version of the problem. By a theoretical analysis and numerical experiments, we show that the convergence is similar to or better than the convergence of the scalar algebraic multigrid algorithm on which it is based. The algorithm benefits from several possibilities for implementation optimization. This results in a computational complexity which, for a modest number of stages, scales almost linearly as a function of the number of variables.     

Numerical computation of the genus of an irreducible curve within an algebraic set

The common zero locus of a set of multivariate polynomials (with complex coefficients) determines an algebraic set. Any algebraic set can be decomposed into a union of irreducible components. Given a one-dimensional irreducible component, i.e. a curve, it is useful to understand its invariants. The most important invariants of a curve are the degree, the arithmetic genus and the geometric genus (where the geometric genus denotes the genus of a desingularization of the projective closure of the curve). This article presents a numerical algorithm to compute the geometric genus of any one-dimensional irreducible component of an algebraic set.     

Geometric and algebraic multigrid techniques for fluid dynamics problems on unstructured grids

Issues concerning the implementation and practical application of geometric and algebraic multigrid techniques for solving systems of difference equations generated by the finite volume discretization of the Euler and Navier–Stokes equations on unstructured grids are studied. The construction of prolongation and interpolation operators, as well as grid levels of various resolutions, is discussed. The results of the application of geometric and algebraic multigrid techniques for the simulation of inviscid and viscous compressible fluid flows over an airfoil are compared. Numerical results show that geometric methods ensure faster convergence and weakly depend on the method parameters, while the efficiency of algebraic methods considerably depends on the input parameters.     

The Computation of Invariant Fields and a Constructive Version of a Theorem by Rosenlicht

Let G be an algebraic group acting on an irreducible variety X. We present an algorithm for computing the invariant field k(X)^G. Moreover, we give a constructive version of a theorem of Rosenlicht, which says that almost all orbits can be separated by rational invariants. More precisely, we give an algorithm for computing a nonempty open subset of X with a geometric quotient.     

Bootstrap Algebraic Multigrid: Status Report,Open Problems,and Outlook

This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process. We review some recent research in the development, analysis and application of bootstrap algebraic multigrid and point to open problems in these areas. Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.