156 research outputs found
Collocation methods for a class of second order initial value problems with oscillatory solutions
We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given
On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems
Multi-frequency, highly-oscillatory Hamiltonian problems derive from the
mathematical modelling of many real life applications. We here propose a
variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to
efficiently deal with the numerical solution of such problems.Comment: 28 pages, 4 figures (a few typos fixed
Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems
On computers, discrete problems are solved instead of continuous ones. One
must be sure that the solutions of the former problems, obtained in real time
(i.e., when the stepsize h is not infinitesimal) are good approximations of the
solutions of the latter ones. However, since the discrete world is much richer
than the continuous one (the latter being a limit case of the former), the
classical definitions and techniques, devised to analyze the behaviors of
continuous problems, are often insufficient to handle the discrete case, and
new specific tools are needed. Often, the insistence in following a path
already traced in the continuous setting, has caused waste of time and efforts,
whereas new specific tools have solved the problems both more easily and
elegantly. In this paper we survey three of the main difficulties encountered
in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed
Decentralized Weakly Convex Optimization Over the Stiefel Manifold
We focus on a class of non-smooth optimization problems over the Stiefel
manifold in the decentralized setting, where a connected network of agents
cooperatively minimize a finite-sum objective function with each component
being weakly convex in the ambient Euclidean space. Such optimization problems,
albeit frequently encountered in applications, are quite challenging due to
their non-smoothness and non-convexity. To tackle them, we propose an iterative
method called the decentralized Riemannian subgradient method (DRSM). The
global convergence and an iteration complexity of for forcing a natural stationarity measure below
are established via the powerful tool of proximal smoothness from
variational analysis, which could be of independent interest. Besides, we show
the local linear convergence of the DRSM using geometrically diminishing
stepsizes when the problem at hand further possesses a sharpness property.
Numerical experiments are conducted to corroborate our theoretical findings.Comment: 27 pages, 6 figures, 1 tabl
The application of generalized, cyclic, and modified numerical integration algorithms to problems of satellite orbit computation
Generalized, cyclic, and modified multistep numerical integration methods are developed and evaluated for application to problems of satellite orbit computation. Generalized methods are compared with the presently utilized Cowell methods; new cyclic methods are developed for special second-order differential equations; and several modified methods are developed and applied to orbit computation problems. Special computer programs were written to generate coefficients for these methods, and subroutines were written which allow use of these methods with NASA's GEOSTAR computer program
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