Showing papers in "Computing in 2002"
TL;DR: The basic ideas of ℋ- andℋ2-matrices are introduced and an algorithm that adaptively computes approximations of general matrices in the latter format is presented.
Abstract: A class of matrices (H2-matrices) has recently been introduced for storing discretisations of elliptic problems and integral operators from the BEM. These matrices have the following properties: (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but after truncation to the H2-matrix format these operations are again of quasi-linear complexity.We introduce the basic ideas of H- and H2-matrices and present an algorithm that adaptively computes approximations of general matrices in the latter format.
247 citations
TL;DR: Numerical examples with postprocessing and error estimation via an averaged stress field illustrate the new Matlab tool and its flexibility.
Abstract: A short Matlab implementation for P1 and Q1 finite elements (FE) is provided for the numerical solution of 2d and 3d problems in linear elasticity with mixed boundary conditions. Any adaptation from the simple model examples provided to more complex problems can easily be performed with the given documentation. Numerical examples with postprocessing and error estimation via an averaged stress field illustrate the new Matlab tool and its flexibility.
126 citations
TL;DR: The gain part of the collision integral is written in a form which allows its numerical computation on the uniform grid to be carried out efficiently and the accuracy and the efficiency of the method in comparison with DSMC computations in the spatially homogeneous case are shown.
Abstract: In the present paper a new numerical method for the Boltzmann equation is developed The gain part of the collision integral is written in a form which allows its numerical computation on the uniform grid to be carried out efficiently The amount of numerical work is shown to be of the order O(n6log(n)) for the most general model of interaction and of the order O(n6) for the Variable Hard Spheres (VHS) interaction model, while the formal accuracy is of the order O(n-2) Here n denotes the number of discretisation points in one direction of the velocity space Some numerical examples for Maxwell pseudo-molecules and for the hard spheres model illustrate the accuracy and the efficiency of the method in comparison with DSMC computations in the spatially homogeneous case
67 citations
TL;DR: Number-theoretic solution techniques are developed for approximating quantities of a surplus process in the presence of a nonlinear dividend barrier and numerical illustrations are given for exponential claim sizes and a parabolic dividend barrier.
Abstract: In the framework of classical risk theory we investigate a surplus process in the presence of a nonlinear dividend barrier and derive equations for two characteristics of such a process, the probability of survival and the expected sum of discounted dividend payments. Number-theoretic solution techniques are developed for approximating these quantities and numerical illustrations are given for exponential claim sizes and a parabolic dividend barrier.
63 citations
TL;DR: The present paper shows that the latter approach satisfies the inf-sup condition as well for k≥2 in any space dimension.
Abstract: One of the most popular pairs of finite elements for solving mixed formulations of the Stokes and Navier-Stokes problem is the Qk - Pk-1disc element. Two possible versions of the discontinuous pressure space can be considered: one can either use an unmapped version of the Pk-1disc space consisting of piecewise polynomial functions of degree at most k - 1 on each cell or define a mapped version where the pressure space is defined as the image of a polynomial space on a reference cell. Since the reference transformation is in general not affine but multilinear, the two variants are not equal on arbitrary meshes. It is well-known, that the inf-sup condition is satisfied for the first variant. In the present paper we show that the latter approach satisfies the inf-sup condition as well for k ≥ 2 in any space dimension.
62 citations
TL;DR: The results settle several open questions in NP-hardness in sequencing problems on a single machine with controllable job processing times and present several polynomial time results.
Abstract: We discuss sequencing problems on a single machine with controllable job processing times. For the maximum job cost criterion, we present several polynomial time results. For the total weighted job completion time criterion, we present an NP-hardness result. Our results settle several open questions in this area.
48 citations
TL;DR: This work considers a general framework for analysing the convergence of multi-grid solvers applied to finite element discretisations of mixed problems, both of conforming and nonconforming type.
Abstract: We consider a general framework for analysing the convergence of multi-grid solvers applied to finite element discretisations of mixed problems, both of conforming and nonconforming type. As a basic new feature, our approach allows to use different finite element discretisations on each level of the multi-grid hierarchy. Thus, in our multi-level approach, accurate higher order finite element discretisations can be combined with fast multi-level solvers based on lower order (nonconforming) finite element discretisations. This leads to the design of efficient multi-level solvers for higher order finite element discretisations.
41 citations
TL;DR: Nitsche type mortaring as a finite element method for treating non-matching meshes of triangles at the interface of some domain decomposition shows that appropriate mesh grading yields convergence rates as known for the classical FEM in presence of regular solutions.
Abstract: The paper deals with Nitsche type mortaring as a finite element method (FEM) for treating non-matching meshes of triangles at the interface of some domain decomposition. The approach is applied to the Poisson equation with Dirichlet boundary conditions (as a model problem) under the aspect that the interface passes re-entrant corners of the domain. For such problems and non-matching meshes with and without local refinement near the re-entrant corner, some properties of the finite element scheme and error estimates are proved. They show that appropriate mesh grading yields convergence rates as known for the classical FEM in presence of regular solutions. Finally, a numerical example illustrates the approach and the theoretical results.
40 citations
TL;DR: A fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time is introduced and analyzed.
Abstract: In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blowup of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme.
39 citations
TL;DR: A cascadic multigrid method for elliptic problems with strong material jumps is proposed and analyzed and numerical results illustrate the efficiency of the new subspace CMG algorithm.
Abstract: A cascadic multigrid (CMG) method for elliptic problems with strong material jumps is proposed and analyzed. Non-matching grids at interfaces between subdomains are allowed and treated by mortar elements. The arising saddle point problems are solved by a subspace confined conjugate gradient method as smoother for the CMG. Details of algorithmic realization including adaptivity are elaborated. Numerical results illustrate the efficiency of the new subspace CMG algorithm.
32 citations
IRIN1
TL;DR: A new factorization scheme based on well-known symbolic identities is introduced in order to handle the dependency problem of interval arithmetic and shows an improvement of 25% of the width of computed intervals with respect to Horner's rule.
Abstract: Interval arithmetic can be used to enclose the range of a real function over a domain. However, due to some weak properties of interval arithmetic, a computed interval can be much larger than the exact range. This phenomenon is called dependency problem. In this paper, Horner's rule for polynomial interval evaluation is revisited. We introduce a new factorization scheme based on well-known symbolic identities in order to handle the dependency problem of interval arithmetic. The experimental results show an improvement of 25% of the width of computed intervals with respect to Horner's rule.
TL;DR: In the standard step-by-step cubic spline collocation method for Volterra integral equations an initial condition is replaced by a not-a-knot boundary condition at the other end of the interval, this method is stable in the same region of collocation parameter as in the step- by-step implementation with linear splines.
Abstract: In the standard step-by-step cubic spline collocation method for Volterra integral equations an initial condition is replaced by a not-a-knot boundary condition at the other end of the interval. Such a method is stable in the same region of collocation parameter as in the step-by-step implementation with linear splines. The results about stability and convergence are based on the uniform boundedness of corresponding cubic spline interpolation projections. The numerical tests given at the end completely support the theoretical analysis.
Journal Article•
TL;DR: In this paper, the authors describe a business case where due to organisational and confidentiality reasons a single configurator approach is not applicable and several configurators along the supply chain must cooperate in finding correct product configurations and in presenting them to an online customer.
Abstract: Knowledgebased product configurators support their users in tailoring configurable products according to their specific demands and these systems have been successfully applied in many industrial sectors over the last decades. However, within today’s networked economy, the complex solutions of fered to the customers are in many cases assembled from configurable subproducts themselves. Within this paper we describe a business case where due to organisational and confidentiality reasons a singleconfigurator approach is not applicable and several configurators along the supply chain must cooperate in finding correct product configurations and in presenting them to an online customer. We present an algorithm based on Constraint Satisfaction that takes the specific characteristics of the problem domain into account and compare our approach to other work in the field of Distributed Problem Solving. The implementation framework for distributed configuration which is currently developed in the EUfunded project CAWICOMS1 is discussed in the final sections.
TL;DR: This work proposes a successively coupled finite-boundary element method for the solution of inhomogeneous boundary value problems in complex three-dimensional domains by using a finite element method in a simpler auxiliary domain.
Abstract: For the solution of inhomogeneous boundary value problems in complex three-dimensional domains we propose a successively coupled finite-boundary element method. By using a finite element method in a simpler auxiliary domain we first compute a particular solution of the inhomogeneous partial differential equation. This solution is used in a second step to approximate the Newton potential in the boundary integral formulation which is related to the original boundary value problem. A rigorous error analysis and a numerical example are given.
TL;DR: A number of results on the combinatorics, the algorithmics, and the complexity of subcolorings are derived, including asymptotically best possible upper bounds on the subchromatic number of interval graphs, chordal graphs, and permutation graphs in terms of the number of vertices.
Abstract: A subcoloring is a vertex coloring of a graph in which every color class induces a disjoint union of cliques. We derive a number of results on the combinatorics, the algorithmics, and the complexity of subcolorings.On the negative side, we prove that 2-subcoloring is NP-hard for comparability graphs, and that 3-subcoloring is NP-hard for AT-free graphs and for complements of planar graphs. On the positive side, we derive polynomial time algorithms for 2-subcoloring of complements of planar graphs, and for r-subcoloring of interval and of permutation graphs. Moreover, we prove asymptotically best possible upper bounds on the subchromatic number of interval graphs, chordal graphs, and permutation graphs in terms of the number of vertices.
Metz1
TL;DR: The aim of the analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion equation cux−ɛuxx=f, is to take the average onto the same mesh of the two equations of the mixed form, the conservation law div p=f and the constitutive law p=ϕ(u,∇u).
Abstract: ,∇u)=f, is to take the average onto the same mesh of the two equations of the mixed form, the conservation law div p=f and the constitutive law p=ϕ(u,∇u). In this paper, we perform the numerical analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion equation cu
x
−ɛu
xx
=f. In the first one, introduced by B. Courbet in [9,10], the numerical average of the diffusive flux is upwinded along the sign of the velocity, giving a first order accurate scheme. The second one is fourth order accurate. It is based onto the Euler-MacLaurin quadrature formula for the average of the diffusive flux. We emphasize in each case the link with the SUPG finite element method.
TL;DR: In this paper, the explicit expressions for the sixfold volume integrals arising for the Newton potential for a 1/r integrand are described. But they do not describe the explicit expression for the threefold integral arising from collocation.
Abstract: In boundary element methods, the evaluation of the weakly singular integrals can be performed either a) numerically, b) symbolically, i.e., by explicit expressions, or c) in a combined manner. The explicit integration is of particular interest, when the integrals contain the singularity or if the singularity is rather close to the integration domain. In this paper we describe the explicit expressions for the sixfold volume integrals arising for the Newton potential, i.e., for a 1/r integrand. The volume elements are axi-parallel bricks. The sixfold integrals are typical for the Galerkin method. However, the threefold integral arising from collocation methods can be derived in the same way.
TL;DR: An algorithm is presented allowing it to transform linearizable equations by hyperexponential transformations into linear form from which solutions may be obtained more easily.
Abstract: The subject of this article are third-order differential equations that may be linearized by a variable change. To this end, at first the equivalence classes of linear equations are completely described. Thereafter it is shown how they combine into symmetry classes that are determined by the various symmetry types. An algorithm is presented allowing it to transform linearizable equations by hyper-exponential transformations into linear form from which solutions may be obtained more easily. Several examples are worked out in detail.
TL;DR: Applications of the FORTRAN algorithm include multivariate permutation analyses of experimental design and prediction models which are based on Euclidean distance.
Abstract: A FORTRAN package is presented for multivariate Least sum of Absolute Deviations (LAD) regression with respect to the l1 of l2 norm, and this regression method is compared with l2 of l2 regression and l1 of l1 regression. Applications of the algorithm include multivariate permutation analyses of experimental design and prediction models which are based on Euclidean distance.
TL;DR: In this article, the authors consider a singularly perturbed reaction diffusion problem and investigate whether adaptive approaches are successful in designing robust solution procedures, using the energy norm as a posteriori error estimator.
Abstract: This paper considers a singularly perturbed reaction diffusion problem. It is investigated whether adaptive approaches are successful to design robust solution procedures. A key ingredient is the a posteriori error estimator. Since robust and mathematically analysed error estimation is possible in the energy norm, the focus is on this choice of norm and its implications. The numerical performance for several model problems confirms that the proposed adaptive algorithm (in conjunction with an energy norm error estimator) produces optimal results. Hence the energy norm is suitable for the purpose considered here. The investigations also provide valuable justification for forthcoming research.
TL;DR: The boundary concentrated FEM, a variant of the hp-version of the finite element method, is proposed for the numerical treatment of elliptic boundary value problems and it is shown that the Cholesky factorization of the resulting stiffness matrix requires O(Nlog4N) units of storage and can be computed with O( Nlog8N) work.
Abstract: The boundary concentrated FEM, a variant of the hp-version of the finite element method, is proposed for the numerical treatment of elliptic boundary value problems. It is particularly suited for equations with smooth coefficients and non-smooth boundary conditions. In the two-dimensional case it is shown that the Cholesky factorization of the resulting stiffness matrix requires O(Nlog4N) units of storage and can be computed with O(Nlog8N) work, where N denotes the problem size. Numerical results confirm theoretical estimates.
TL;DR: A new domain decomposition method based on natural boundary reduction is devised for the solution of nonlinear time-dependent exterior wave problems and the Schwarz alternating method is proposed.
Abstract: A new domain decomposition method based on natural boundary reduction is devised for the solution of nonlinear time-dependent exterior wave problems. The two-dimensional nonlinear scalar wave equation is taken as a model to illustrate the method. The governing equation is first discretized in time, leading to a time-stepping scheme, where a nonlinear exterior elliptic problem has to be solved at each time step. Two artificial boundaries are introduced. The Schwarz alternating method is proposed. The convergence of this algorithm is given. The contraction factor for exterior circular domain is also discussed. Numerical results are presented for the nonlinear wave equation to demonstrate the performance of the method.
TL;DR: The transport net corresponding to an undirected biconnected graph on a distributed or network model of computation is implemented, capable of handling topology changes in a transient manner.
Abstract: A self-stabilizing algorithm is presented in this paper that constructs a transport net corresponding to an undirected biconnected graph on a distributed or network model of computation. The algorithm is resilient to transient faults and does not require initialization. In addition, it is capable of handling topology changes in a transient manner. The paper includes a correctness proof of the algorithm. Finally, it concludes with some final remarks.
TL;DR: Instead of directly inverting the cumulative distribution function (CDF), this work provides an approximation of the inverse function which is simple to obtain by standard numerical methods and which is fast to compute.
Abstract: It has been observed by E. Eberlein and U. Keller that the hyperbolic distribution fits logarithmic rates of returns of a stock much better than the normal distribution. We give a method for sampling from the hyperbolic distribution by the inversion method, which is suited for simulation using low discrepancy point sets.Instead of directly inverting the cumulative distribution function (CDF) we provide an approximation of the inverse function which is simple to obtain by standard numerical methods and which is fast to compute.
TL;DR: A new method for storing sparse matrices as a block-matrix graph where the block are stored as local valuue arrays which are indexed via a set of a compact row-ordered schemes.
Abstract: We present a new method for storing sparse matrices as a block-matrix graph where the block are stored as local valuue arrays which are indexed via a set of a compact row-ordered schemes. This combines the flexibility of the graph structure with higher efficiency due to higher data locality. The inner compact pattern also allows identification of entries, which can lead to further advantages with respect to memory and computing time. To examine the efficiency of this new method, we study of a model case the performance of a matrix-vector multiplication which is the basic building block for most iterative methods. It turns out that our technique is competitive except for very small inner blocks. We also present two more realistic reactive flow applications to which this technique was applied.
TL;DR: In this paper, the authors used quadratic forms in d + 1 variables over finiter fields to scatter points the surface of the unit shpere S,d 1.
Abstract: We use solutions to quadratic forms in d + 1 variables over finiter fields to scatter points the surface of the unit shpere S ,d 1. Applications are given for spherical t designs and generalized s energies.
TL;DR: Formulae for calculating tight error bounds for CPM when this technique is applied to second order linear ODEs, and apply in particular to the Tau Method and to any approximation procedure equivalent to it.
Abstract: The coefficients perturbation method (CPM) is a numerical technique for solving ordinary differential equations (ODE) associated with initial or boundary conditions. The basic principle of CPM is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as the conditions associated to it. In this paper we shall develop formulae for calculating tight error bounds for CPM when this technique is applied to second order linear ODEs. Unlike results reported in the literature, ours do not require any a priori information concerning the exact error function or its derivative. The results of this paper apply in particular to the Tau Method and to any approximation procedure equivalent to it. The convergence of the derived bounds is also discussed, and illustrated numerically.
TL;DR: Performance bounds for “Ahrens” algorithm, that can generate random variates from continuous distributions with monotonically decreasing density, are discussed, with bounds on the rejection constant of this algorithm depending only on the number of intervals.
Abstract: This short note discusses performance bounds for "Ahrens" algorithm, that can generate random variates from continuous distributions with monotonically decreasing density. This rejection algorithm uses constant hat-functions and constant squeezes over many small intervals. The choice of these intervals is important. Ahrens has demonstrated that the equal area rule that uses strips of constant area leads to a very simple algorithm. We present bounds on the rejection constant of this algorithm depending only on the number of intervals.
TL;DR: Based on asymptotic error expansions of the Adini's nonconforming finite element for solving the biharmonic equation with the Dirchlet boundary value condition, a series of sharp error estimates and extrapolations are obtained.
Abstract: In this paper, we present the asymptotic error expansions of the Adini's nonconforming finite element for solving the biharmonic equation with the Dirchlet boundary value condition. Based on these expansions, we have obtained a series of sharp error estimates and extrapolations.
TL;DR: There are new estimates in the L2-norm for the additional error term originated by lumping in the FEM with lumping that can be applied directly the proof technique of error estimates known for conforming FEMs.
Abstract: The paper is devoted to error estimates of a special FEM with lumping for parabolic PDEs in the L norm. This well-known FELM is given by the use of the vertical line method and conforming linear finite elements on a triangulation. The main result of the paper are new estimates in the L-norm for the additional error term originated by lumping. Using these ones, for the FEM with lumping we can apply directly the proof technique of error estimates known for conforming FEMs.