Showing papers in "Numerische Mathematik in 1982"
TL;DR: An error bound is given that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation) and for thePDE in Lagrangian form.
Abstract: This paper deals with an algorithm for the solution of diffusion and/or convection equations where we mixed the method of characteristics and the finite element method. Globally it looks like one does one step of transport plus one step of diffusion (or projection) but the mathematics show that it is also an implicit time discretization of thePDE in Lagrangian form. We give an error bound (h+Δt+h×h/Δt in the interesting case) that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation).
697 citations
TL;DR: In this paper, a comparative study of nonlinear optimization algorithms is presented, and it is shown that quadratic approximation methods, characterized by solving a sequence of quadratically subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present.
Abstract: The paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms. This study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present. The purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail. In Part 1, the exactL 1-penalty function of Han and Powell is replaced by a differentiable augmented Lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution. In Part 2, the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency, and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems.
247 citations
TL;DR: In this article, a method for computing the exact rational solution to a regular system of linear equations with integer coefficients is described, which can be implemented in time O(n 3(logn)2 ) for matrices A and b with entries of bounded size and dimensions n×n and n×1.
Abstract: A method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA for some primep; (ii) using successive refinements to compute an integer vector $$\bar x$$ such that $$A\bar x \equiv b$$ (modp m ) for a suitably large integerm; and (iii) deducing the rational solutionx from thep-adic approximation $$\bar x$$ . For matricesA andb with entries of bounded size and dimensionsn×n andn×1, this method can be implemented in timeO(n 3(logn)2) which is better than methods previously used.
196 citations
TL;DR: A minimization method based on the idea of partitioned updating of the Hessian matrix in the case where the objective function can be decomposed in a sum of convex “element” functions is presented.
Abstract: This paper presents a minimization method based on the idea of partitioned updating of the Hessian matrix in the case where the objective function can be decomposed in a sum of convex "element" functions. This situation occurs in a large class of practical problems including nonlinear finite elements calculations. Some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented.
160 citations
TL;DR: In this article, a method for the construction of a set of data of interpolation in several variables is given, which are either function values or directional derivatives values, give rise to a space of polynomials, in such a way that unisolvence is guaranteed.
Abstract: A method for the construction of a set of data of interpolation in several variables is given. The resulting data, which are either function values or directional derivatives values, give rise to a space of polynomials, in such a way that unisolvence is guaranteed. The interpolating polynomial is calculated using a procedure which generalizes the Newton divided differences formula for a single variable.
160 citations
TL;DR: In this paper, the authors consider the convergence properties of inexact partitioned quasi-Newton algorithms for the solution of certain non-linear equations and, in particular, the optimization of partially separable objective functions.
Abstract: This paper considers local convergence properties of inexact partitioned quasi-Newton algorithms for the solution of certain non-linear equations and, in particular, the optimization of partially separable objective functions. Using the bounded deterioration principle, one obtains local and linear convergence, which impliesQ-superlinear convergence under the usual conditions on the quasi-Newton updates. For the optimization case, these conditions are shown to be satisfied by any sequence of updates within the convex Broyden class, even if some Hessians are singular at the minimizer. Finally, local andQ-superlinear convergence is established for an inexact partitioned variable metric method under mild assumptions on the initial Hessian approximations.
148 citations
TL;DR: In this article, the authors present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction and generalize all convergence results of those algorithms which are based on the vectorial norm hypothesis.
Abstract: Nous presentons dans cet article des resultats de convergence des algorithmes asynchrones bases essentiellement sur la notion classique de contraction.
Nous generalisons, en particulier, tous les resultats de convergence de ces algorithmes qui font l'hypothese de contraction en norme vectorielle qui recemment a ete tres souvant utilisee.
Par ailleurs, l'hypothese de contraction en norme vectorielle peut se trouver difficile, voire impossible a verifier pour certains problemes qui peuvent etre cependant abordes dans le cadre de la contraction classique que nous adoptons.
In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction.
We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently.
Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.
144 citations
TL;DR: An efficient algorithm for the solution of linear equations arising in a finite element method for the Dirichlet problem and the cost is proportional to N2log2N (N=1/h) where the cost of solving the capacitance matrix equations is N log2N on regular grids and N3/2log 2N on irregular ones.
Abstract: An efficient algorithm for the solution of linear equations arising in a finite element method for the Dirichlet problem is given. The cost of the algorithm is proportional toN 2log2 N (N=1/h) where the cost of solving the capacitance matrix equations isNlog2 N on regular grids andN 3/2log2 N on irregular ones.
106 citations
TL;DR: In this paper, an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns is described, and the algorithm is shown to be efficient.
Abstract: This paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns.
102 citations
TL;DR: In this paper, the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems is discussed, and three possibilities are investigated, their O(h2)-convergence established and illustrated by numerical examples.
Abstract: We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x?y?)?=f(x,y), y(0)=A, y(1)=B, 0
93 citations
TL;DR: In this paper, a posteriori estimation of the space discretization error in the finite element method of lines solution of parabolic equations is analyzed for time-independent space meshes, and the effectiveness of the estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size.
Abstract: In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.
TL;DR: In this paper, a comparative study of nonlinear optimization algorithms was conducted and it was shown that quadratic approximation methods, which are characterized by solving a sequence of subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present.
Abstract: SummaryThe paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms. This study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present. The purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail. In Part 1, the exactL1-penalty function of Han and Powell is replaced by a differentiable augmented Lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution. In Part 2, the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency, and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems.
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TL;DR: In this paper, it was shown that the bisection algorithm remains optimal even if general information on f is allowed by permitting the adaptive evaluations of n arbitrary linear functionals, i.e., if f is infinitely many times differentiable on [a, b] and has exactly one simple zero.
Abstract: We seek an approximation to a zero of a continuous functionf:[a,b]?? such thatf(a)?0 andf(b)?0. It is known that the bisection algorithm makes optimal use ofn function evaluations, i.e., yields the minimal error which is (b?a)/2 n+1, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals. They conjectured [5, p. 170] that the bisection algorithm remains optimal even if these general evaluations are permitted. This paper affirmatively proves this conjecture. In fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on [a, b] and has exactly one simple zero.
TL;DR: In this paper, the authors show that the scaled stability region of a method, satisfying some reasonable conditions, cannot be properly contained in the scaled stabilizer region of another method, for general nonlinear ordinary differential systems, for systems obtained from parabolic problems, and for hyperbolic problems.
Abstract: Stability regions of explicit "linear" time discretization methods for solving initial value problems are treated. If an integration method needsm function evaluations per time step, then we scale the stability region by dividing bym. We show that the scaled stability region of a method, satisfying some reasonable conditions, cannot be properly contained in the scaled stability region of another method. Bounds for the size of the stability regions for three different purposes are then given: for "general" nonlinear ordinary differential systems, for systems obtained from parabolic problems and for systems obtained from hyperbolic problems. We also show how these bounds can be approached by high order methods.
TL;DR: In this paper, a finite element approximation for the stationary Stokes and Navier Stokes equations in a bounded domain in?3 is introduced, which can be used tetahedrons or cubes.
Abstract: We introduce some new families of finite element approximation for the stationary Stokes and Navier Stokes equations in a bounded domain in ?3. These elements can used tetahedrons or cubes. The approximation satisfie exactly the incompressibility condition.
TL;DR: In this article, the use of the condition number of a problem, as defined by Rice in 1966, is discussed and lower and upper bounds for condition numbers are derived for the eigenvalue, eigenvector, and linear least squares problems.
Abstract: In this paper the use of the condition number of a problem, as defined by Rice in 1966, is discussed. For the eigenvalue, eigenvector, and linear least squares problems either condition numbers according to various norms are determined or lower and upper bounds for them are derived.
TL;DR: The analysis of a posteriori estimates of the space discretization error presented in a previous paper for time-independent space meshes is extended and a procedure based upon this theory is presented for the adaptive construction of time-dependent meshes.
Abstract: : We extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper (3) for time-independent space meshes. In the context of the model problem studied there, results are given relating the effectivity of the error estimator to properties of the solution, space, meshes, and manner in which the meshes change. A procedure based upon this theory is presented for the adaptive construction of time-dependent meshes. The results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems. (Author)
TL;DR: In this paper, a special case of a generalization of the Richardson extrapolation process is considered, and its complete solution is given in closed form, using this, an algorithm for implementing the extrapolation is devised.
Abstract: A special case of a generalization of the Richardson extrapolation process is considered, and its complete solution is given in closed form. Using this, an algorithm for implementing the extrapolation is devised. It is shown that this algorithm needs a very small amount of arithmetic operations and very little storage. Convergence and stability properties for some cases are also considered.
TL;DR: In this paper, a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist is presented, and Ukita's theorem is extended to characterize the class of matrices with tridagonal inverses.
Abstract: This paper presents a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist Ukita's theorem is extended to characterize the class of matrices that have tridiagonal inverses
TL;DR: It is proved that optimal error estimates hold inL2,H1 andL∞, provided that certain relationships hold between the frequency, mesh size and outer radius.
Abstract: The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL 2,H 1 andL ? , provided that certain relationships hold between the frequency, mesh size and outer radius.
TL;DR: In this article, the Von Karman equations were approximated by the mixed finite element scheme of Miyoshi and the solution arcs at a neighbourhood of the first eigenvalue of the linearized problem were followed by a continuation method.
Abstract: The purpose of this paper is to study the approximation of the Von Karman equations by the mixed finite element scheme of Miyoshi and to follow the solutions arcs at a neighbourhood of the first eigenvalue of the linearized problem. This last problem is solved by a continuation method.
TL;DR: The treatment of general linear discretization methods for initial value problems is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general ‘method-free’ statements are again obtained.
Abstract: This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general `method-free' statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.
TL;DR: The theory of general Runge-Kutta methods for Volterra integrodifferential equations is developed and the local order is characterized in terms of the coefficients of the method.
Abstract: The present paper develops the theory of general Runge-Kutta methods for Volterra integrodifferential equations. The local order is characterized in terms of the coefficients of the method. We investigate the global convergence of mixed and extended Runge-Kutta methods and give results on asymptotic error expansions. In a further section we construct examples of methods up to order 4.
TL;DR: An algorithm for the stable evaluation of the weights of interpolatory quadratures with prescribed simple or multiple knots is presented and the results indicate that the accuracy of the method presented is much higher than that achieved by solving the Vandermonde system directly.
Abstract: We present an algorithm for the stable evaluation of the weights of interpolatory quadratures with prescribed simple or multiple knots and compare its performance with that obtained by directly solving, using the method proposed by Galimberti and Pereyra [1], the confluent Vandermonde system of linear equations satisfied by the weights. Elsewhere Kautsky [5] has described a property which relates the weights of interpolatory quadratures to the principal vectors of certain non-derogatory matrices. Using this property one can get the information about the weight functionw of the approximated integral implicitly through the (symmetric tridiagonal) Jacobi matrix associated with the polynomials orthonormal with respect tow. The results indicate that the accuracy of the method presented is much higher than that achieved by solving the Vandermonde system directly.
TL;DR: In this article, error estimates for the infinite element method used in the approximation of solutions of interface problems are derived, and approximations of stress intensity factors are given, and an approximate solution which has a singularity at the singular point can be also obtained.
Abstract: In this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems. Furthermore, approximations of stress intensity factors are given. The infinite element method may be considered as a certain scheme of mesh refinement, but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently, and that an approximate solution which has a singularity at the singular point can be also obtained.
TL;DR: In this paper, Lagrange interpolation involving trigonometric polynomials of degree n in one space direction, and piecewise polynomial over a finite element decomposition of mesh size n in the other space direction is considered.
Abstract: We consider Lagrange interpolation involving trigonometric polynomials of degree ?N in one space direction, and piecewise polynomials over a finite element decomposition of mesh size ?h in the other space directions. We provide error estimates in non-isotropic Sobolev norms, depending additively on the parametersh andN. An application to the convergence analysis of an elliptic problem, with some numerical results, is given.
TL;DR: In this article, the authors studied the existence, uniqueness and convergence of discrete cubic splines which interpolate to a given function at one interior point of each mesh interval, including continuous periodic cubic spline.
Abstract: In the present paper we study the existence, uniqueness and convergence of discrete cubic spline which interpolate to a given function at one interior point of each mesh interval. Our result in particular, includes the interpolation problems concerning continuous periodic cubic splines and discrete cubic splines with boundary conditions considered respectively in Meir and Sharma (1968) and Lyche (1976) for the case of equidistant knots.
TL;DR: In this paper, the authors approximate the solution of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration, which ensures the same order of convergence as usual conforming finite element methods.
Abstract: In this paper, we approximate the solution of a problem of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration. Compatibility conditions which have to be satisfied at the mesh points are given. These conditions ensure for this method the same order of convergence as usual conforming finite element methods.
TL;DR: Fast Givens rotations with half as many multiplications for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily to offer distinct advantages for sparse matrices.
Abstract: Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.
TL;DR: In this article, a priori and a posteriori error bounds for the secant method for solving non-linear equations in Banach spaces were given, and the numerical stability of this method was also investigated.
Abstract: Sharp a priori and a posteriori error bounds are given for the secant method for solving non-linear equations in Banach spaces. The numerical stability of this method is also investigated. The stability results are analogous to those obtained by Lancaster for Newton's method.