Showing papers in "Computing in 2003"
TL;DR: The dimension–adaptive quadrature method is developed and presented, based on the sparse grid method, which tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators, and leads to an approach which is based on generalized sparse grid index sets.
Abstract: We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.
578 citations
TL;DR: The proposed algorithm which uses the ℋ-matrix format is purely algebraic and relies on a small part of the collocation matrix for its blockwise approximation by low-rank matrices.
Abstract: This article deals with the solution of integral equations using collocation methods with almost linear complexity. Methods such as fast multipole, panel clustering and H-matrix methods gain their efficiency from approximating the kernel function. The proposed algorithm which uses the H-matrix format, in contrast, is purely algebraic and relies on a small part of the collocation matrix for its blockwise approximation by low-rank matrices. Furthermore, a new algorithm for matrix partitioning that significantly reduces the number of blocks generated is presented.
520 citations
TL;DR: This paper presents a construction of the hierarchical matrix format for standard finite element and boundary element applications for which two criteria, the sparsity and idempotency, are sufficient to give the desired bounds.
Abstract: In previous papers hierarchical matrices were introduced which are data-sparse and allow an approximate matrix arithmetic of nearly optimal complexity. In this paper we analyse the complexity (storage, addition, multiplication and inversion) of the hierarchical matrix arithmetics. Two criteria, the sparsity and idempotency, are sufficient to give the desired bounds. For standard finite element and boundary element applications we present a construction of the hierarchical matrix format for which we can give explicit bounds for the sparsity and idempotency.
456 citations
TL;DR: Modifications of the BiCGStab(ℓ) method are developed which allow to solve the seed and the shifted system at the expense of just the matrix-vector multiplications needed to solve Ax = b via BiCGstab( ℓ), showing that in the case that A is positive real and σ ≥ 0, the resulting method is still a well-smoothed variant of BiCG.
Abstract: We consider a seed system Ax = b together with a shifted linear system of the form (A + σI)x = b, σ ∈ C, A ∈ Cn×n, b ∈ Cn. We develop modifications of the BiCGStab(l) method which allow to solve the seed and the shifted system at the expense of just the matrix-vector multiplications needed to solve Ax = b via BiCGStab(l). On the shifted system, these modifications do not perform the corresponding BiCGStab(l)-method, but we show, that in the case that A is positive real and σ ≥ 0, the resulting method is still a well-smoothed variant of BiCG. Numerical examples from an application arising in quantum chromodynamics are given to illustrate the efficiency of the method developed.
128 citations
TL;DR: This work investigates a new approach to exploit the ℋ-matrix structure for the solution of large scale Lyapunov and Riccati equations as they typically arise for optimal control problems where the constraint is a partial differential equation of elliptic type.
Abstract: In previous papers, a class of hierarchical matrices (H-matrices) is introduced which are data-sparse and allow an approximate matrix arithmetic of almost optimal complexity. Here, we investigate a new approach to exploit the H-matrix structure for the solution of large scale Lyapunov and Riccati equations as they typically arise for optimal control problems where the constraint is a partial differential equation of elliptic type. This approach leads to an algorithm of linear-logarithmic complexity in the size of the matrices.
117 citations
TL;DR: The first numerical results confirm the efficiency and the robustness predicted by the analysis, and there is an unified framework for coupling, handling, and analyzing both methods.
Abstract: In this paper we introduce the Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established Finite Element Tearing and Interconnecting (FETI) methods. In some practical important applications such as far field computations, handling of singularities and moving parts etc., BETI methods have certainly some advantages over their finite element counterparts. This claim is especially true for the sparse versions of the BETI preconditioners resp. methods. Moreover, there is an unified framework for coupling, handling, and analyzing both methods. In particular, the FETI methods can benefit from preconditioning components constructed by boundary element techniques. The first numerical results confirm the efficiency and the robustness predicted by our analysis.
109 citations
TL;DR: The classical approach to the initial value problem leads to an algorithm with very high arithmetic complexity, so an alternative that leads to lower complexity without sacrificing too much precision is derived.
Abstract: We investigate strategies for the numerical solution of the initial value problem y(αv)(x) = f(x, y(x), y(α1)(x),..., y(αv-1)(x)) with initial conditions y(k)(0) = y0(k) (k = 0, 1,..., ⌈αv⌉ - 1), where 0 0 (not necessarily αj ∈ N) in the sense of Caputo. The methods are based on numerical integration techniques applied to an equivalent nonlinear and weakly singular Volterra integral equation. The classical approach leads to an algorithm with very high arithmetic complexity. Therefore we derive an alternative that leads to lower complexity without sacrificing too much precision.
104 citations
TL;DR: The sparse grid approach, based upon a direct higher order discretization on the sparse grid, overcomes this dilemma to some extent, and introduces additional flexibility with respect to both the order of the 1 D quadrature rule applied (in the sense of Smolyak's tensor product decomposition) and the placement of grid points.
Abstract: In this paper, we study the potential of adaptive sparse grids for multivariate numerical quadrature in the moderate or high dimensional case, i.e. for a number of dimensions beyond three and up to several hundreds. There, conventional methods typically suffer from the curse of dimension or are unsatisfactory with respect to accuracy. Our sparse grid approach, based upon a direct higher order discretization on the sparse grid, overcomes this dilemma to some extent, and introduces additional flexibility with respect to both the order of the 1 D quadrature rule applied (in the sense of Smolyak's tensor product decomposition) and the placement of grid points. The presented algorithm is applied to some test problems and compared with other existing methods.
92 citations
TL;DR: It is proved that the k-th moment solves a deterministic problem in Dk⊂ℝdk, for which well-posedness and regularity are discussed and an efficient algorithm is proposed for solving the resulting system.
Abstract: We define the higher order moments associated to the stochastic solution of an elliptic BVP in D ⊂ Rd with stochastic input data. We prove that the k-th moment solves a deterministic problem in Dk ⊂ Rdk, for which we discuss well-posedness and regularity. We discretize the deterministic k-th moment problem using sparse grids and, exploiting a spline wavelet basis, we propose an efficient algorithm, of logarithmic-linear complexity, for solving the resulting system.
75 citations
TL;DR: The error estimator can be applied also to problems and discretizations where the Galerkin orthogonality is not available and captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.
Abstract: The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive the reliable and efficient a posteriori error estimator that measures the error in the energy norm. The estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H1. independently of the discretization method chosen. In particular, our error estimator can be applied also to problems and discretizations where the Galerkin orthogonality is not available. We will present different strategies for the evaluation of the error estimator. Only one constant appears in its definition which is the one from Friedrichs' inequality; that constant depends solely on the domain geometry, and the estimator is quite non-sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.
64 citations
TL;DR: In this paper, a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions was studied using the duality technique, and the authors derived the reliable and efficient estimator.
Abstract: The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive the reliable and efficient...
TL;DR: A flexible efficient and accurate inverse Laplace transform algorithm is developed that computes the coefficients of the continued fractions needed for the inversion process by combining diagonalwise operations and recursion relations in the quotient-difference schemes.
Abstract: A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotient-difference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the quotient-difference schemes, the algorithm controls the dimension of the inverse Laplace transform approximation automatically. Application of the algorithm to the solute transport equations in porous media is explained in a general setting. Also, a numerical simulation is performed to show the accuracy and efficiency of the developed algorithm.
TL;DR: Despite the theoretical reliability-efficiency gap for the relaxed problem, numerical evidence supports that adaptive mesh-refining algorithms generate efficient triangulations and improve the experimental convergence rates optimally.
Abstract: Macroscopic simulations of non-convex minimisation problems with enforced microstructures encounter oscillations on finest length scales - too fine to be fully resolved. The numerical analysis must rely on an essentially equivalent relaxed mathematical model. The paper addresses a prototype example, the scalar 2-well minimisation problem and its convexification and introduces a benchmark problem with a known (generalised) solution. For this benchmark, the stress error is studied empirically to asses the performance of adaptive finite element methods for the relaxed and the original minimisation problem. Despite the theoretical reliability-efficiency gap for the relaxed problem, numerical evidence supports that adaptive mesh-refining algorithms generate efficient triangulations and improve the experimental convergence rates optimally. Moreover, the averaging error estimators perform surprisingly accurate.
TL;DR: This paper investigates the following scheduling problem: The authors have two sets of identical machines, the jobs have two processing times one for each set of machines, and considers two different objective functions.
Abstract: In this paper we investigate the following scheduling problem: We have two sets of identical machines, the jobs have two processing times one for each set of machines. We consider two different objective functions, in the first model the goal is to minimize the maximum of the makespans on the sets, in the second model we minimize the sum of the makespans. We consider the online, semi online and offline versions of these problems.
TL;DR: This work presents a method for discretizing and solving general elliptic partial differential equations on sparse grids employing higher order finite elements, and finds it to be highly efficient, yielding balanced errors on the computational domain.
Abstract: We present a method for discretizing and solving general elliptic partial differential equations on sparse grids employing higher order finite elements. On the one hand, our approach is charactarized by its simplicity. The calculation of the occurring functionals is composed of basic pointwise or unidirectional algorithms. On the other hand, numerical experiments prove our method to be robust and accurate. Discontinuous coefficients can be treated as well as curvilinearly bounded domains. When applied to adaptively refined sparse grids, our discretization results to be highly efficient, yielding balanced errors on the computational domain.
TL;DR: These results form the basis for obtaining insight into the analogous properties of numerical solutions generated by continuous Runge-Kutta or collocation methods, where these methods are applied to a suitable reformulation of the given initial-value problem.
Abstract: In this paper we study asymptotic stability and contractivity properties of solutions of a class of delay functional integro-differential equations. These results form the basis for obtaining insight into the analogous properties of numerical solutions generated by continuous Runge-Kutta or collocation methods, where these methods are applied to a suitable reformulation of the given initial-value problem.
TL;DR: This paper surveys the main polynomial time approximation algorithms for the minimum graph-coloring and discusses their approximation performance and their complexity, and improves the approximation ratio for graph- Coloring.
Abstract: Consider a graph G = (V,E) of order n. In the minimum graph-coloring problem we try to color V with as few colors as possible so that no two adjacent vertices receive the same color. This problem is among the first ones proved to be intractable, and hence, it is very unlikely that an optimal polynomial-time algorithm could ever be devised for it. In this paper, we survey the main polynomial time approximation algorithms (the ones for which theoretical approximability bounds have been studied) for the minimum graph-coloring and we discuss their approximation performance and their complexity. Finally, we further improve the approximation ratio for graph-coloring.
TL;DR: This paper considers modifications of those methods, which under certain assumptions on the starting vector deliver nested sequences converging to [x*], and shows, that [ x*] is optimal in a precisely defined sense.
Abstract: In this paper we introduce the total step method, the single step method and the symmetric single step method for linear complementarity problems with interval data. They are applied to an interval matrix [A] and an interval vector [b]. If all A ∈ [A] are H-matrices with positive diagonal elements, these methods are all convergent to the same interval vector [x*]. This interval vector includes the solution x of the linear complementarity problem defined by any fixed A ∈ [A] and any fixed b ∈ [b]. If all A ∈ [A] are M-matrices, then we will show, that [x*] is optimal in a precisely defined sense. We also consider modifications of those methods, which under certain assumptions on the starting vector deliver nested sequences converging to [x*]. We close our paper with some examples which illustrate our theoretical results.
TL;DR: Several cases of the server scheduling problem to be solved in O(n log n) by known algorithms, namely, finding a schedule feasible with respect to a given set of deadlines, minimizing the maximum lateness and, if the job processing times are agreeable, minimize the total completion time.
Abstract: We study the problem of scheduling a single server that processes n jobs in a two-machine flow shop environment A machine dependent setup time is needed whenever the server switches from one machine to the other The problem with a given job sequence is shown to be reducible to a single machine batching problem This result enables several cases of the server scheduling problem to be solved in O(n log n) by known algorithms, namely, finding a schedule feasible with respect to a given set of deadlines, minimizing the maximum lateness and, if the job processing times are agreeable, minimizing the total completion time Minimizing the total weighted completion time is shown to be NP-hard in the strong sense Two pseudopolynomial dynamic programming algorithms are presented for minimizing the weighted number of late jobs Minimizing the number of late jobs is proved to be NP-hard even if setup times are equal and there are two ditstinct due dates This problem is solved in O(n3) time when all job processing times on the first machine are equal, and it is solved in O(n4) time when all processing times on the second machine are equal
TL;DR: H–p–adaptive projection with respect to any prescribed threshold value for the visual error is presented, which can then be processed by various local rendering methods, e.g. color coding of data or isosurface extraction.
Abstract: We propose an appropriate and efficient multiresolution visualization method for piecewise higher order polynomial data on locally refined computational grids. Given some suitable error indicators, we efficiently extract a continuous h-p-adaptive projection with respect to any prescribed threshold value for the visual error. This projection can then be processed by various local rendering methods, e.g. color coding of data or isosurface extraction. Especially for color coding purposes modern texture capabilities are used to directly render higher polynomial data by superposition of polynomial basis function textures and final color look-up tables. Numerical experiments from CFD clearly demonstrate the applicability and efficiency of our approach.
TL;DR: A singularly perturbed reaction-diffusion elliptic problem in two dimensions, with strongly anisotropic coefficients and line interface, is considered, and finite volume difference schemes on condensed Shihskin meshes are constructed and ɛ-uniform convergence in discrete energy and maximum norms are proved.
Abstract: We consider a singularly perturbed reaction-diffusion elliptic problem in two dimensions (x,y), with strongly anisotropic coefficients and line interface. The second order derivative with respect to x is multiplied by a small parameter e2. We construct finite volume difference schemes on condensed Shihskin meshes and prove e-uniform convergence in discrete energy and maximum norms. Numerical experiments that agree with the theoretical results are given.
TL;DR: A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented.
Abstract: A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.
TL;DR: Another algorithm is provided which is then extented such that quality criteria for the splitting of faces are respected and the paper provides another algorithm for transforming hexahedral finite element meshes into tetrahedral meshes without introducing new nodes.
Abstract: The paper is concerned with algorithms for transforming hexahedral finite element meshes into tetrahedral meshes without introducing new nodes. Known algorithms use only the topological structure of the hexahedral mesh but no geometry information. The paper provides another algorithm which is then extented such that quality criteria for the splitting of faces are respected.
TL;DR: A modified partitioning and admissibility condition is derived that ensures good convergence also for the singularly perturbed case of L∞-coefficients in the case of increasing convection.
Abstract: Hierarchical matrices provide a technique for the sparse approximation and matrix arithmetic of large, fully populated matrices. This technique has been proven to be applicable to matrices arising in the boundary and finite element method for uniformly elliptic operators with L∞-coefficients. This paper analyses the application of hierarchical matrices to the convection-dominant convection-diffusion equation with constant convection. In the case of increasing convection, the convergence of a standard H-matrix approximant towards the original matrix will deteriorate. We derive a modified partitioning and admissibility condition that ensures good convergence also for the singularly perturbed case.
TL;DR: In this paper the expansion of a polynomial into Bernstein polynomials over an interval I is considered and it is shown that the so-called Bernstein form is inclusion isotone, which was shown by a longish proof in 1995 in this journal by Hong and Stahl.
Abstract: In this paper the expansion of a polynomial into Bernstein polynomials over an interval I is considered. The convex hull of the control points associated with the coefficients of this expansion encloses the graph of the polynomial over I. By a simple proof it is shown that this convex hull is inclusion isotonic, i.e. if one shrinks I then the convex hull of the control points on the smaller interval is contained in the convex hull of the control points on I. From this property it follows that the so-called Bernstein form is inclusion isotone, which was shown by a longish proof in 1995 in this journal by Hong and Stahl. Inclusion isotonicity also holds for multivariate polynomials on boxes. Examples are presented which document that two simpler enclosures based on only a few control points are in general not inclusion isotonic.
TL;DR: The mixed covolume method is considered for a system of first order partial differential equations resulting from the mixed formulation of a general self-adjoint elliptic problem with a variable full diffusion tensor to model the transport of a contaminant carried by a flow.
Abstract: We consider a mixed covolume method for a system of first order partial differential equations resulting from the mixed formulation of a general self-adjoint elliptic problem with a variable full diffusion tensor. The system can be used to model the transport of a contaminant carried by a flow. We use the lowest order Raviart-Thomas mixed finite element space. We show the first order convergence in L2 norm and the superconvergence in certain discrete norms both for the pressure and velocity. Finally some numerical examples illustrating the error behavior of the scheme are provided.
TL;DR: A fast and accurate resolution method, the pseudo-Lévy series method, which has higher computational accuracy and efficiency than existing Fourier series and pseudo-spectral methods as well as other numerical methods and can be used to solve complex surface blending problems which cannot be tackled by the closed form resolution method.
Abstract: In our previous work, a more general fourth order partial differential equation (PDE) with three vector-valued parameters was introduced. This equation is able to generate a superset of the blending surfaces of those produced by other existing fourth order PDEs found in the literature. Since it is usually more difficult to solve PDEs analytically than numerically, many references are only concerned with numerical solutions, which unfortunately are often inefficient. In this paper, we have developed a fast and accurate resolution method, the pseudo-Levy series method. Due to its analytical nature, the comparison with other existing methods indicates that the developed method can generate blending surfaces almost as quickly and accurately as the closed form resolution method, and has higher computational accuracy and efficiency than existing Fourier series and pseudo-spectral methods as well as other numerical methods. In addition, it can be used to solve complex surface blending problems which cannot be tackled by the closed form resolution method. To demonstrate the potential of this new method we have applied it to various surface blending problems, including the generation of the blending surface between parametric primary surfaces, general second and higher degree surfaces, and surfaces defined by explicit equations.
TL;DR: It is shown that the First In First Out (FIFO) heuristic has a tight worst-case performance of 3−2/m, when jobs processing times and costs are set as in some optimal preemptive schedule.
Abstract: In a scheduling problem with controllable processing times the job processing time can be compressed through incurring an additional cost. We consider the identical parallel machines max flow time minimization problem with controllable processing times. We address the preemptive and non-preemptive version of the problem. For the preemptive case, a linear programming formulation is presented which solves the problem optimally in polynomial time. For the non-preemptive problem it is shown that the First In First Out (FIFO) heuristic has a tight worst-case performance of 3 - 2/m, when jobs processing times and costs are set as in some optimal preemptive schedule.
TL;DR: The performance of parallel implementation measured in terms of speedup and efficiency factors is found to be good and the influence of consideration of terminal resistance on pressure and velocity waveforms have been analyzed.
Abstract: In this study, parallel computation of blood flow in a 1-D model of human arterial network has been carried out employing a Taylor Galerkin Finite Element Method Message passing interface libraries have been used on Origin 2000 SGI machine A Greedy strategy for load-distribution has been devised and data-flow graphs necessary for parallelization have been generated The performance of parallel implementation measured in terms of speedup and efficiency factors is found to be good Further, the parallel code is used in simulating the propagation of pressure and velocity waveforms in our 1-D arterial model for two different inflow pressure pulses Also, the influence of consideration of terminal resistance on pressure and velocity waveforms have been analyzed
TL;DR: An explicit one-step method that can be used for solving stiff problems and is viewed as a modification of the explicit Euler method that allows to reduce the stiffness in some sense.
Abstract: In this paper we introduce an explicit one-step method that can be used for solving stiff problems. This method can be viewed as a modification of the explicit Euler method that allows to reduce the stiffness in some sense. Some numerical experiments on linear stiff problems and on the Van der Pol's equation show the effectiveness of the method.