Showing papers in "Advances in Computational Mathematics in 1997"

Regularity of multiwavelets

Abstract: The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets. The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Holder regularity in arbitrary L p spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Holder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields.

Spectral factorization of Laurent polynomials

Abstract: We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly influenced by the variation in magnitude of the coefficients of the Laurent polynomial, by the closeness of the zeros of this polynomial to the unit circle, and by the spacing of these zeros.

The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes

Abstract: This paper continues the theme of the recent work [Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal., to appear] and further develops the Petrov–Galerkin method for Fredholm integral equations of the second kind. Specifically, we study wavelet Petrov–Galerkin schemes based on discontinuous orthogonal multiwavelets and prove that the condition number of the coefficient matrix for the linear system obtained from the wavelet Petrov–Galerkin scheme is bounded. In addition, we propose a truncation strategy which forms a basis for fast wavelet algorithms and analyze the order of convergence and computational complexity of these algorithms.

Parallel Linear System Solvers for Runge-Kutta Methods

Abstract: If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form I-A ⊗ hJ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form I - B ⊗ hJ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LU-decompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor s3 . It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode.

Parallel methods for ODEs

Abstract: In recent years there has been considerable interest in exploiting parallel computers for the efficient solution of ordinary differential equations. Because initial value problems are by their very nature sequential it can require considerable ingenuity to construct and implement efficient methods which take advantage of parallelism (see, for example, the monograph of Burrage (Oxford University Press, 1995)). Although it is not possible to cover all of these new approaches in one special issue, this issue does attempt to convey the flavour of this research across a wide variety of techniques. Parallel techniques for solving differential systems such as

Numerical treatment of retarded differential–algebraic equations by collocation methods

Abstract: This paper investigates retarded differential–algebraic equations of index zero to two with state-dependent delay. The theory needed to understand the numerical approach and analyze the numerical treatment by collocation methods is developed. Different strategies for tracking the jump discontinuities are considered and numerical examples are presented to support the convergence results.

A new theoretically motivated higher order upwind scheme on unstructured grids of simplices

Abstract: Many approaches exist to define a cell-centered upwind finite volume scheme of higher order on an unstructured grid of simplices. However, real theoretical motivation in the form of a convergence result does not exist for these approaches. Furthermore, some theoretical results of convergence exist for higher order finite volume methods, where no description of the numerical implementation is given to realize the necessary requirements for the convergence theory. Therefore we present in this paper a new limiter function which is motivated by these requirements and ensures a convergent scheme in the theoretical context: The approximated solution converges to the entropy solution in the case of scalar conservation laws in two space dimensions. This new limiter function is combined with a typical class of reconstruction functions very efficiently, which is illustrated by several test examples for scalar conservation laws as well as systems of such laws. In connection with the requirements to be fulfilled, a proof of a maximum principle of the finite volume scheme applied to simplices and dual cells is given. So for the approach of the higher order upwind finite volume scheme on dual cells, as used in several papers, a missing proof is now given. The ideas in this proof are also applied to the discontinuous Galerkin method, so that an existing maximum principle can be improved considerably. The main advantage comes from the fact that no requirements on the discretization of the domain are necessary: no B-triangulations or Delaunay triangulation are needed.

Stepsize selection for tolerance proportionality in explicit Runge-Kutta codes

Abstract: The potential for adaptive explicit Runge–Kutta (ERK) codes to produce global errors that decrease linearly as a function of the error tolerance is studied. It is shown that this desirable property may not hold, in general, if the leading term of the locally computed error estimate passes through zero. However, it is also shown that certain methods are insensitive to a vanishing leading term. Moreover, a new stepchanging policy is introduced that, at negligible extra cost, ensures a robust global error behaviour. The results are supported by theoretical and numerical analysis on widely used formulas and test problems. Overall, the modified stepchanging strategy allows a strong guarantee to be attached to the complete numerical process.

Parallel iterated methods based on multistep Runge-Kutta methods of Radau type

Abstract: This paper investigates iterated Multistep Runge-Kutta methods of Radau type as a class of explicit methods suitable for parallel implementation. Using the idea of van der Houwen and Sommeijer [18], the method is designed in such a way that the right-hand side evaluations can be computed in parallel. We use stepsize control and variable order based on iterated approximation of the solution. A code is developed and its performance is compared with codes based on iterated Runge-Kutta methods of Gauss type and various Dormand and Prince pairs [15]. The accuracy of some of our methods are comparable with the PIRK10 methods of van der Houwen and Sommeijer [18], but require fewer processors. In addition at very stringent tolerances these new methods are competitive with RK78 pairs in a sequential implementation.

Order and stability of parallel methods for stiff problems

Abstract: This paper explores methods of DIMSIM structure, with coefficient matrix of the form A = λ I, allowing the stages to be evaluated in parallel. It is found that many of these methods possess a strong A-stable property making them suitable for the solution of large stiff problems.

DIMSEMs — Diagonally Implicit Single-Eigenvalue Methods for the Numerical Solution of Stiff ODEs on Parallel Computers

Abstract: This paper derives a new class of general linear methods (GLMs) intended for the solution of stiff ordinary differential equations (ODEs) on parallel computers. Although GLMs were introduced by Butcher in the 1960s, the task of deriving formulas from the class with properties suitable for specific applications is far from complete. This paper is a contribution to that work. Our new methods have several properties suited for the solution of stiff ODEs on parallel computers. They are strictly diagonally implicit, allowing parallelism in the Newton iteration used to solve the nonlinear equations arising from the implicitness of the formula. The stability matrix has no spurious eigenvalues (that is, only one eigenvalue of the stability matrix is non-zero), resulting in a solution free from contamination from spurious solutions corresponding to non-dominant, non-zero eigenvalues. From these two properties arises the name DIMSEM, for Diagonally IMplicit Single-Eigenvalue Method. The methods have high stage order, avoiding the phenomenon of order reduction that occurs, for example, with some Runge-Kutta methods. The methods are L-stable, with the result that the chosen stepsize is dictated by convergence requirements rather than stability considerations imposed by the stiffness of the problem. An introduction to GLMs is given and some order barriers for DIMSEMs are presented. DIMSEMs of orders 2-6 are derived, as well as an L-stable class of diagonal methods of all orders which do not, however, possess the single-eigenvalue property. A fixed-order, variable-stepsize implementation of the DIMSEMs is described, including the derivation of local error estimators, and the results of testing on both sequential and parallel computers is presented. The testing shows the DIMSEMs to be competitive with fixed-order versions of the popular solver LSODE on a practical test problem.

Discrete qualocation methods for logarithmic-kernel integral equations on a piecewise smooth boundary

Abstract: We consider a fully discrete qualocation method for Symm’s integral equation. The method is that of Sloan and Burn (1992), for which a complete analysis is available in the case of smooth curves. The convergence for smooth curves can be improved by a subtraction of singularity (Jeon and Kimn, 1996). In this paper we extend these results for smooth boundaries to polygonal boundaries. The analysis uses a mesh grading transformation method for Symm’s integral equation, as in Elschner and Graham (1995) and Elschner and Stephan (1996), to overcome the singular behavior of solutions at corners.

Interior estimates for a low order finite element method for the Reissner–Mindlin plate model

Abstract: Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner–Mindlin plates It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method As an application, we show that for the soft simply supported plate, the Arnold–Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer Numerical results are given which support our conclusion

Parallel ODE solvers based on block BVMs

Abstract: In this paper we deal with Boundary Value Methods (BVMs), which are methods recently introduced for the numerical approximation of initial value problems for ODEs. Such methods, based on linear multistep formulae (LMF), overcome the stability limitations due to the well-known Dahlquist barriers, and have been the subject of much research in the last years. This has led to the definition of a new stability framework, which generalizes the one stated by Dahlquist for LMF. Moreover, several aspects have been investigated, including the efficient stepsize control [17,25,26] and the application of the methods for approximating different kinds of problems such as BVPs, PDEs and DAEs [7,23,41]. Furthermore, a block version of such methods, recently proposed for approximating Hamiltonian problems [24], is able to provide an efficient parallel solver for ODE systems [3].

Spectral properties of matrix continuous refinement operators

Abstract: A complete description of the spectrum of the matrix form of the continuous refinement operators on a subspace of compactly supported functions in L p (ℝ d ) is given. Properties of the compactly supported solutions of matrix refinement equations are derived from the spectral properties of the corresponding operators.

Parallel iterated method based on multistep Runge-Kutta of Radau type for stiff problems

Abstract: Research on parallel iterated methods based on Runge-Kutta formulas both for stiff and non-stiff problems has been pioneered by van der Houwen et al., for example see [8-11]. Burrage and Suhartanto have adopted their ideas and generalized their work to methods based on Multistep Runge-Kutta of Radau type [2] for non-stiff problems. In this paper we discuss our methods for stiff problems and study their performance.

A parallel stiff ODE solver based on MIRKs

Abstract: A parallel, "across the method" implementation of a stiff ODE solver is presented. The construction of the methods is outlined and the main implementational issues are discussed. Performance results, using MPI on the IBM SP-2, are presented and they indicate that a speed-up between 3 and 5 typically can be obtained compared to state-of-the-art sequential codes.

Sensitivity analysis of the differential matrix Riccati equation based on the associated linear differential system

Abstract: Perturbation bounds are given for the solution of the nth order differential matrix Riccati equation using the associated linear 2nth order differential system. The new bounds are alternative to those existing in the literature and are sharper in some cases.

Remarks on the optimal convolution kernel for CSOR waveform relaxation

Abstract: The convolution SOR waveform relaxation method is a numerical method for solving large-scale systems of ordinary differential equations on parallel computers. It is similar in spirit to the SOR acceleration method for solving linear systems of algebraic equations, but replaces the multiplication with an overrelaxation parameter by a convolution with a time-dependent overrelaxation function. Its convergence depends strongly on the particular choice of this function. In this paper, an analytic expression is presented for the optimal continuous-time convolution kernel and its relation to the optimal kernel for the discrete-time iteration is derived. We investigate whether this analytic expression can be used in actual computations. Also, the validity of the formulae that are currently used to determine the optimal continuous-time and discrete-time kernels is extended towards a larger class of ODE systems.

The compass (star) identity for vector-valued rational interpolants

Abstract: The compass identity (Wynn's five point star identity) for Pade approximants connects neighbouring elements called N, S, E, W and C in the Pade table. Its form has been extended to the cases of rational interpolation of ordinary (scalar) data and interpolation of vector-valued data. In this paper, full specifications of the associated five point identity for the scalar denominator polynomials and the vector numerator polynomials of the vector-valued rational interpolants on real data points are given, as well as the related generalisations of Frobenius' identities. Unique minimal forms of the polynomials constituting the interpolants and results about unattainable points correspond closely to their counterparts in the scalar case.

Error analysis of mixed finite elements for cylindrical shells

Abstract: In this paper, based on the Naghdi shell model, we analyze the uniform convergence of mixed finite element methods for cylindrical shell problems using macroelement techniques We show that Taylor–Hood elements p 2-P 1 and P 1 iso P 2 are locking free elements for the model problems Optimal error estimates are presented with these elements

Waveform relaxation methods for implicit differential equations

Abstract: We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying Runge-Kutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initial-value problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.

Adaptive finite element for semi-linear convection–diffusion problems

Abstract: In this article a strategy of adaptive finite element for semi-linear problems, based on minimizing a residual-type estimator, is reported. We get an a posteriori error estimate which is asymptotically exact when the mesh size h tends to zero. By considering a model problem, the quality of this estimator is checked. It is numerically shown that without constraint on the mesh size h, the efficiency of the a posteriori error estimate can fail dramatically. This phenomenon is analysed and an algorithm which equidistributes the local estimators under the constraint h ⩽ h max is proposed. This algorithm allows to improve the computed solution for semi-linear convection–diffusion problems, and can be used for stabilizing the Lagrange finite element method for linear convection–diffusion problems.

Distances between oriented curves in geometric modeling

Abstract: We consider the choice of a functional to measure the distance between two parametric curves. We identify properties of such a distance functional that are important for geometric design. Several popular definitions of distance are examined, and new functionals are presented which satisfy the desired properties.