On block minimal residual methods

In the present work, we give some new results for block minimal residual methods when applied to multiple linear systems. Using the Schur complement, we develop new expressions for the approximation obtained, for the corresponding residual and for the Frobenius residual norm. These results could be used to derive new convergence properties for the block minimal residual methods.     

Conjugate gradient and minimal residual methods for solving symmetric indefinite systems

Norm-minimizing-type methods for solving large sparse linear systems with symmetric and indefinite coefficient matrices are considered. The Krylov subspace can be generated by either the Lanczos approach, such as the methods MINRES, GMRES and QMR, or by a conjugate-gradient approach. Here, we propose an algorithm based on the latter approach. Some relations among the search directions and the residuals, and how the search directions are related to the Krylov subspace are investigated. Numerical experiments are reported to verify the convergence properties.     

A numerical comparison of two minimal residual methods for linear polynomials in unitary matrices

Two minimal residual methods for solving linear systems of the form (αU + βI)x = b, where U is a unitary matrix, are compared numerically. The first method uses conventional Krylov subspaces, while the second involves generalized Krylov subspaces. Experiments favor the second method if |α| > |β|. Moreover, the greater the ratio |α|/|β|, the higher the superiority of the second method.     

Analysis of the convergence of the minimal and the orthogonal residual methods

We consider two Krylov subspace methods for solving linear systems, which are the minimal residual method and the orthogonal residual method. These two methods are studied without referring to any particular implementations. By using the Petrov–Galerkin condition, we describe the residual norms of these two methods in terms of Krylov vectors, and the relationship between there two norms. We define the Ritz singular values, and prove that the convergence of these two methods is governed by the convergence of the Ritz singular values. AMS subject classification 65F10     

A fast minimal residual algorithm for shifted unitary matrices

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.     

On the local convergence of inexact Newton-type methods under residual control-type conditions

A local convergence analysis of inexact Newton-type methods using a new type of residual control was recently presented by C. Li and W. Shen. Here, we introduce the center-Hölder condition on the operator involved, and use it in combination with the Hölder condition to provide a new local convergence analysis with the following advantages: larger radius of convergence, and tighter error bounds on the distances involved. These results are obtained under the same hypotheses and computational cost. Numerical examples further validating the theoretical results are also provided in this study.     

Groups with minimal conditions

The author establishes a number of new propositions asserting that groups with various minimal conditions are Chernikov under some additional extremely weak finiteness conditions. These propositions include a great many known theorems on groups with minimal conditions.     

Penalty methods for minimal surfaces with obstacles

Summary In this paper, we discuss an approach to the obstacle problem for minimal boundaries via penalty techniques. After investigating some classes of penalized problems, a general method is introduced, based on the minimization of a suitable functional containing an extra term related to the mean curvature of the given obstacle.     

A minimal residual method for linear polynomials in unitary matrices

A minimal residual method, called MINRES-N2, that is based on the use of unconventional Krylov subspaces was previously proposed by the authors for solving a system of linear equations Ax = b with a normal coefficient matrix whose spectrum belongs to an algebraic second-degree curve Γ. However, the computational scheme of this method does not cover matrices of the form A = αU + βI, where U is an arbitrary unitary matrix; for such matrices, Γ is a circle. Systems of this type are repeatedly solved when the eigenvectors of a unitary matrix are calculated by inverse iteration. In this paper, a modification of MINRES-N2 suitable for linear polynomials in unitary matrices is proposed. Numerical results are presented demonstrating the significant superiority of the modified method over GMRES as applied to systems of this class.     

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     

The convergence rate of the minimal residual method for the Stokes problem

Summary. Discretisation of the classical Stokes problem gives rise to symmetric indefinite matrices with eigenvalues which, in a precise way, are not symmetric about the origin, but which do depend on a mesh size parameter. Convergence estimates for the Conjugate Residual or Minimum Residual iterative solution of such systems are given by best minimax polynomial approximations on an inclusion set for the eigenvalues. In this paper, an analytic convergence estimate for such problems is given in terms of an asymptotically small mesh size parameter. Received November 16, 1993 / Revised version received August 2, 1994     

A Berry-Esseen theorem for hypergeometric probabilities under minimal conditions

In this paper, we consider simple random sampling without replacement from a dichotomous finite population and derive a necessary and sufficient condition on the finite population parameters for a valid large sample Normal approximation to Hypergeometric probabilities. We then obtain lower and upper bounds on the difference between the Normal and the Hypergeometric distributions solely under this necessary and sufficient condition.

     

Convergence conditions for secant-type methods

We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.     

A minimal stabilisation procedure for mixed finite element methods

Summary. Stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods. Stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties. It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed methods, that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem. To do so, we rely on the fact that some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present two classes of stabilisations for the inf-sup condition in mixed problems. We present many examples, most arising from the discretisation of flow problems. Section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems. Received August 9, 1999 / Revised version received May 19, 2000 / Published online March 20, 2001     

Sufficiency conditions with minimal regularity assumptions

The aim of this article is to provide second-order sufficiency criteria that extend known ones in [6] and [7] to the case where the control candidate and some of the data are merely essentially bounded, and/or the control setU is any convex subset of ^m . In the classical setting, where a restriction on the velocity is imposed , it is shown that whenU is compact the known strengthened Weierstrass condition is equivalent to the Weierstrass condition with strict inequality.The research was supported by an NSERC Grant and by GNAFA of CNR, which are gratefully acknowledged.     

Conjugate residual methods for almost symmetric linear systems

This paper concerns the use of conjugate residual methods for the solution of nonsymmetric linear systems arising in applications to differential equations. We focus on an application derived from a seismic inverse problem. The linear system is a small perturbation to a symmetric positive-definite system, the nonsymmetries arising from discretization errors in the solution of certain boundary-value problems. We state and prove a new error bound for a class of generalized conjugate residual methods; we show that, in some cases, the perturbed symmetric problem can be solved with an error bound similar to the one for the conjugate residual method applied to the symmetric problem. We also discuss several applications for special distributions of eigenvalues.This work was supported in part by the National Science Foundation, Grants DMS-84-03148 and DCR-81-16779, and by the Office of Naval Research, Contract N00014-85-K-0725.     

Minimal residual methods for large scale Lyapunov equations

Projection methods have emerged as competitive techniques for solving large scale matrix Lyapunov equations. We explore the numerical solution of this class of linear matrix equations when a Minimal Residual (MR) condition is used during the projection step. We derive both a new direct method, and a preconditioned operator-oriented iterative solver based on CGLS, for solving the projected reduced least squares problem. Numerical experiments with benchmark problems show the effectiveness of an MR approach over a Galerkin procedure using the same approximation space.     

Weighted residual classification of approximate methods

A rational classification of the different approximating methods which points out their fundamental differences is presented here. The classification is based on the weighted residual formulation and the type of approximating and weighting functions required.Laplace's equation is used throughout the paper to illustrate the different techniques and to permit critical comparisons, but the conclusions are equally applicable to any weighted residual formulation. A fluid flow application is presented and solved numerically using three different approximate techniques, each of them belonging to one of the three groups into which each of the methods have been classified.     

Order conditions for partitioned Runge-Kutta methods

We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations.