A Dual-Primal FETI method for incompressible Stokes equations

In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed finite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, the solution of an indefinite Stokes problem is reduced to solving a symmetric positive definite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by the conjugate gradient method with a Dirichlet preconditioner. In each iteration step, both subdomain problems and a coarse level problem are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf-sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains. Numerical experiments demonstrate the scalability of this new method. This work is based on a doctoral dissertation completed at Courant Institute of Mathematical Sciences, New York University. This work was supported in part by the National Science Foundation under Grants NSF-CCR-9732208, and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127.     

A stationary iterative pseudoinverse algorithm

Iterative methods applied to the normal equationsA ^T Ax=A ^T b are sometimes used for solving large sparse linear least squares problems. However, when the matrix is rank-deficient many methods, although convergent, fail to produce the unique solution of minimal Euclidean norm. Examples of such methods are the Jacobi and SOR methods as well as the preconditioned conjugate gradient algorithm. We analyze here an iterative scheme that overcomes this difficulty for the case of stationary iterative methods. The scheme combines two stationary iterative methods. The first method produces any least squares solution whereas the second produces the minimum norm solution to a consistent system. This work was supported by the Swedish Research Council for Engineering Sciences, TFR.     

Approximation by interpolating variational splines

In this paper we present an approximation problem of parametric curves and surfaces from a Lagrange or Hermite data set. In particular, we study an interpolation problem by minimizing some functional on a Sobolev space that produces the new notion of interpolating variational spline. We carefully establish a convergence result. Some specific cases illustrate the generality of this work.     

Quadrature formulas descending from BS Hermite spline quasi-interpolation

Two new classes of quadrature formulas associated to the BS Boundary Value Methods are discussed. The first is of Lagrange type and is obtained by directly applying the BS methods to the integration problem formulated as a (special) Cauchy problem. The second descends from the related BS Hermite quasi-interpolation approach which produces a spline approximant from Hermite data assigned on meshes with general distributions. The second class formulas is also combined with suitable finite difference approximations of the necessary derivative values in order to define corresponding Lagrange type formulas with the same accuracy.     

A class of Uzawa-SOR methods for saddle point problems

In this paper, we consider a class of Uzawa-SOR methods for saddle point problems, and prove the convergence of the proposed methods. We solve a lower triangular system per iteration in the proposed methods, instead of solving a linear equation Az=b. Actually, the new methods can be considered as an inexact iteration method with the Uzawa as the outer iteration and the SOR as the inner iteration. Although the proposed methods cannot achieve the same convergence rate as the GSOR methods proposed by Bai et al. [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1-38], but our proposed methods have less workloads per iteration step. Experimental results show that our proposed methods are feasible and effective.     

Solution of augmented linear systems using orthogonal factorizations

We consider solvingx+Ay=b andA ^T x=c for givenb, c andm ×n A of rankn. This is called the augmented system formulation (ASF) of two standard optimization problems, which include as special cases the minimum 2-norm of a linear underdetermined system (b=0) and the linear least squares problem (c=0), as well as more general problems. We examine the numerical stability of methods (for the ASF) based on the QR factorization ofA, whether by Householder transformations, Givens rotations, or the modified Gram-Schmidt (MGS) algorithm, and consider methods which useQ andR, or onlyR. We discuss the meaning of stability of algorithms for the ASF in terms of stability of algorithms for the underlying optimization problems.We prove the backward stability of several methods for the ASF which useQ andR, inclusing a new one based on MGS, and also show under what circumstances they may be regarded as strongly stable. We show why previous methods usingQ from MGS were not backward stable, but illustrate that some of these methods may be acceptable-error stable. We point out that the numerical accuracy of methods that do not useQ does not depend to any significant extent on which of of the above three QR factorizations is used. We then show that the standard methods which do not useQ are not backward stable or even acceptable-error stable for the general ASF problem, and discuss how iterative refinement can be used to counteract these deficiencies.Dedicated to Carl-Eric Fröberg on the occasion of his 75th birthdayThis research was partially supported by NSERC of Canada Grant No. A9236.     

Linear M-estimation with bounded variables

A subproblem in the trust region algorithm for non-linear M-estimation by Ekblom and Madsen is to find the restricted step. It is found by calculating the M-estimator of the linearized model, subject to anL ₂-norm bound on the variables. In this paper it is shown that this subproblem can be solved by applying Hebden-iterations to the minimizer of the Lagrangian function. The new method is compared with an Augmented Lagrange implementation.     

On algebraic multi-level methods for non-symmetric systems - Comparison results

We establish theoretical comparison results for algebraic multi-level methods applied to non-singular non-symmetric M-matrices. We consider two types of multi-level approximate block factorizations or AMG methods, the AMLI and the MAMLI method. We compare the spectral radii of the iteration matrices of these methods. This comparison shows, that the spectral radius of the MAMLI method is less than or equal to the spectral radius of the AMLI method. Moreover, we establish how the quality of the approximations in the block factorization effects the spectral radii of the iteration matrices. We prove comparisons results for different approximations of the fine grid block as well as for the used Schur complement. We also establish a theoretical comparison between the AMG methods and the classical block Jacobi and block Gauss-Seidel methods.     

A tearing-based hybrid parallel banded linear system solver

A new parallel algorithm for the solution of banded linear systems is proposed. The scheme tears the coefficient matrix into several overlapped independent blocks in which the size of the overlap is equal to the system’s bandwidth. A corresponding splitting of the right-hand side is also provided. The resulting independent, and smaller size, linear systems are solved under the constraint that the solutions corresponding to the overlap regions are identical. This results in a linear system whose size is proportional to the sum of the overlap regions which we refer to as the “balance” system. We propose a solution strategy that does not require obtaining this “balance” system explicitly. Once the balance system is solved, retrieving the rest of the solution can be realized with almost perfect parallelism. Our proposed algorithm is a hybrid scheme that combines direct and iterative methods for solving a single banded system of linear equations on parallel architectures. It has broad applications in finite-element analysis, particularly as a parallel solver of banded preconditioners that can be used in conjunction with outer Krylov iterative schemes.     

Eigenvalue bounds for the Schur complement with a pressure convection–diffusion preconditioner in incompressible flow computations

If the stationary Navier–Stokes system or an implicit time discretization of the evolutionary Navier–Stokes system is linearized by a Picard iteration and discretized in space by a mixed finite element method, there arises a saddle point system which may be solved by a Krylov subspace method or an Uzawa type approach. For each of these resolution methods, it is necessary to precondition the Schur complement associated to the saddle point problem in question. In the work at hand, we give upper and lower bounds of the eigenvalues of this Schur complement under the assumption that it is preconditioned by a pressure convection–diffusion matrix.     

Extensions of block-projections methods with relaxation parameters to inconsistent and rank-deficient least-squares problems

There exist many classes of block-projections algorithms for approximating solutions of linear least-squares problems. Generally, these methods generate sequences convergent to the minimal norm least-squares solution only for consistent problems. In the inconsistent case, which usually appears in practice because of some approximations or measurements, these sequences do no longer converge to a least-squares solution or they converge to the minimal norm solution of a “perturbed” problem. In the present paper, we overcome this difficulty by constructing extensions for almost all the above classes of block-projections methods. We prove that the sequences generated with these extensions always converge to a least-squares solution and, with a suitable initial approximation, to the minimal norm solution of the problem. Numerical experiments, described in the last section of the paper, confirm the theoretical results obtained.     

On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.     

BTTB preconditioners for BTTB least squares problems

In this paper, we consider solving the least squares problem min_x‖b-Tx‖₂ by using preconditioned conjugate gradient (PCG) methods, where T is a large rectangular matrix which consists of several square block-Toeplitz-Toeplitz-block (BTTB) matrices and b is a column vector. We propose a BTTB preconditioner to speed up the PCG method and prove that the BTTB preconditioner is a good preconditioner. We then discuss the construction of the BTTB preconditioner. Numerical examples, including image restoration problems, are given to illustrate the efficiency of our BTTB preconditioner. Numerical results show that our BTTB preconditioner is more efficient than the well-known Level-1 and Level-2 circulant preconditioners.     

On the convergence of splittings for semidefinite linear systems

Recently, Lee et al. [Young-ju Lee, Jinbiao Wu, Jinchao Xu, Ludmil Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl. 28 (2006) 634-641] introduce new criteria for the semi-convergence of general iterative methods for semidefinite linear systems based on matrix splitting. The new conditions generalize the classical notion of P-regularity introduced by Keller [H.B. Keller, On the solution of singular and semidefinite linear systems by iterations, SIAM J. Numer. Anal. 2 (1965) 281-290]. In view of their results, we consider here stipulations on a splitting A=M-N, which lead to fixed point systems such that, the iterative scheme converges to a weighted Moore-Penrose solution to the system Ax=b. Our results extend the result of Lee et al. to a more general case and we also show that it requires less restrictions on the splittings than Keller’s P-regularity condition to ensure the convergence of iterative scheme.     

Implementation of a modified Marder–Weitzner method for solving nonlinear eigenvalue problems

Our goal is to propose four versions of modified Marder–Weitzner methods and to present the implementation of the new-type methods with incremental unknowns for solving nonlinear eigenvalue problems. By combining with compact schemes and modified Marder–Weitzner methods, six schemes well suited for the calculation of unstable solutions are obtained. We illustrate the efficiency of the new algorithms by using numerical computations and by comparing them with existing methods for some two-dimensional problems.     

Matrix enlarging methods and their application

This paper explores several methods for matrix enlarging, where an enlarged matrixà is constructed from a given matrixA. The methods explored include matrix primitization, stretching and node splitting. Graph interpretations of these methods are provided. Solving linear problems using enlarged matrices yields the answer to the originalAx=b problem.à can exhibit several desirable properties. For example,à can be constructed so that the valence of any row and/or column is smaller than some desired number (≥4). This is beneficial for algorithms that depend on the square of the number of entries of a row or column. Most particularly, matrix enlarging can results in a reduction of the fill-in in theR matrix which occurs during orthogonal factorization as a result of dense rows. Numerical experiments support these conjectures.     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

Incomplete block-matrix factorization iterative methods for convection-diffusion problems

Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems.This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter.In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter =1.The 2 ^nd author's research was sponsored by Control Data Corporation through its PACER fellowship program.The 3 ^rd author's research was supported by the Netherlands organization for scientific research (NWO).On leave from the Institute of Mathematics, Academy of Science, 1090 Sofia, P.O. Box 373, Bulgaria.     

Third-order iterative methods with applications to Hammerstein equations: A unified approach

The geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in [S. Amat, S. Busquier, J.M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1) (2003) 197-205]. This family includes, as particular cases, some of the most famous third-order iterative methods: Chebyshev methods, Halley methods, super-Halley methods, C-methods and Newton-type two-step methods. The aim of the present paper is to analyze the convergence of this family for equations defined between two Banach spaces by using a technique developed in [J.A. Ezquerro, M.A. Hernández, Halley’s method for operators with unbounded second derivative. Appl. Numer. Math. 57(3) (2007) 354-360]. This technique allows us to obtain a general semilocal convergence result for these methods, where the usual conditions on the second derivative are relaxed. On the other hand, the main practical difficulty related to the classical third-order iterative methods is the evaluation of bilinear operators, typically second-order Fréchet derivatives. However, in some cases, the second derivative is easy to evaluate. A clear example is provided by the approximation of Hammerstein equations, where it is diagonal by blocks. We finish the paper by applying our methods to some nonlinear integral equations of this type.     

Solving sparse linear systems by an abs-method that corresponds toLU-decomposition

Abaffy, Broyden, and Spedicato (ABS) have recently proposed a class of direct methods for solving nonsparse linear systems. It is the purpose of this paper to demonstrate that with proper choice of parameters, ABS methods exploit sparsity in a natural way. In particular, we study the choice of parameters which corresponds to an LU-decomposition of the coefficient matrix. In many cases, the fill-in, represented by the nonzero elements of the deflection matrix, uses less storage than the corresponding fill-in of an explicit LU factorization. Hence the above can be a viable and simple method for solving sparse linear systems. A simple reordering scheme for the coefficient matrix is also proposed for the purpose of reducing fill-in of the deflection matrices.