732 research outputs found
Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem
We compare the effectiveness of solving Dirichlet-Neumann problems via the
Craig-Sulem (CS) expansion, the Ablowitz-Fokas-Musslimani (AFM) implicit
formulation, the dual AFM formulation (AFM*), a boundary integral collocation
method (BIM), and the transformed field expansion (TFE) method. The first three
methods involve highly ill-conditioned intermediate calculations that we show
can be overcome using multiple-precision arithmetic. The latter two methods
avoid catastrophic cancellation of digits in intermediate results, and are much
better suited to numerical computation.
For the Craig-Sulem expansion, we explore the cancellation of terms at each
order (up to 150th) for three types of wave profiles, namely band-limited,
real-analytic, or smooth. For the AFM and AFM* methods, we present an example
in which representing the Dirichlet or Neumann data as a series using the AFM
basis functions is impossible, causing the methods to fail. The example
involves band-limited wave profiles of arbitrarily small amplitude, with
analytic Dirichlet data. We then show how to regularize the AFM and AFM*
methods by over-sampling the basis functions and using the singular value
decomposition or QR-factorization to orthogonalize them. Two additional
examples are used to compare all five methods in the context of water waves,
namely a large-amplitude standing wave in deep water, and a pair of interacting
traveling waves in finite depth.Comment: 31 pages, 18 figures. (change from version 1: corrected error in
table on page 12
Optimal realizations of floating-point implemented digital controllers with finite word length considerations.
The closed-loop stability issue of finite word length (FWL) realizations is
investigated for digital controllers implemented in floating-point arithmetic.
Unlike the existing methods which only address the effect of the mantissa bits
in floating-point implementation to the sensitivity of closed-loop stability,
the sensitivity of closed-loop stability is analysed with respect to both the
mantissa and exponent bits of floating-point implementation. A computationally
tractable FWL closed-loop stability measure is then defined, and the method of
computing the value of this measure is given. The optimal controller realization
problem is posed as searching for a floating-point realization that maximizes
the proposed FWL closed-loop stability measure, and a numerical optimization
technique is adopted to solve for the resulting optimization problem. Simulation
results show that the proposed design procedure yields computationally efficient
controller realizations with enhanced FWL closed-loop stability performance
Computation of the inverse Laplace Transform based on a Collocation method which uses only real values
We develop a numerical algorithm for inverting a Laplace transform (LT), based on Laguerre polynomial series expansion of the
inverse function under the assumption that the LT is known on the real axis only. The method belongs to the class of Collocation
methods (C-methods), and is applicable when the LT function is regular at infinity. Difficulties associated with these problems are due
to their intrinsic ill-posedness. The main contribution of this paper is to provide computable estimates of truncation, discretization,
conditioning and roundoff errors introduced by numerical computations. Moreover, we introduce the pseudoaccuracy which will be
used by the numerical algorithm in order to provide uniform scaled accuracy of the computed approximation for any x with respect
to ex . These estimates are then employed to dynamically truncate the series expansion. In other words, the number of the terms of
the series acts like the regularization parameter which provides the trade-off between errors.
With the aim to validate the reliability and usability of the algorithm experiments were carried out on several test functions
Finite Wordlength Controller Realizations using the Specialized Implicit Form
Une forme d'état implicite spécialisée est présentée pour étudier les effets de l'implantation en précision finie des régulateurs. Cette forme permet une description macroscopique des algorithmes à implanter. Elle constitue un canevas unificateur permettant de décrire les différentes structures utilisées pour l'implantation, telles que les réalisations avec l'opérateur delta, la forme directe II en rho, la forme d'état-observateur et bien d'autres formes qui sont d'habitude traitées séparément dans la littérature. Différentes mesures quantifiant les effets de l'implantation sur le comportement en boucle fermée sont définis dans ce contexte. Elles concernent aussi bien la stabilité que la performance. L'écart entre la réalisation à précision infinie et la réalisation à précision finie est évaluée selon la mesure de sensibilité des coefficients et la mesure du bruit de quantification. Le problème consistant à trouver une réalisation dont l'implantation amène un minimum de dégradation peut alors est résolut numériquement. Cette approche est illustrée avec deux exemples
- …