Showing papers presented at "International Conference on Numerical Analysis and Its Applications in 2000"
11 Jun 2000
TL;DR: A nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes is proposed and Numerical results are given to illustrate the effectiveness of this method.
Abstract: We consider the solution of ordinary differential equations (ODEs) using implicit linear multistep formulae (LMF). More precisely, here we consider Boundary Value Methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. I n [6], Chan et al. proposed using Strang block-circulant preconditioners for solving these linear systems. However, as observed in [1], Strang preconditioners can be often ill-conditioned or singular even when the given system is well-conditioned. In this paper, we propose a nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes. Numerical results are given to illustrate the effectiveness of our method.
21 citations
11 Jun 2000
TL;DR: HOC (High Order Compact) compact monotone finite difference schemes, defined on a priori Shishkin meshes, uniformly convergent with respect the diffusion parameter ?
Abstract: The central difference scheme for reaction-diffusion problems, when fitted Shishkin type meshes are used, gives uniformly convergent methods of almost second order. In this work, we construct HOC (High Order Compact) compact monotone finite difference schemes, defined on a priori Shishkin meshes, uniformly convergent with respect the diffusion parameter ?, which have order three and four except for a logarithmic factor. We show some numerical experiments which support the theoretical results.
20 citations
11 Jun 2000
TL;DR: The general nonlinear matrix equation X + A* X-n A = I is discussed and two methods for iterative computing a positive definite solution are investigated.
Abstract: The general nonlinear matrix equation X + A* X-n A = I is discussed (n is a positive integer). Some necessary and sufficient conditions for existence a solution are given. Two methods for iterative computing a positive definite solution are investigated. Numerical experiments to illustrate the performance of the methods are reported.
19 citations
11 Jun 2000
TL;DR: Finite-difference schemes approximating the one-dimensional initial-boundary value problems for the heat equation with concentrated capacity are derived and an abstract operator's method is developed for studying such problems.
Abstract: In this paper finite-difference schemes approximating the one-dimensional initial-boundary value problems for the heat equation with concentrated capacity are derived. An abstract operator's method is developed for studying such problems. Convergence rate estimates consistent with the smoothness of the data are obtained.
15 citations
11 Jun 2000
TL;DR: It is shown, theoretically and through numerical tests, that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the corresponding Monte Carlo methods.
Abstract: The convergence of Monte Carlo method for numerical integration can often be improved by replacing pseudorandom numbers (PRNs) with more uniformly distributed numbers known as quasirandom numbers (QRNs). Standard Monte Carlo methods use pseudorandom sequences and provide a convergence rate of O(N-1/2) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)k) N-1).In this paper we study the possibility of using QRNs for computing matrix-vector products, solving systems of linear algebraic equations and calculating the extreme eigenvalues of matrices. Several algorithms using the same Markov chains with different random variables are described. We have shown, theoretically and through numerical tests, that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the corresponding Monte Carlo methods. Numerical tests are performed on sparse matrices using PRNs and Sobol, Halton, and Faure QRNs.
13 citations
11 Jun 2000
TL;DR: The implementation and performance of a novel fill-minimization ordering technique for sparse LU factorization with partial pivoting based on a nested-dissection ordering of AT A, which reduces the LU running time on some very large matrices by more than a factor of 2.
Abstract: We describe the implementation and performance of a novel fill-minimization ordering technique for sparse LU factorization with partial pivoting. The technique was proposed by Gilbert and Schreiber in 1980 but never implemented and tested. Like other techniques for ordering sparse matrices for LU with partial pivoting, our new method preorders the columns of the matrix (the row permutation is chosen by the pivoting sequence during the numerical factorization). Also like other methods, the column permutation Q that we select is a permutation that minimizes the fill in the Cholesky factor of QT AT AQ. Unlike existing column-ordering techniques, which all rely on minimum-degree heuristics, our new method is based on a nested-dissection ordering of AT A. Our algorithm, however, never computes a representation of ATA, which can be expensive. We only work with a representation of A itself. Our experiments demonstrate that the method is efficient and that it can reduce fill significanly relative to the best existing methods. The method reduces the LU running time on some very large matrices (tens of millions of nonzeros in the factors) by more than a factor of 2.
13 citations
11 Jun 2000
TL;DR: A soft rock model is presented which is capable of taking into account the rate sensitivity, time effects and creep rupture, and to extend the finite element code Lagamine (MSM-ULg) for viscoplastic problems in geomechanics.
Abstract: A constitutive model is developed within the framework of Perzyna's viscoplasticity for predicting the stress-strain-time behaviour of soft porous rocks. The model is based on the hyperelasticity and multisurface viscoplasticity with hardening. A time-stepping algorithm is presented for integrating the creep sensitive law. An example of application to one-dimensional consolidation is presented. The objectives are to: 1. present a soft rock model which is capable of taking into account the rate sensitivity, time effects and creep rupture; 2. to discuss the use of an incremental procedure for time stepping using large time increments and 3. to extend the finite element code Lagamine (MSM-ULg) for viscoplastic problems in geomechanics.
11 citations
11 Jun 2000
TL;DR: This paper turns to an application of security pricing where the pros and cons of the different generators of Monte Carlo methods are investigated, and assesses the potentials of parallel MC in the computational finance framework.
Abstract: Monte Carlo (MC) methods have proved to be flexible, robust and very useful techniques in computational finance. Several studies have investigated ways to achieve greater efficiency of such methods for serial computers. In this paper, we concentrate on the parallelization potentials of the MC methods. While MC is generally thought to be "embarrassingly parallel", the results eventually depend on the quality of the underlying parallel pseudo-random number generators. There are several methods for obtaining pseudo-random numbers on a parallel computer and we briefly present some alternatives. Then, we turn to an application of security pricing where we empirically investigate the pros and cons of the different generators. This also allows us to assess the potentials of parallel MC in the computational finance framework.
11 citations
11 Jun 2000
TL;DR: Numerical experiments show that parallel implicit Runge-Kutta methods can attain speed-ups close to the ideal values when the initial value problem is stiff and has between ten and hundred equations.
Abstract: We describe a Maple package named D-NODE (Distributed Numerical solver for ODEs), implementing a number of difference methods for initial value problems. The distribution of the computational effort follows the idea of parallelism across method. We have benchmark the package in a cluster environment. Distributed Maple ensures the inter-processor communications. Numerical experiments show that parallel implicit Runge-Kutta methods can attain speed-ups close to the ideal values when the initial value problem is stiff and has between ten and hundred equations. The stage equations of the implicit methods are solved on different processors using Maple's facilities.
10 citations
11 Jun 2000
TL;DR: Estimates for the rate of convergence in discrete energetic Sobolev's norms compatible with the smoothness of the solution are presented.
Abstract: We first discuss the difficulties that arise at the construction of difference schemes on uniform meshes for a specific elliptic interface problem. Estimates for the rate of convergence in discrete energetic Sobolev's norms compatible with the smoothness of the solution are also presented.
10 citations
11 Jun 2000
TL;DR: Different preconditioning techniques of both implicit and explicit form are considered in connection with Krylov methods for the solution of large dense complex symmetric non-Hermitian systems of equations arising in computational electromagnetics, focusing on sparse approximate inverse techniques that use a static nonzero pattern selection.
Abstract: We consider different preconditioning techniques of both implicit and explicit form in connection with Krylov methods for the solution of large dense complex symmetric non-Hermitian systems of equations arising in computational electromagnetics. We emphasize in particular sparse approximate inverse techniques that use a static nonzero pattern selection. By exploiting geometric information from the underlying meshes, a very sparse but effective preconditioner can be computed. In particular our strategies are applicable when fast multipole methods are used for the matrix-vector products on parallel distributed memory computers.
11 Jun 2000
TL;DR: A new quadratically convergent algorithm is proposed, having a low computational cost per step and good numerical stability properties, that allows the computation of the maximal solutions of the matrix equations.
Abstract: We propose a new quadratically convergent algorithm, having a low computational cost per step and good numerical stability properties, that allows the computation of the maximal solutions of the matrix equations X + C* X-1 C = Q, X - C* X-1 C = Q, X + C* (R + B* X B)-1 C = Q. The algorithm is based on the cyclic reduction method.
11 Jun 2000
TL;DR: The authors prove the stability property of such implementation of GSA provided the hyperbolic transformations are performed in factored form, as shown in this paper.
Abstract: The generalized Schur algorithm (GSA) is a fast method to compute the Cholesky factorization of a wide variety of structured matrices. The stability property of the GSA depends on the way it is implemented. In [15] GSA was shown to be as stable as the Schur algorithm, provided one hyperbolic rotation in factored form [3] is performed at each iteration. Fast and efficient algorithms for solving Structured Total Least Squares problems [14,15] are based on a particular implementation of GSA requiring two hyperbolic transformations at each iteration. In this paper the authors prove the stability property of such implementation provided the hyperbolic transformations are performed in factored form [3].
11 Jun 2000
TL;DR: Various variants of Schwarz methods for a singularly perturbed two dimensional stationary convection-diffusion problem are constructed and analysed.
Abstract: Various variants of Schwarz methods for a singularly perturbed two dimensional stationary convection-diffusion problem are constructed and analysed. The iteration counts, the errors in the discrete solutions and the convergence behaviour of the numerical solutions are analysed in terms of their dependence on the singular perturbation parameter of the Schwarz methods. Conditions for the methods to converge parameter uniformly and for the number of iterations to be independent of the perturbation parameter are discussed.
11 Jun 2000
TL;DR: This paper considers the first initial-boundary value problem for the heat equation with variable coeficients in the domain ( 0, 1)2 × (0, T] and assumes that the solution of the problem and the coefficients of equation belong to the corresponding anisotropic Sobolev spaces.
Abstract: In this paper we consider the first initial-boundary value problem for the heat equation with variable coeficients in the domain (0, 1)2 × (0, T]. We assume that the solution of the problem and the coefficients of equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimates consistent with the smoothness of the data are obtained.
11 Jun 2000
TL;DR: In this article, necessary and sufficient conditions for solvability of the matrix equation sinh X = A in the complex and real cases were given and some algorithms for computing one of these solutions were presented.
Abstract: We give necessary and sufficient conditions for solvability of the matrix equation sinh X = A in the complex and real cases and present some algorithms for computing one of these solutions. The numerical features of the algorithms are analysed along with some numerical tests.
11 Jun 2000
TL;DR: A subspace linesearch strategy for the globalization of Newton-GMRES method is proposed, which is the simple and inexpensive way to determine descent directions in the low-dimensional subspaces generated by GMRES.
Abstract: A subspace linesearch strategy for the globalization of Newton-GMRES method is proposed. The main feature of our proposal is the simple and inexpensive way we determine descent directions in the low-dimensional subspaces generated by GMRES. Global and local quadratic convergence is established under standard assumptions.
11 Jun 2000
TL;DR: The paper presents a recursive version of LU factorization algorithm for general matrices using FORTRAN 90 and some modification of the LAPACK algorithms and they transform some basic linear algebra operations from BLAS level 2 toBLAS level 3.
Abstract: The effective use of the cache memories of the processors is a key component of obtaining high performance algorithms and codes, including here algorithms and codes for parallel computers with shared and distributed memories. The recursive algorithms seem to be a tool for such an action. Unfortunately, worldwide used programming language FORTRAN 77 does not allow explicit recursion.The paper presents a recursive version of LU factorization algorithm for general matrices using FORTRAN 90. FORTRAN 90 allows writing recursive procedures and the recursion is automatic as it is a duty of the compiler. Usually, recursion speeds up the algorithms. The recursive versions reported in the paper are some modification of the LAPACK algorithms and they transform some basic linear algebra operations from BLAS level 2 to BLAS level 3.
11 Jun 2000
TL;DR: New 6-th and 8-th order explicit symplectic Runge-Kutta-Nystrom methods for Hamiltonian systems which are more efficient than other previously known algorithms are presented.
Abstract: We present new 6-th and 8-th order explicit symplectic Runge-Kutta-Nystrom methods for Hamiltonian systems which are more efficient than other previously known algorithms. The methods use the processing technique and non-trivial flows associated with different elements of the Lie algebra involved in the problem. Both the processor and the kernel are compositions of explicitly computable maps.
11 Jun 2000
TL;DR: A singularly perturbed time-dependent convection-diffusion problem is examined on non-rectangular domains and the numerical solutions are shown to converge independently of the singular perturbation parameter.
Abstract: A singularly perturbed time-dependent convection-diffusion problem is examined on non-rectangular domains The nature of the boundary and interior layers that arise depends on the geometry of the domains For problems with different types of layers, various numerical methods are constructed to resolve the layers in the solutions and the numerical solutions are shown to converge independently of the singular perturbation parameter
11 Jun 2000
TL;DR: This paper studies the application of preconditioned conjugate gradient methods in high-resolution color image reconstruction problems and uses the weighted H1 regularization functional in the Tikhonov regularization to capture the changes of reflectivity across color channels.
Abstract: This paper studies the application of preconditioned conjugate gradient methods in high-resolution color image reconstruction problems. The high-resolution color images are reconstructed from multiple undersampled, shifted, degraded color frames with subpixel displacements. The resulting degradation matrices are spatially variant. To capture the changes of reflectivity across color channels, the weighted H1 regularization functional is used in the Tikhonov regularization. The Neumann boundary condition is also employed to reduce the boundary artifacts. The preconditioners are derived by taking the cosine transform approximation of the degradation matrices. Numerical examples are given to illustrate the fast convergence of the preconditioned conjugate gradient method.
11 Jun 2000
TL;DR: It was shown that the preconditioners are nonsingular when the BVM is A?, µ-?-stable, and the eigenvalues of preconditionsed matrices are clustered, so the number of iterations for solving the precONDitioned systems by the GMRES method is bounded by a constant that is independent of the discretization mesh.
Abstract: We consider linear constant coefficient differential-algebraic equations (DAEs) Ax?(t) + Bx(t) = f(t) where A, B are square matrices and A is singular. If det(?A + B) with ? ? C is not identically zero, the system of DAEs is solvable and can be separated into two uncoupled subsystems. One of them can be solved analytically and the other one is a system of ordinary differential equations (ODEs). We discretize the ODEs by boundary value methods (BVMs) and solve the linear system by using the generalized minimal residual (GMRES) method with Strang-type block-circulant preconditioners. It was shown that the preconditioners are nonsingular when the BVM is A?, µ-?-stable, and the eigenvalues of preconditioned matrices are clustered. Therefore, the number of iterations for solving the preconditioned systems by the GMRES method is bounded by a constant that is independent of the discretization mesh. Numerical results are also given.
11 Jun 2000
TL;DR: In the present paper some simplifications in numerical model are performed and the finite element method is applied for solid and shell model of pelvic bone.
Abstract: Numerical modeling of human pelvic bone makes possibilities to determine the stress and strain distribution in bone tissue. The general problems are: complex geometry, material structure and boundary conditions. In the present paper some simplifications in numerical model are performed. Homogeneous elastic properties of bone tissue are assumed. The shell model and solid model of pelvic bone are analyzed. The finite element method is applied. Some numerical results for solid and shell model are presented.
11 Jun 2000
TL;DR: This work presents numerical experiments obtained with the multigrid method for this class of linear systems and sees that modifying only the restriction operator in an appropriate form, the algorithm is convergent, the CPU time increases linearly with the discretization parameter and the number of cycles is independent of the mesh sizes.
Abstract: Solving the algebraic linear systems proceeding from the discretization on some condensed meshes of 2D singularly perturbed problems, is a difficult task. In this work we present numerical experiments obtained with the multigrid method for this class of linear systems. On Shishkin meshes, the classical multigrid algorithm is not convergent. We see that modifying only the restriction operator in an appropriate form, the algorithm is convergent, the CPU time increases linearly with the discretization parameter and the number of cycles is independent of the mesh sizes.
11 Jun 2000
TL;DR: To calculate distribution of displacements and stresses in planet wheels with cooperating elements FEM has been implemented and forces and strains calculated with FEM have been used to check the assumptions which have been applied only in analytical so far.
Abstract: Implementation of high speed engines requires application of high ratio mechanical gears. Relatively, the smallest mechanical gear is the cycloidal planetary gear known as Cyclo gear [2, 8- 11]. The complex construction of planet wheels in cycloidal planetary gear (Cyclo) practically makes impossible its optimal design. To calculate distribution of displacements and stresses in planet wheels with cooperating elements FEM has been implemented. There were series of numerical models of planet wheels generated and for example of real model of gear it has been calculated proper values of forces, strains and stresses. In the paper forces and strains calculated with FEM have been used to check the assumptions which have been applied only in analytical so far.
11 Jun 2000
TL;DR: The numerical results show that the derived theoretical estimates can be used to predict the behavior of a wide class of elliptic boundary value problems.
Abstract: In this work a grid free Monte Carlo algorithm for solving elliptic boundary value problems is investigated. The proposed Monte Carlo approach leads to a random process called a ball process.In order to generate random variables with the desired distribution, rejection techniques on two levels are used.Varied numerical tests on a Sun Ultra Enterprise 4000 with 14 Ultra-SPARC processors were performed. The code which implemented the new algorithm was written in JAVA.The numerical results show that the derived theoretical estimates can be used to predict the behavior of a wide class of elliptic boundary value problems.
11 Jun 2000
TL;DR: A generic alternative through the concept of random arithmetic is presented and two alternatives are developped around the well know Sylvester equations.
Abstract: Classical accuracy estimation in problem solving is basically based upon sensitivity analysis and conditionning computation. Such an approach is frequently much more difficult than solving the problem itself. Here a generic alternative through the concept of random arithmetic is presented. These two alternatives are developped around the well know Sylvester equations. Matlab implentation as a new object class is discussed and numerically illustrated.
11 Jun 2000
TL;DR: A new algorithm for solving Toeplitz linear least squares problems is developed and it is shown that the normwise backward stability is independent of the condition number of the ToePlitz matrix.
Abstract: We develop a new algorithm for solving Toeplitz linear least squares problems. The Toeplitz matrix is first embedded into a circulant matrix. The linear least squares problem is then transformed into a discrete least squares approximation problem for polynomial vectors. Our implementation shows that the normwise backward stability is independent of the condition number of the Toeplitz matrix.
11 Jun 2000
TL;DR: Matrices are investigated that are Hankel matrices in bases of orthogonal polynomials that are fast LU-factorization algorithms and superfast solvers are constructed.
Abstract: Matrices are investigated that are Hankel matrices in bases of orthogonal polynomials. With the help of 3 equivalent definitions of this class fast LU-factorization algorithms and superfast solvers are constructed.
11 Jun 2000
TL;DR: The classical Kantorovich theorem for Newton's method assumes that the derivative of the involved operator satisfies a Lipschitz condition, with a special emphasis in the center-Lipschitzer condition.
Abstract: The classical Kantorovich theorem for Newton's method assumes that the derivative of the involved operator satisfies a Lipschitz condition ?F? (x0-1 [F? (x) - F? (y)] ? ? L?x - y? In this communication, we analyse the different modifications of this condition, with a special emphasis in the center-Lipschitz condition: ?F? (x0)-1 [F? (x) - F? (x0] ? ? ?(?x - x0?) being ? a positive increasing real function and x0 the starting point for Newton's iteration