Preconditioning projection methods for solving algebraic linear systems

Numerical experiments have shown that projection methods are robust for solving the problem of finding a point satisfying a linear system of n variables and m equations; however, their qualities of convergence depend on certain parameters: an n × n symmetric positive definite matrix M, and a vector u with m components. We are concerned here with the choice of M. Through a link with Conjugate Gradient methods we determine an expedient M. Preliminary numerical results on a hard 3D partial differential equation are highly promising. We solve a discretized system that could not be solved by conventional methods. We also give hints on how to adapt our findings to the solution of a linear system of inequalities. This is the first stage of a forthcoming research. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical methods for solving linear least squares problems

A common problem in a Computer Laboratory is that of finding linear least squares solutions. These problems arise in a variety of areas and in a variety of contexts. Linear least squares problems are particularly difficult to solve because they frequently involve large quantities of data, and they are ill-conditioned by their very nature. In this paper, we shall consider stable numerical methods for handling these problems. Our basic tool is a matrix decomposition based on orthogonal Householder transformations.Reproduction in Whole or in Part is permitted for any Purpose of the United States government. This report was supported in part by Office of Naval Research Contract Nonr-225(37) (NR 044-11) at Stanford University.     

Synthesis scheme for hybrid methods of solving boundary problems

A scheme is proposed for the development of a numericoanalytic boundary problem technique that can synthesize the mesh-projective method with the integral transform method to yield an effective hybrid method for the determination of the temperature field in a body from a moving heat source with given dependences of the thermophysical parameters on temperature.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 19–22, 1987.     

Computational methods for solving two-stage stochastic linear programming problems

Approximating a given continuous probability distribution of the data of a linear program by a discrete one yields solution methods for the stochastic linear programming problem with complete fixed recourse. For a procedure along the lines of [8], the reduction of the computational amount of work compared to the usual revised simplex method is figured out. Furthermore, an alternative method is proposed, where by refining particular discrete distributions the optimal value is approximated.

---

Zusammenfassung Für das zweistufige Modell der stochastischen linearen Programmierung mit vollständiger Kompensation werden Verfahren untersucht, die sich aus der Annäherung einer gegebenen stetigen Wahrscheinlichkeitsverteilung der Daten durch endlich diskrete Verteilungen ergeben. Beim Vorgehen nach [8] wird die Reduktion des Rechenaufwandes im Vergleich zur üblichen revidierten Simplexmethode ermittelt. Als Alternative wird ein Verfahren vorgeschlagen, in dem durch sukzessive Verfeinerung speziell gewählter diskreter Verteilungen der Optimalwert monoton angenähert wird.

---

---

Herrn Prof. Dr. Eduard Stiefel in kollegialer Verbundenheit     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000     

Nonlinear relaxation methods for solving symmetric linear complementarity problems

A family of iterative algorithms is presented for the solution of the symmetric linear complementarity problem,

Preconditioned GAOR methods for solving weighted linear least squares problems

In this paper, we present the preconditioned generalized accelerated overrelaxation (GAOR) method for solving linear systems based on a class of weighted linear least square problems. Two kinds of preconditioning are proposed, and each one contains three preconditioners. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the convergence rate of the preconditioned GAOR methods is indeed better than the rate of the original method, whenever the original method is convergent. Finally, a numerical example is presented in order to confirm these theoretical results.     

On a modification of minimal iteration methods for solving systems of linear algebraic equations

Modifications of certain minimal iteration methods for solving systems of linear algebraic equations are proposed and examined. The modified methods are shown to be superior to the original versions with respect to the round-off error accumulation, which makes them applicable to solving ill-conditioned problems. Numerical results demonstrating the efficiency of the proposed modifications are given.     

Iterative methods for solving linear equations

This paper presents some of the original versions of the conjugate-gradient method for solving a system of linear equations of the formAx=k.This paper originally appeared as NAML Report No. 52-9, 1951. Its preparation was supported in part by the Office of Naval Research.     

An iterative method for solving systems of linear algebraic equations

An iterative solution process for systems of linear algebraic equations is proposed. It converges starting from any initial approximation and theoretically does not require preliminary transformation of the input data.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 64–68, 1985.     

An approach to solving multiparameter algebraic problems

An approach to solving the following multiparameter algebraic problems is suggested: (1) spectral problems for singular matrices polynomially dependent on q≥2 spectral parameters, namely: the separation of the regular and singular parts of the spectrum, the computation of the discrete spectrum, and the construction of a basis that is free of a finite regular spectrum of the null-space of polynomial solutions of a multiparameter polynomial matrix; (2) the execution of certain operations over scalar and matrix multiparameter polynomials, including the computation of the GCD of a sequence of polynomials, the division of polynomials by their common divisor, and the computation of relative factorizations of polynomials; (3) the solution of systems of linear algebraic equations with multiparameter polynomial matrices and the construction of inverse and pseudoinverse matrices. This approach is based on the so-called ΔW-q factorizations of polynomial q-parameter matrices and extends the method for solving problems for one- and two-parameter polynomial matrices considered in [1–3] to an arbitrary q≥2. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 191–246. Translated by V. N. Kublanovskaya.     

Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important to select fast, robust and reliable methods for their solution, also in the case where fast modern computers are available. Since the coefficient matrices of the systems are normally sparse (i.e. most of their elements are zeros), the first requirement is to efficiently exploit the sparsity. However, this is normally not sufficient when the systems are very large. The computation of preconditioners based on approximate LU-factorizations and their use in the efforts to increase further the efficiency of the calculations will be discussed in this paper. Computational experiments based on comprehensive comparisons of many numerical results that are obtained by using ten well-known methods for solving systems of linear algebraic equations (the direct Gaussian elimination and nine iterative methods) will be reported. Most of the considered methods are preconditioned Krylov subspace algorithms.     

An algebraic foundation for factoring linear boundary problems

Motivated by boundary problems for linear differential equations, we define an abstract boundary problem as a pair consisting of a surjective linear map (“differential operator”) and an orthogonally closed subspace of the dual space (“boundary conditions”). Defining the composition of boundary problems corresponding to their Green’s operators in reverse order, we characterize and construct all factorizations of a boundary problem from a given factorization of the defining operator. For the case of ordinary differential equations, the main results can be made algorithmic. We conclude with a factorization of a boundary problem for the wave equation. This work was supported by the Austrian Science Fund (FWF) under the SFB grant F1322.     

Applying mod-k-cuts for solving linear ordering problems

The linear ordering problem consists of finding an ordering of the nodes of the weighted complete digraph on n nodes such that the sum of the weights of the arcs compatible with the ordering is maximized. In this paper, we report about the usefulness of mod-k cuts in a branch-and-cut algorithm for solving linear ordering problems to optimality.      

An efficient discretization scheme for solving ill-posed problems

In this paper, we consider a finite-dimensional approximation scheme combined with Tikhonov regularization for solving ill-posed problems. Error estimates are obtained by an a priori parameter choice strategy and the results show that the amount of discrete information required for solving the problem is far less than the traditional finite-dimensional approach.     

A numerical scheme for solving nonhomogeneous time-dependent problems

A numerical technique for solving time-dependent problems with variable coefficient governed by the heat, convection diffusion, wave, beam and telegraph equations is presented. The Sinc–Galerkin method is applied to construct the numerical solution. The method is tested on three problems and comparisons are made with the exact solutions. The numerical results demonstrate the reliability and efficiency of using the Sinc–Galerkin method to solve such problems.     

A numerical scheme for solving nonhomogeneous time-dependent problems

A numerical technique for solving time-dependent problems with variable coefficient governed by the heat, convection diffusion, wave, beam and telegraph equations is presented. The Sinc–Galerkin method is applied to construct the numerical solution. The method is tested on three problems and comparisons are made with the exact solutions. The numerical results demonstrate the reliability and efficiency of using the Sinc–Galerkin method to solve such problems. Received: January 18, 2005     

On modification of certain methods of the conjugate direction type for solving systems of linear algebraic equations

A modification of certain well-known methods of the conjugate direction type is proposed and examined. The modified methods are more stable with respect to the accumulation of round-off errors. Moreover, these methods are applicable for solving ill-conditioned systems of linear algebraic equations that, in particular, arise as approximations of ill-posed problems. Numerical results illustrating the advantages of the proposed modification are presented.