A discrete collocation method for Symm's integral equation on curves with corners

This paper is devoted to the approximate solution of the classical first-kind boundary integral equation with logarithmic kernel (Symm's equation) on a closed polygonal boundary in ℝ². We propose a fully discrete method with a trial space of trigonometric polynomials, combined with a trapezoidal rule approximation of the integrals. Before discretization the equation is transformed using a nonlinear (mesh grading) parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions. The convergence results are illustrated with some numerical examples.     

On the mini-computer solution of large systems of linear equations arising from the finite element method

The efficiency of the application of mini-computers to the solution of many problems by means of the finite element method is well recognized, and the importance of the percentage of time devoted to the solution of the resultant systems of linear equations is also noteworthy.Here, we present a comparative study of six programs of the solution of systems of linear equations, implemented in a 64-Kbyte mini-computer (HP1000F). The methods and algorithms on which the programs are based are described.Then, six finite element problems are presented with various degrees of discretization, giving rise to 19 different cases. In each case computation times, the number of disk input-output transfers and the number of arithmetic operations carried out are compared. It is concluded that the programs based on algorithms of variable bandwidth are always superior to those of constant bandwidth. In some cases, the optimum results are obtained with the algorithm of hypermatrices.     

An algorithm for the solution of ill-posed linear systems arising from the discretization of Fredholm integral equation of the first kind

An algorithm for obtaining approximate solutions of ill-posed systems of linear equations arising from the discretization of Fredholm integral equation of the first kind is described. The ill-posed system is first replaced by an equivalent consistent system of linear equations. The method calculates the minimum length least squares solution of the consistent system. Starting from rank = 1 of the consistent system, the rank is increased by one in succession and a new solution is calculated. This is repeated until a certain simple criterion is satisfied. Linear programming techniques are used for which successive solutions are the basic solutions in the successive simplex tableaux. The algorithm is numerically stable. Numerical results show that this method compares favorably with other direct methods.     

A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains

The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.     

Fundamental solution of the stationary Dirac equation with a linear potential

Theoretical and Mathematical Physics - We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier...     

Preconditioning and iterative solution of symmetric indefinite linear systems arising from interior point methods for linear programming

We propose to compute the search direction at each interior-point iteration for a linear program via a reduced augmented system that typically has a much smaller dimension than the original augmented system. This reduced system is potentially less susceptible to the ill-conditioning effect of the elements in the (1,1) block of the augmented matrix. A preconditioner is then designed by approximating the block structure of the inverse of the transformed matrix to further improve the spectral properties of the transformed system. The resulting preconditioned system is likely to become better conditioned toward the end of the interior-point algorithm. Capitalizing on the special spectral properties of the transformed matrix, we further proposed a two-phase iterative algorithm that starts by solving the normal equations with PCG in each IPM iteration, and then switches to solve the preconditioned reduced augmented system with symmetric quasi-minimal residual (SQMR) method when it is advantageous to do so. The experimental results have demonstrated that our proposed method is competitive with direct methods in solving large-scale LP problems and a set of highly degenerate LP problems. Research supported in parts by NUS Research Grant R146-000-076-112 and SMA IUP Research Grant.     

On the solution of linear equation/inequality systems

A pivotal algebra algorithm is given and finite convergence shown for finding a vector which satisfies an arbitrary system of linear equations and/or inequalities. A modified form of the algorithm, obtained by introducing a redundant equation, is then shown to be a way to describe phase I of the simplex method without reference to artificial variables or an artificial objective function.The hypothesis is introduced that in each pivot stage each row of the tableau has equal probability of being chosen as the pivot row. Under this assumption the expected value of the ratio of the number of pivot stages to the number of rows should grow with the natural Log of the number of rows.Use of the algorithm in proving theorems of the alternative is indicated.This paper was presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.     

Projected equation methods for approximate solution of large linear systems

We consider linear systems of equations and solution approximations derived by projection on a low-dimensional subspace. We propose stochastic iterative algorithms, based on simulation, which converge to the approximate solution and are suitable for very large-dimensional problems. The algorithms are extensions of recent approximate dynamic programming methods, known as temporal difference methods, which solve a projected form of Bellman’s equation by using simulation-based approximations to this equation, or by using a projected value iteration method.     

Efficient and stable solution of structured Hessenberg linear systems arising from difference equations

This paper is concerned with the efficient solution of (block) Hessenberg linear systems whose coefficient matrix is a Toeplitz matrix in (block) Hessenberg form plus a band matrix. Such problems arise, for instance, when we apply a computational scheme based on the use of difference equations for the computation of many significant special functions and quantities occurring in engineering and physics. We present a divide‐and‐conquer algorithm that combines some recent techniques for the numerical treatment of structured Hessenberg linear systems. Our approach is computationally efficient and, moreover, in many practical cases it can be shown to be componentwise stable. Copyright © 2000 John Wiley & Sons, Ltd.     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

A four-step method for the numerical solution of the Schrödinger equation

A new four-step exponentially-fitted method is developed in this paper. The expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue Schrödinger equation (i.e., equivalent to positive energy).     

Block-circulant preconditioners for systems arising from discretization of the three-dimensional convection–diffusion equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection–diffusion model problem. This system is typically nonsymmetric. The GMRES method with the Strang block-circulant preconditioner is proposed for solving this linear system. We show that our preconditioners are invertible and study the spectra of the preconditioned matrices. Numerical results are reported to illustrate the effectiveness of our methods.     

Iterative solution of large systems of normal equations arising from the fitting of surfaces to irregularely spaced data

Summary The Gauss-Seidel iteration process is specialised to fit unique surfaces in the least square sense to irregularely spaced data. The Fortran IV version of the specialised algorithm, called iterative one parameter least square process, is described for immediate computer use. The new algorithm requires so few storage locations that a 32 K computer memory without any peripheric device is sufficient to store 3000 data points and to solve a system of normal equations of order up to 1500. The surface fitting is illustrated by a numerical example.

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Zusammenfassung Das Gauss-Seidel Iterationsverfahren wird derart spezialisiert, daß eine eindeutige Fläche im Sinne der kleinsten Quadrate an unregelmäßig angeordnete Daten angepasst wird. Das Fortran IV Programm des spezialisierten Algorithmus, welchen wir als Verfahren der kleinsten Quadrate mit einem Parameter bezeichnen, wird für unmittelbaren Gebrauch beschrieben. Der neue Algorithmus benötigt so wenig Speicher, daß ein Computer mit 32 K ohne periphere Einrichtungen genügt, um 3000 Werte (x-, y-, z-koordinaten) aufzunehmen und ein System von bis zu 1500 Normalgleichungen zu läsen. Die Flächenanpassung wird an einem Beispiel veranschaulicht.

     

Iterative variants of the Nyström method for the numerical solution of integral equations

Some iterative variants of the Nyström method for the numerical solution of linear and nonlinear integral equations are introduced and discussed. Numerical examples are given; some are for integral equations with singular kernel functions.     

High order implicit collocation method for the solution of two‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving the two dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives of linear hyperbolic equation and collocation method for the time component. The resulted method is unconditionally stable and solves the two‐dimensional linear hyperbolic equation with high accuracy. In this technique, the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Generalized method of conjugate gradients for the solution of linear systems

We obtain convergence criteria for the generalized method of conjugate gradients for solving systems of linear algebraic equations; they imply the convergence of the method in a finite number of steps. The theorem proved in the paper allows at the same time to consider the known variants of the generalized method of conjugate gradients, as well as to devise new modifications for the convergence of this method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 80, pp. 181–188, 1978.