Showing papers in "Computing in 1999"
TL;DR: This paper is the first of a series and is devoted to the first introduction of the $\Cal H$-matrix concept, which allows the exact inversion of tridiagonal matrices.
Abstract: A class of matrices (
$\Cal H$
-matrices) is introduced which 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 almost linear complexity (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the $\Cal H$
-matrix format are again of almost linear complexity (iv) The same statement holds for the inverse of an $\Cal H$
-matrix This paper is the first of a series and is devoted to the first introduction of the $\Cal H$
-matrix concept Two concret formats are described The first one is the simplest possible Nevertheless, it allows the exact inversion of tridiagonal matrices The second one is able to approximate discrete integral operators
1,106 citations
TL;DR: Conditions on the mesh-characterizing function are derived that are sufficient for the convergence of the method, uniformly with respect to the perturbation parameter, and enable one to immediately deduce the rate of convergence.
Abstract: We study convergence properties of the simple upwind difference scheme and a Galerkin finite element method on generalized Shishkin grids. We derive conditions on the mesh-characterizing function that are sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. These conditions are easy to check and enable one to immediately deduce the rate of convergence. Numerical experiments support these theoretical results and indicate that the estimates are sharp. The analysis is set in one dimension, but can be easily generalized to tensor product meshes in 2D.
154 citations
TL;DR: An a-posteriori error estimate is derived which bounds the error on the subdomains by the interface error of the subdomain solutions of the stabilized finite element method for elliptic boundary value problems.
Abstract: The application of a non-overlapping domain decomposition method to the solution of a stabilized finite element method for elliptic boundary value problems is considered. We derive an a-posteriori error estimate which bounds the error on the subdomains by the interface error of the subdomain solutions. As a by-product, some foundation is given to the design of the interface transmission condition. Numerical results support the theoretical results. Furthermore, we adapt a recent result on a-posteriori estimates for singular perturbation problems in order to obtain an a-posteriori estimate for the discrete subdomain solutions. — Authors' Abstract
116 citations
TL;DR: It is shown that one has a best possible algorithm with worst case ratio 4/3 while LS is still the best possible for the other problem with ratio $(r+1)/2$ which is still $3/2$ in the worst case $r=2$.
Abstract: This paper investigates two different semi on-line scheduling problems on a two-machine system. In the first case, we assume that all jobs have their processing times in between p and rp
$(p>0, r\geq1)$
. In the second case, we assume that the largest processing time is known in advance. We show that one has a best possible algorithm with worst case ratio 4/3 while LS is still the best possible for the other problem with ratio $(r+1)/2$
which is still $3/2$
in the worst case $r=2$
.
104 citations
TL;DR: A new algorithm is presented for feature point based motion tracking in long image sequences that efficiently resolve trajectory ambiguities and compares favorably to efficient alternative algorithms selected and tested in a performance evaluation study.
Abstract: A new algorithm is presented for feature point based motion tracking in long image sequences. Dynamic scenes with multiple, independently moving objects are considered in which feature points may temporarily disappear, enter and leave the view field. This situation is typical for surveillance and scene monitoring applications.
Most of the existing approaches to feature point tracking have limited capabilities in handling incomplete trajectories, especially when the number of points and their speeds are large, and trajectory ambiguities are frequent. The proposed algorithm was designed to efficiently resolve these ambiguities. Correspondences between moving points are established in a competitive linking process that develops as the trajectories grow. Appearing and disappearing points are treated in a natural way as the points that do not link.
The proposed algorithm compares favorably to efficient alternative algorithms selected and tested in a performance evaluation study.
74 citations
TL;DR: This work uses an appearance based object representation, namely the parametric eigenspace, but the planning algorithm is actually independent of the details of the specific object recognition environment, so that the probabilistic implementation always outperforms the other approaches.
Abstract: One major goal of active object recognition systems is to extract useful information from multiple measurements. We compare three frameworks for information fusion and view-planning using different uncertainty calculi: probability theory, possibility theory and Dempster-Shafer theory of evidence. The system dynamically repositions the camera to capture additional views in order to improve the classification result obtained from a single view. The active recognition problem can be tackled successfully by all the considered approaches with sometimes only slight differences in performance. Extensive experiments confirm that recognition rates can be improved considerably by performing active steps. Random selection of the next action is much less efficient than planning, both in recognition rate and in the average number of steps required for recognition. As long as the rate of wrong object-pose classifications stays low the probabilistic implementation always outperforms the other approaches. If the outlier rate increases averaging fusion schemes outperform conjunctive approaches for information integration. We use an appearance based object representation, namely the parametric eigenspace, but the planning algorithm is actually independent of the details of the specific object recognition environment.
67 citations
TL;DR: Discrete-time relaxation methods based on direct quadrature methods for the numerical solution of second kind Volterra systems are proposed and the convergence of the discrete-time iterations is analyzed.
Abstract: Discrete-time relaxation methods based on direct quadrature methods for the numerical solution of second kind Volterra systems are proposed. The convergence of the discrete-time iterations is analyzed.
60 citations
TL;DR: An improved dynamic programming algorithm for the bounded knapsack problem is given and a general approach based on dynamic programming is presented to reduce the storage requirements for combinatorial optimization problems where it is computationally more expensive to compute the explicit solution structure than the optimal solution value.
Abstract: The contribution of this paper is twofold: At first an improved dynamic programming algorithm for the bounded knapsack problem is given. It decreases the running time for an instance with n items and capacity c from \(O(nc\log c)\) to \(O(nc)\), which is the same pseudopolynomial complexity as usually given for the 0--1 knapsack problem. In the second part a general approach based on dynamic programming is presented to reduce the storage requirements for combinatorial optimization problems where it is computationally more expensive to compute the explicit solution structure than the optimal solution value. Among other applications of this scheme it is shown that the 0--1 knapsack problem as well as the bounded knapsack problem can be solved in \(O(nc)\) time and \(O(n+c)\) space.
55 citations
TL;DR: This paper presents a self-stabilizing algorithm that finds the bridge-connected components of a connected undirected graph on an asynchronous distributed or network model of computation.
Abstract: The basic idea of our new approach is to determine in a first step for each node those pairs of nodes which allow a good interpolation of the unknowns located at this node. These pairs of neighbor nodes (in some cases only one node) are called parent nodes. This is done by solving a local minimization problem which, in addition, yields the interpolation and restriction coefficients. The construction scheme has been generalized to systems of convection-diffusion-reaction equations using a point-block approach. After these suitable pairs of parent nodes have been determined, the nodes are labeled as C- and F-nodes such that each F-node can be interpolated using one of these suitable pairs of parent nodes and the already computed coefficients. Additionally, a simple heuristic algorithm tries to minimize the number of C-nodes and the number of non-zero entries in the coarse grid matrix. The algorithm has been parallelized and shows mesh size independent convergence for standard model problems. Realistic numerical experiments confirm the efficiency of the presented algorithm.
47 citations
TL;DR: The results show that there exist classes of matrices for which it is possible to find the minimum bandwidth in a small amount of time, and provide additional experimental evidence of the effectiveness of the perimeter search technique.
Abstract: Given an $m\times m$
sparse symmetric matrix, we consider the problem of finding the permutation of rows and columns which minimizes the bandwidth. This problem is known to be NP-complete therefore no algorithm polynomial in m is likely to exist. We present two algorithms which exhaustively enumerate all permutations trying to discard as early as possible those which cannot lead to an optimal ordering. Our first algorithm uses a depth first search strategy, while our second algorithm uses a recently developed technique called perimeter search. We have tested the performance of these algorithms solving problems of size ranging from 40 to 100. Our results show that there exist classes of matrices for which it is possible to find the minimum bandwidth in a small amount of time. Our results also provide additional experimental evidence of the effectiveness of the perimeter search technique: we found that, compared to depth-first search, perimeter search can reduce the running time up to a factor 100.
46 citations
TL;DR: A black-box parallel iterative method suitable for solving both elliptic and certain non-elliptic problems discretized on unstructured meshes is proposed and the validity of the proved convegence estimate is confirmed.
Abstract: We propose a black-box parallel iterative method suitable for solving both elliptic and certain non-elliptic problems discretized on unstructured meshes. The method is analyzed in the case of the second order elliptic problems discretized on quasiuniform P1 and Q1 finite element meshes. The numerical experiments confirm the validity of the proved convegence estimate and show that the method can successfully be used for more difficult problems (e.g. plates, shells and Helmholtz equation in high-frequency domain.)
TL;DR: A new and more general bounding procedure based on the dual of the linearization of Adams and Johnson is proposed and the computational results indicate that the new bound competes well with existing linearization bounds and yields a good trade off between computation time and bound quality.
Abstract: A dual framework allowing the comparison of various bounds for the quadratic assignment problem (QAP) based on linearization, e.g. the bounds of Adams and Johnson, Carraresi and Malucelli, and Hahn and Grant, is presented. We discuss the differences of these bounds and propose a new and more general bounding procedure based on the dual of the linearization of Adams and Johnson. The new procedure has been applied to problems of dimension up to \(n=72\), and the computational results indicate that the new bound competes well with existing linearization bounds and yields a good trade off between computation time and bound quality.
TL;DR: In this paper, a general scheme is proposed, which guarantees the approximation property of nonconforming finite element discretisations, and is applied to the discretisation by non-matching grids (mortar elements).
Abstract: Nonconforming finite element discretisations require special care in the construction of the prolongation and restriction in the multigrid process. In this paper, a general scheme is proposed, which guarantees the approximation property. As an example, the technique is applied to the discretisation by non-matching grids (mortar elements).
TL;DR: A highly accurate spectral Legendre--Galerkin method is used to solve the biharmonic eigenvalue problem and the buckling plate problem and shows that the asymptotic expansion is highly accurate already from the first sign change.
Abstract: We solve the biharmonic eigenvalue problem $\Delta^2u = \lambda u$
and the buckling plate problem ${\Delta}^2u = - {\lambda}\Delta u$
on the unit square using a highly accurate spectral Legendre--Galerkin method. We study the nodal lines for the first eigenfunction near a corner for the two problems. Five sign changes are computed and the results show that the eigenfunction exhibits a self similar pattern as one approaches the corner. The amplitudes of the extremal values and the coordinates of their location as measured from the corner are reduced by constant factors. These results are compared with the known asymptotic expansion of the solution near a corner. This comparison shows that the asymptotic expansion is highly accurate already from the first sign change as we have complete agreement between the numerical and the analytical results. Thus, we have an accurate description of the eigenfunction in the entire domain.
TL;DR: In this paper, a feature point based motion tracking algorithm for long image sequences is presented. But this algorithm is not suitable for dynamic scenes with multiple, independently moving objects. And feature points may not be used in the tracking process.
Abstract: A new algorithm is presented for feature point based motion tracking in long image sequences. Dynamic scenes with multiple, independently moving objects are considered in which feature points may t...
TL;DR: New complexity results are derived for single-machine problems with constant positive time-lags which also lead to new results for flow-shop problems with unit processing times and job precedences.
Abstract: In a single-machine problem with time-lags a set of jobs has to be processed on a single machine in such a way that certain timing restrictions between the finishing and starting times of the jobs are satisfied and a given objective function is minimized. We consider the case of positive finish-start time-lags \(l_{ij}\) which mean that between the finishing time of job i and the starting time of job j the minimal distance \(l_{ij}\) has to be respected. New complexity results are derived for single-machine problems with constant positive time-lags \(l_{ij}=l\) which also lead to new results for flow-shop problems with unit processing times and job precedences.
TL;DR: An algorithm for computing all the zeros of f that lie in the interior of γ that is self-starting in the sense that it does not require initial approximations and is based on the theory of formal orthogonal polynomials.
Abstract: Let W be a simply connected region in \(\Bbb C\), \(\) analytic in W and γ a positively oriented Jordan curve in W that does not pass through any zero of f. We present an algorithm for computing all the zeros of f that lie in the interior of γ. It proceeds by evaluating certain integrals along γ numerically and is based on the theory of formal orthogonal polynomials. The algorithm requires only f and not its first derivative f'. We have found that it gives accurate approximations for the zeros. Moreover, it is self-starting in the sense that it does not require initial approximations. The algorithm works for simple zeros as well as multiple zeros, although it is unable to compute the multiplicity of a zero explicitly. Numerical examples illustrate the effectiveness of our approach.
TL;DR: A new rule based method on dual graph contraction for transforming the run graph and its dual into MLPP graphs is presented and results given.
Abstract: In line image understanding a minimal line property preserving (MLPP) graph of the image compliments the structural information in geometric graph representations like the run graph. With such a graph and its dual it is possible to efficiently detect topological features like loops and holes and to make use of relations like containment. We present a new rule based method on dual graph contraction for transforming the run graph and its dual into MLPP graphs. A parallel O(log(longest curve)) algorithm is presented and results given.
TL;DR: In this article, a solution algorithm is described that determines its general solution in closed form by reducing it to a linear third-order equation, based on the results obtained by Sophus Lie in the last century.
Abstract: Second order ordinary differential equations of the form $y''+A{y'}^3+B{y'}^2+Cy'+D$
with $y\equiv y(x)$
and A, B, C and D functions of x and y are of special interest because they may allow the largest possible group of point symmetries if its coefficients satisfy certain constraints. For large classes of these equations a solution algorithm is described that determines its general solution in closed form by reducing it to a linear third-order equation. If the results obtained by Sophus Lie in the last century are supplemented by more recent concepts like Janet bases and Loewy decompositions, a systematic solution procedure is obtained that is easily implemented in a computer algebra system.
TL;DR: This paper presents the convergence analysis for the total-step and the single-step methods with Newton's corrections for Halley's type iteration methods for the simultaneous inclusion of all zeros of a polynomial.
Abstract: In this paper we present iteration methods of Halley's type for the simultaneous inclusion of all zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis for the total-step and the single-step methods with Newton's corrections. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. A numerical example is given.
TL;DR: An a posteriori estimate is derived which bounds the errors on the subdomains by the difference of traces of the subdomain solutions when the domain decomposition method is discretized by finite elements.
Abstract: We consider a non-overlapping domain decomposition method for diffusion-reaction problems which is known to converge strongly from previous work. We derive an a posteriori estimate which bounds the errors on the subdomains by the difference of traces of the subdomain solutions. If the domain decomposition method is discretized by finite elements we can adapt the techniques of the usual a posteriori error analysis for finite elements to get an a posteriori estimate for the discrete subdomain solutions.
TL;DR: A proof of convergence for a preconditionsed subspace method which shows the dependency of the convergence rate on the preconditioner used, which means that, for example, multigrid precondITIONers can be used to find eigenvalues of elliptic PDE's at a grid-independent rate.
Abstract: In this paper we present a proof of convergence for a preconditioned subspace method which shows the dependency of the convergence rate on the preconditioner used. This convergence rate depends only on the condition of the pre-conditioned system $ \kappa _{2}(MA) $
and the relative separation of the first two eigenvalues $ 1-\lambda _{1}/\lambda _{2} $
. This means that, for example, multigrid preconditioners can be used to find eigenvalues of elliptic PDE's at a grid-independent rate.
TL;DR: A modification of Newton's method for computing multiple roots of systems of analytic equations is proposed in which not only an approximation for the root is refined, but also approximations for these constants.
Abstract: We propose a modification of Newton's method for computing multiple roots of systems of analytic equations. Under mild assumptions the iteration converges quadratically. It involves certain constants whose product is a lower bound for the multiplicity of the root. As these constants are usually not known in advance, we devise an iteration in which not only an approximation for the root is refined, but also approximations for these constants. Numerical examples illustrate the effectiveness of our approach.
TL;DR: It is shown that in this case the semidefinite program can be solved through an ordinary linear program, where the underlying graph arises from an association scheme.
Abstract: We consider semidefinite programs, where the matrices defining the problem all arise from some association scheme. We show that in this case the semidefinite program can be solved through an ordinary linear program. As an application, we consider the max-cut problem, where the underlying graph arises from an association scheme.
TL;DR: Only a small number of arithmetical operations is needed to construct the cubature formulas using Smolyak's technique and different methods to obtain formulas with high trigonometric degree are compared.
Abstract: We study cubature formulas for d-dimensional integrals with a high trigonometric degree. To obtain a trigonometric degree \(\ell\) in dimension d, we need about \(d^\ell/\ell !\) function values if d is large. Only a small number of arithmetical operations is needed to construct the cubature formulas using Smolyak's technique. We also compare different methods to obtain formulas with high trigonometric degree.
TL;DR: A set containing eigenpairs is constructed which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space by using a finite element approximation and constructive error estimates, and is proved to prove the local uniqueness separately of eigenvalues and eigenfunctions.
Abstract: We propose a numerical method to enclose the eigenvalues and eigenfunctions of second-order elliptic operators with local uniqueness. We numerically construct a set containing eigenpairs which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space by using a finite element approximation and constructive error estimates. We then prove the local uniqueness separately of eigenvalues and eigenfunctions. This local uniqueness assures the simplicity of the eigenvalue. Numerical examples are presented.
TL;DR: The proposed communication scheme has the advantage that 1) it is the same for all three conditions; 2) each node of a WK-recursive network, to transmit messages, does not require any information about their presence or location; and 3) this routing algorithm is able to tolerate up to up to $\big[{N_d(N-d-3)\over2}+1\big]{N _d^l-1\over N_d-1}
Abstract: This paper presents an easy and straightforward routing algorithm for WK-recursive topologies The algorithm, based on adaptive routing, takes advantage of the geometric properties of such topologies Once a source node S and destination node D have been determined for a message communication, they characterize, at some level l, two virtual nodes \(hl\_vn(S_D)\) and \(hl\_vn(D_S)\) that respectively contain S but not D and D but not S Such virtual nodes characterize other \(N_d-2\) (where \(N_d\) is the node degree for a fixed topology) virtual nodes \(hl\_vn(I_{SD})\) of the same level that contain neither S nor D Consequently, it is possible to locate \(N_d-2\) triangles whose vertices are these virtual nodes with property to share the same path, called the self-routing path, directly connecting \(hl\_vn(S_D)\) to \(hl\_vn(D_S)\) When the self-routing path is unavailable to transmit a message from S to D because of deadlock, fault, and congestion conditions, the routing strategy can follow what we call the triangle rule to deliver it The proposed communication scheme has the advantage that 1) it is the same for all three conditions; 2) each node of a WK-recursive network, to transmit messages, does not require any information about their presence or location Furthermore, this routing algorithm is able to tolerate up to \(\big[{N_d(N_d-3)\over2}+1\big]{N_d^l-1\over N_d-1}\) faulty links
TL;DR: The existence theorem is given, an efficient algorithm for searching the parameters is proposed and some numerical examples are given.
Abstract: In this paper we consider a special nonlinear total least squares problem, where the model function is of the form \(f(x;a,b)=\phi^{-1}(ax+b)\). Using the fact that after an appropriate substitution, the model function becomes linear in parameters, and that the symmetry preserves the distances, this nonlinear total least squares problem can be greatly simplified. In this paper we give the existence theorem, propose an efficient algorithm for searching the parameters and give some numerical examples.
TL;DR: In this article, the local Cauchy problem for nonlinear functional differential systems is investigated and a general class of difference methods for this problem is investigated. And the stability of the stability is based on functional difference inequalities.
Abstract: The paper deals with the local Cauchy problem for nonlinear functional differential systems We investigate a general class of difference methods for this problem We construct interpolating operators on the Haar pyramid and we give an error estimate for approximate solutions We adopt nonlinear estimates of the Perron type for given functions with respect to the functional variable The proof of the stability is based on functional difference inequalities
TL;DR: The symmetric coupling of nonconforming finite elements and boundary elements is established and a priori error estimates are shown and numerical examples prove the new scheme useful in practice.
Abstract: Nonconforming finite element methods are sometimes considered as a variational crime and so we may regard its coupling with boundary element methods. In this paper, the symmetric coupling of nonconforming finite elements and boundary elements is established and a priori error estimates are shown. The coupling involves a further continuous layer on the interface in order to separate the nonconformity in the domain from its boundary data which are required to be continuous. Numerical examples prove the new scheme useful in practice. A posteriori error control and adaptive algorithms will be studied in the forthcoming Part II.