Showing papers in "Computing in 1992"
TL;DR: The Sandwich algorithm approximates a convex function of one variable over an interval by evaluating the function and its derivative at a sequence of points and it is shown that the global approximation error withn evaluation points decreases by the order ofO(1/n2), which is optimal.
Abstract: The Sandwich algorithm approximates a convex function of one variable over an interval by evaluating the function and its derivative at a sequence of points. The connection of the obtained points is a piecewise linear upper approximation, and the tangents yield a piecewise linear lower approximation. Similarly, a planar convex figure can be approximated by convex polygons.
120 citations
TL;DR: An algorithm is described that determines for any system of algebraic partial differential equations the number of parameters if the symmetry group is finite, and theNumber of unspecified functions and its arguments if it is infinite.
Abstract: To determine the symmetry group of pointor Lie-symmetries of a differential equation is of great theoretical and practical importance, in particular for determining closed form solutions. There does not seem to exist an algorithm that finds this group in general. However, it is always possible to determine thesize of the symmetry group. In this article an algorithm is described that determines for any system of algebraic partial differential equations the number of parameters if the symmetry group is finite, and the number of unspecified functions and its arguments if it is infinite. To this end the so calleddetermining system is transformed into aninvolutive system by means of a critical-pair/completion algorithm similar like it is applied for computing Grobner bases in polynomial ideal theory. The foundation for obtaining this form is the theory of Riquier and Janet for partial differential equations. The algorithmInvolution System has been implemented in several computer algebra systems as part of the packageSPDE. Various results that have been obtained by applying it are presented as well. If symmetry analysis is considered as part of the more general process of obtaining the best possible information on the solutions of a differential equation, the algorithm described in this article removes the heuristics which is usually involved in making the transition from analytical to numerical methods.
58 citations
TL;DR: A domain splitting scheme is proposed which is locally implicit on slightly overlapping subdomains but propagates the corresponding boundary data by a simple explicit process and can be effectively parallelized.
Abstract: In the parallel implementation of solution methods for parabolic problems one has to find a proper balance between the parallel efficiency of a fully explicit scheme and the need for stability and accuracy which requires some degree of implicitness. As a compromise a domain splitting scheme is proposed which is locally implicit on slightly overlapping subdomains but propagates the corresponding boundary data by a simple explicit process. The analysis of this algorithm shows that it has satisfactory stability and approximation properties and can be effectively parallelized. These theoretical results are confirmed by numerical tests on a transputer system.
37 citations
TL;DR: This paper provides a method using Householder transformations to construct matrices of trigonometric series which leads to multivariate wavelet decompositions.
Abstract: In this paper we provide a method using Householder transformations to construct matrices of trigonometric series which leads to multivariate wavelet decompositions.
36 citations
TL;DR: Duffy's triangular or local polar coordinates in conjunction with tensor product Gaussian quadrature are efficient and reliable for bothh-andp-boundary elements.
Abstract: The numerical integration of all singular surface integrals arising in 3-d boundary element methods is analyzed theoretically and computationally. For all weakly singular integrals arising in BEM, Duffy's triangular or local polar coordinates in conjunction with tensor product Gaussian quadrature are efficient and reliable for bothh-andp-boundary elements. Cauchy- and hypersingular surface integrals are reduced to weakly singular ones by analytic regularization which is done automatically by symbolic manipulation.
34 citations
TL;DR: The convergence of some quasi-Newton methods for solving nonlinear equationAx+g(x)=0 in a domainD⊄Rn, whereA is ann×n matrix andg is a nondifferentiable but Lipschitz continuous operator, is studied.
Abstract: In this paper, we study the convergence of some quasi-Newton methods for solving nonlinear equationAx+g(x)=0 in a domainD⊄R
n
, whereA is ann×n matrix andg is a nondifferentiable but Lipschitz continuous operator. By interval analysis, we give a new convergence theorem of the methods.
28 citations
TL;DR: In this article, a boundary element method is applied to solve the governing Stokes creeping flow equations for an arbitrarily initial shaped fluid region, and it is shown that the viscous sintering problem is well-conditioned from an evolutionary point of view.
Abstract: By viscous sintering it is meant the process of bringing a granular compact to a temperature at which the viscosity of the material becomes low enough for surface tension to cause the particles to deform and coalesce, whereby the material transport can be modelled as a viscous incompressible newtonian volume flow. Here a two-dimensional model is considered. A Boundary Element Method is applied to solve the governing Stokes creeping flow equations for an arbitrarily initial shaped fluid region. In this paper we show that the viscous sintering problem is well-conditioned from an evolutionary point of view. However as boundary value problem at each time step, the problem is ill-conditioned when the contact surfaces of the particles are small, i.e. in the early stages of the coalescence. This is because the curvature of the boundary at those places can be very large. This ill-conditioning is illustrated by an example: the coalescence of two equal circles. This example demonstrates the main evolutionary features of the sintering phenomenon very well. A numerical consequence of this ill-conditioning is that special care has to be taken for distributing and redistributing the nodal points at these boundary parts. Therefore an algorithm for this node redistribution is outlined. Several numerical examples sustain the analysis.
27 citations
TL;DR: A new half-explicit Runge-Kutta method for the numerical integration of differential-algebraic systems of index 2 is constructed and is particularly efficient for the solution of the equations of motion of constrained mechanical systems.
Abstract: A new half-explicit Runge-Kutta method for the numerical integration of differential-algebraic systems of index 2 is constructed. It is particularly efficient for the solution of the equations of motion of constrained mechanical systems. Numerical experiments and comparisons with other codes (DASSL, MEXX) demonstrate the efficiency of the new method.
26 citations
TL;DR: A general nonmonotone trust region method with curvilinear path for unconstrained optimization problem is presented and convergence properties similar to those for the usual trust region methods withCurvil inear path are proved under certain conditions.
Abstract: A general nonmonotone trust region method with curvilinear path for unconstrained optimization problem is presented. Although this method allows the sequence of the objective function values to be nonmonotone, convergence properties similar to those for the usual trust region methods with curvilinear path are proved under certain conditions. Some numerical results are reported which show the superiority of the nonmonotone trust region method with respect to the numbers of gradient evaluations and function evaluations.
25 citations
TL;DR: The optimality of the quadrature rules will be examined and the numerical calculation of singular integrals in the sense of Hadamard by simple integration methods is dealt with.
Abstract: In this paper we deal with the numerical calculation of singular integrals in the sense of Hadamard by simple integration methods. Error estimates and numerical examples are given. The optimality of the quadrature rules will be examined.
21 citations
TL;DR: A method for finding all global minimizers of a real-valued objective function of several variables is presented and an algorithm for interpolation and approximation in multidimensional spaces is developed.
Abstract: A method for finding all global minimizers of a real-valued objective function of several variables is presented For this purpose a problem-oriented type of number is used: the set of real compact intervals The range of the objective function over a rectangular set is estimated by natural interval extension of a suitable modelling function An algorithm for interpolation and approximation in multidimensional spaces is developed This optimization method can be applied successfully to conventionally, eg with real arithmetic, programmed functions
TL;DR: This modified version of the recent divide-and-conquer algorithms of [3] is presented, avoiding the numerical stability problems of the algorithms but preserving their insensitivity to clustering the eigenvalues and the possibility to give a priori upper bounds on their computational cost for any input matrix.
Abstract: We present a practical modification of the recent divide-and-conquer algorithms of [3] for approximating the eigenvalues of a real symmetric tridiagonal matrix. In this modified version, we avoid the numerical stability problems of the algorithms of [3] but preserve their insensitivity to clustering the eigenvalues and the possibility to give a priori upper bounds on their computational cost for any input matrix. We confirm the theoretical effectiveness of our algorithms by numerical experiments.
TL;DR: In this article, the shape preserving C1-interpolation of data sets given on rectangular grids was studied and rational biquadratic splines were derived which are sufficient for the positivity, monotonicity, and S-convexity.
Abstract: This paper is concerned with shape preservingC1-interpolation of data sets given on rectangular grids. Using special rational biquadratic splines, criteria are derived which are sufficient for the positivity, monotonicity, andS-convexity and which, in addition, are satisfied for sufficiently large rationality parameters.
TL;DR: A theory is developed for this class of methods based on known results from group representation theory and algebras of finite groups that if the problem splits into subproblems based on isomorphic subdomains, then the group of symmetries must be commutative.
Abstract: The domain reduction method uses a finite group of symmetries of a system of linear equations arising by discretization of partial differential equations to obtain a decomposition into independent subproblems, which can be solved in parallel. This paper develops a theory for this class of methods based on known results from group representation theory and algebras of finite groups. The main theoretical result is that if the problem splits into subproblems based on isomorphic subdomains, then the group of symmetries must be commutative. General decompositions are then obtained by nesting decompositions based on commutative groups of symmetries.
TL;DR: For smoothing histograms under constraints like convexity or monotonicity, the functionalsK2 andK∞ are proposed which can be considered as extensions of the Schoenberg functional known from data smoothing.
Abstract: For smoothing histograms under constraints like convexity or monotonicity, in this paper the functionalsK
2 andK
∞ are proposed which can be considered as extensions of the Schoenberg functional known from data smoothing When using quadratic splines we are led to structured finite dimensional programming problems Occuring partially separable convex programs can be solved effectively via dualization
TL;DR: A method is given to extract the eigenvalues of standard and generalised eigenvalue problems involving centrosymmetric matrices.
Abstract: A method is given to extract the eigenvalues of standard and generalised eigenvalue problems involving centrosymmetric matrices.
TL;DR: A quasi steady laminar axisymmetric flow of Bingham type under the assumption that the yield stress increases at a rate proportional to the internal power dissipation has been observed in coal-water slurries.
Abstract: We consider a quasi steady laminar axisymmetric flow of Bingham type under the assumption that the yield stress increases at a rate proportional to the internal power dissipation. Such a behaviour has been observed in coal-water slurries. Many different cases can occur (including the appearance of a rigid shell at the boundary of the pipeline), which are analyzed both theoretically and numerically.
TL;DR: This presentation includes a short-term model for the optimal dispatch of thermal units, which is solved by a specific path following method, a daily model for a generation system consisting of Thermal units, pumped storage plants and an energy contract, which can be solved by standard convex quadratic programming algorithms.
Abstract: Some models for the economic dispatch of electric power are introduced and treated by mathematical programming techniques. In particular, our presentation includes (i) a short-term model for the optimal dispatch of thermal units, which is solved by a specific path following method, (ii) a daily model for a generation system consisting of thermal units, pumped storage plants and an energy contract, which can be solved by standard convex quadratic programming algorithms, and (iii) two stochastic programming models for the optimal daily dispatch, which depend on the (unknown) probability distribution of the electric power demand. One of the latter models can be solved efficiently by combining nonparametric estimation procedures and convex programming methods.
TL;DR: A new approach to detect and locate a singularity is suggested, which does not require the changing of the underlying formula and is comprised of two stages which gathers more information about the existence and location of the singular point.
Abstract: Many authors have worked on approaches for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) whose solutions contain one or more singular points within the interval of integration. Their approaches, however, assumed that the user knows in advance that the problem is singular. Hence they introduced new formulas to cope with this difficulty. In this paper, a new approach to detect and locate a singularity is suggested. This approach, which does not require the changing of the underlying formula, is comprised of two stages. The first is a preliminary singularity detection stage. The second stage is the confirmation stage which gathers more information about the existence and location of the singular point. We justify the first state and introduce three different techniques for confirming the existence of a singularity. The numerical results show that our approach is effective.
TL;DR: It is shown that arbitrarily small and smooth perturbation can cause dramatic instabilities by introducing “small” perturbations of the matrix pencil's generalized eigen values atz=∞, leading to large positive finite eigenvalues.
Abstract: It is well-known that linear time-varying high-index DAEs can be very sensitive to parametric perturbations, [1, p. 31]. Stability is also affected, as is known from singular perturbation theory. In this note, we show that arbitrarily small and smooth perturbations can cause dramatic instabilities by introducing “small” perturbations of the matrix pencil's generalized eigenvalues atz=∞, leading to large positive finite eigenvalues. The smaller the perturbation, the larger is the instability of the perturbed problem, in contrast to the ODE case. Some high-index problems can thus be considered as marginally stable, with neighboring problems (usually of lower index) exhibiting severe instabilities.
TL;DR: An extension of a direct method due to Griewank and Reddien is considered for the characterization and computation of double singular points with corank 2 andSingular points which satisfy certain type of symmetry will also be considered.
Abstract: We will consider an extension of a direct method due to Griewank and Reddien for the characterization and computation of double singular points with corank 2. Singular points which satisfy certain type of symmetry will also be considered. The method used will produce an extended system which does not introduce the null vectors as variables, but gives a good idea bout them. Several numerical examples are presented to demonstrate that the method is efficient.
TL;DR: The Lobatto collocation method is modified for efficiently solving linear boundary value problems of differential-algebraic equations with index 1 and the stability and superconvergence of this method are established.
Abstract: The Lobatto collocation method is modified for efficiently solving linear boundary value problems of differential-algebraic equations with index 1. The stability and superconvergence of this method are established. Numerical implementations are discussed and a numerical example is given.
TL;DR: In this article, a two-phase global random search procedure for solving some computationally intractable discrete optimization problems is proposed, where guarantees for quality of random search results are derived from analysis of nonasymptotic order statistics and distribution-free intervals that are obtainable in this way.
Abstract: A two-phase global random search procedure for solving some computationally intractable discrete optimization problems is proposed. Guarantees for quality of random search results are derived from analysis of non-asymptotic order statistics and distribution-free intervals that are obtainable in this way: the confidence interval for a quantile of given order, or the tolerance interval for the parent distribution of goal function values. It has been shown that results, related to the multiconstrained 0–1 knapsack problem, within a few percentage from the true optimal solution can be obtained.
TL;DR: The long-awaited revision of Fortran 77 is now called Fortran 90, it was finalized on 11 April 1991, published as an ISO Standard in August 1991, and the first compiler is now on the market.
Abstract: Fortran 77 is the most widely used language for scientific programming. Its long-awaited revision is now called Fortran 90. It was finalized (down to the last editorial detail) on 11 April 1991, published as an ISO Standard in August 1991, and the first compiler is now on the market. This seems an appropriate moment to review its history and explain its advantages.
TL;DR: This paper considers the parallel computation of vector norms and inner products in floating-point and a proposed new form of computer arithmetic, the symmetric level-index system, which yields results of comparable accuracy to those offloating-point in cases where the latter system works and still yields Results of high accuracy when the floating- point system fails altogether.
Abstract: In this paper we consider the parallel computation of vector norms and inner products in floating-point and a proposed new form of computer arithmetic, the symmetric level-index system. The vector norms provide an illuminating example of the contrast between the two arithmetic systems under discussion in terms of the ability to program for (complete) robustness and parallelizability. The conflict between robustness of the computation—in the sense of the dual requirements of accuracy and freedom from overflow and underflow—and easy parallelization of the algorithms within a floating-point environment is made plain. It is seen that this conflict disappears if the symmetric level-index system of arithmetic is used. The freedom from overflow and underflow offered by this system allows the programming of the straightforward definitions in a way which is simple, robust and immediately parallelizable. Numerical results are given to illustrate the fact that the symmetric level-index system yields results of comparable accuracy to those of floating-point in cases where the latter system works and still yields results of high accuracy when the floating-point system fails altogether.
TL;DR: This method can be considered as a generalization of the so-called exclusion methods and also as a simplification of a method for finding the global minimum in a compactm-dimensional interval.
Abstract: The paper gives a new method for finding solutions of a nonlinear system of equations in a compactm-dimensional interval (rectangle). This method can be considered as a generalization of the so-called exclusion methods and also as a simplification of a method (given by the author) for finding the global minimum in a compactm-dimensional interval.
TL;DR: A memory-coupled multiprocessor—well suited to bit-wise operation—can be utilized to operate as a 1024 items cellular processing unit.
Abstract: A memory-coupled multiprocessor—well suited to bit-wise operation—can be utilized to operate as a 1024 items cellular processing unit. Each processor is working on 32 bits and 32 such processors are combined to a multiprocessor. The information is stored in “vertical” direction, as it is defined and described in earlier papers [1] on “vertical processing”. The two-dimensional array (32 times 32 bits) is composed of the 32 bit-machine-words of the coupled processors on the one hand and of 32 processors in nearest-neighbour-topology on the other hand. The bit-wise cellular operation at one of the 1024 “points” is realized by the program of the processor—possibly assisted by appropriate microprogam sequences.
TL;DR: Conditions are given which guarantee that a set of points admits unique Lagrange interpolation by quadratic splines on a four-directional mesh and the poisedness of these sets will be proved by reducing one bivariate problem to a finite sequence of univariate problems.
Abstract: We give sufficient conditions which guarantee that a set of points admits unique Lagrange interpolation by quadratic splines on a four-directional mesh. The poisedness of these sets will be proved by reducing one bivariate problem to a finite sequence of univariate problems.
TL;DR: A direct method for determining simple turning points of nonlinear parameterdependent equations is presented and applications are discussed and illustrated by numerical examples.
Abstract: A direct method for determining simple turning points of nonlinear parameterdependent equations is presented. Applications are discussed and illustrated by numerical examples.
TL;DR: A numerical method for the calculation of the deformations and the stresses in a slab using a formulation of the governing equations in mixed Eulerian-Lagrangian coordinates is presented.
Abstract: Bulging of continuously cast slabs can reduce the steel quality quite heavily. We present a numerical method for the calculation of the deformations and the stresses in a slab using a formulation of the governing equations in mixed Eulerian-Lagrangian coordinates. Due to the extreme nonlinearity of the material law, imbedding techniques are necessary to achieve convergence.