Applied Numerical Analysis & Computational Mathematics最新文献

Exponentially fitted versions of the extended one-step methods for first-order ODEs are constructed following the six-step flow chart introduced by Ixaru and Vanden Berghe in Exponential fitting, (Mathematics and Its Application 568, Kluwer Academic Publishers, 2004) and their properties are examined. It is shown how formulae for optimal frequencies can be constructed. Some numerical experiments illustrate the use of the developed algorithms. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We consider a phase space stability error control for numerical simulation of dynamical systems. We illustrate how variable time-stepping algorithms perform poorly for long time computations which pass close to a fixed point. A new error control was introduced in [9], which is a generalization of the error control first proposed in [8]. In this error control, the local truncation error at each step is bounded by a fraction of the solution arc length over the corresponding time interval. We show how this error control can be thought of either a phase space or a stability error control. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In particular, we analyze the forward Euler method applied to the linear system whose coefficient matrix has real and negative eigenvalues. We also consider the dynamics in the neighborhood of saddle points. We introduce a step-size selection scheme which allows this error control to be incorporated within the standard adaptive algorithm as an extra constraint at negligible extra computational cost. Theoretical and numerical results are presented to illustrate the behavior of this error control. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The Nelder–Mead or simplex search algorithm is one of the best known algorithms for unconstrained optimization of non–smooth functions. Even though the basic algorithm is quite simple, it is implemented in many different ways. Apart from some minor computational details, the main difference between various implementations lies in the selection of convergence (or termination) tests, which are used to break the iteration process. A fairly simple efficiency analysis of each iteration step reveals a potential computational bottleneck in the domain convergence test. To be efficient, such a test has to be sublinear in the number of vertices of the working simplex. We have tested some of the most common implementations of the Nelder–Mead algorithm, and none of them is efficient in this sense.

Therefore, we propose a simple and efficient domain convergence test and discuss some of its properties. This test is based on tracking the volume of the working simplex throughout the iterations. Similar termination tests can also be applied in some other simplex–based direct search methods. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, we present a method based on an optimal control technique for numerical computations of geodesic paths between two fixed points of a Riemannian Manifold under the assumption of existence. In this method, the control variable is the tangent vector to the geodesic we are looking for. Defining a cost function corresponding to the requested control, we explain how to derive the optimal control algorithm by the use of an adjoint state method for the calculation of the gradient of that cost function. We then give a geometrical interpretation of the adjoint state. After having introduced the discrete optimal control algorithm, we show an application to wooden roof design. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The logarithmic matrix norm with respect to l₂ matrix norm will be used to investigate mean square stability for a class of second-order weak schemes when applied to 2-dimensional linear stochastic differential systems with one multiplicative noise. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We analyze a class of preconditioners for the mortar method, based on substructuring. After splitting in a suitable way the degrees of freedom in interior, edge and vertex, we study the performance of a block Jacobi type preconditioner for which the condition number of the preconditioned matrix only grows polylogarithmically. Unlike the previous work by Achdou, Maday and Widlund [1], which is restricted to the case of first order finite element, this paper relies on abstract assumptions and therefore applies to finite element of any order. Moreover, the use of a suitable coarse preconditioner (whose effect we analyze) makes this technique more efficient. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper an exponential fitted partitioned linear multistep method is developed for the long term integration of the N-body problem. The new method integrates exactly any linear combination of the functions 1, x, x², x³ ,…, x^2k–1, exp(±wx) for the coordinates and any linear combination of the functions 1, x, x², x³,…, x^2k–3, exp(±wx) for the velocities. Numerical results are produced and compared with a set of well known symplectic and single step methods. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The region occupied by and the contour of a physical object in 3-dimensional space are in a way dual or interchangeable characteristics of the object: the contour is the region's boundary and the region is contained inside the contour. In the same way the characterization of the object's contour by its Fourier descriptors, and the reconstruction of its region from the object's multidimensional moments, are also dual problems. While both problems are well-understood in two dimensions, the complexity increases tremendously when moving to the three-dimensional world.

In Section 2 we discuss how the latest techniques allow to reconstruct an object's shape from the knowledge of its moments. For 2D significantly different techniques must be used, compared to the general 3D case. In Section 3, the parameterization of a 2D contour onto a unit circle and a 3D surface onto a unit sphere is described. Furthermore, the theory of Fourier descriptors for 2D shape representation and the extension to 3D shape analysis are discussed.

The reader familiar with the use of either Fourier descriptors or moments as shape descriptors of physical objects may find the comparative discussion in the concluding section interesting. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We propose a highly flexible sequential methodology for the experimental analysis of optimization algorithms. The proposed technique employs computational statistic methods to investigate the interactions among optimization problems, algorithms, and environments. The workings of the proposed technique are illustrated on the parameterization and comparison of both a population–based and a direct search algorithm, on a well–known benchmark problem, as well as on a simplified model of a real–world problem. Experimental results are reported and conclusions are derived. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Given a family of curves or surfaces in R^s, an important problem is that of finding a member of the family which gives a “best” fit to m given data points. There are many application areas, for example metrology, computer graphics, pattern recognition, and the most commonly used criterion is the least squares norm. However, there may be wild points in the data, and a more robust estimator such as the l₁ norm may be more appropriate. On the other hand, the object of modelling the data may be to assess the quality of a manufactured part, so that accept/reject decisions may be required, and this suggests the use of the Chebyshev norm.

We consider here the use of the l₁ and l_∞ norms in the context of fitting to data curves and surfaces defined parametrically. There are different ways to formulate the problems, and we review here formulations, theory and methods which generalize in a natural way those available for least squares. As well as considering methods which apply in general, some attention is given to a fundamental fitting problem, that of lines in three dimensions. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)