Topic
Piecewise
About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.
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TL;DR: This paper rewrites some of the conditions on the Hermite derivatives that are sufficient for a piecewise bicubic function to be monotonic and presents a much simpler five-step algorithm for satisfying them that produces a visually pleasing monotone interpolant.
Abstract: This paper describes an algorithm for monotone interpolation to monotone data on a rectangular mesh by piecewise bicubic functions. In [SIAM J. Numer. Anal. 22 (1985), pp. 386–400] the authors developed conditions on the Hermite derivatives that are sufficient for such a function to be monotonic. The present paper rewrites some of these conditions and presents a much simpler five-step algorithm for satisfying them that produces a visually pleasing monotone interpolant. The result of the algorithm does not depend on the order of the independent variables nor on whether the inequalities are swept left-to-right or right-to-left.
111 citations
01 Jan 2005
110 citations
TL;DR: Novel PWC denoising methods are introduced, and comparisons between these methods performed on synthetic and real signals are compared, showing that the new understanding of the problem gained in part I leads to new methods that have a useful role to play.
Abstract: Removing noise from piecewise constant (PWC) signals is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need to be separated into stratigraphic zones, and in biophysics, jumps between molecular dwell states have to be extracted from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited. This paper (part I, the first of two) shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following and coordinate descent. In the second paper, part II, we introduce novel PWC denoising methods, and comparisons between these methods performed on synthetic and real signals, showing that the new understanding of the problem gained in part I leads to new methods that have a useful role to play.
110 citations
TL;DR: A time-varying positive definite Lyapunov function is used to develop a novel piecewise continuous model-reference adaptive law and a dwell-time switching law for adaptive asymptotic tracking control of uncertain switched linear systems.
Abstract: This technical note establishes a novel result for adaptive asymptotic tracking control of uncertain switched linear systems. The result exploits a recently proposed stability condition for switched systems. In particular, a time-varying positive definite Lyapunov function is used to develop a novel piecewise continuous model-reference adaptive law and a dwell-time switching law. In contrast with previous research, where asymptotic tracking was possible only in the presence of a common Lyapunov function for the reference models, in this work asymptotic tracking is shown in a more general setting. Additionally, in the presence of persistence of excitation, the controller parameter estimation errors will converge to zero asymptotically. The main contribution of this work consists in establishing a symmetry between adaptive control of classical non-switched linear systems and adaptive control of switched linear systems. A practical example with an electro-hydraulic system is adopted to illustrate the results.
110 citations
TL;DR: Baladi et al. as discussed by the authors showed that the average of a smooth function with respect to the SRB measure is not always Lipschitz (Baladi 2007 Commun. Math. Phys. 275 839?59, Mazzolena 2007 Master's Thesis Rome 2, Tor Vergata).
Abstract: The average of a smooth function with respect to the SRB measure ?t of a smooth one-parameter family ft of piecewise expanding interval maps is not always Lipschitz (Baladi 2007 Commun. Math. Phys. 275 839?59, Mazzolena 2007 Master's Thesis Rome 2, Tor Vergata). We prove that if ft is tangent to the topological class of f, and if ?t ft|t = 0 = X f, then is differentiable at zero, and coincides with the resummation proposed (Baladi 2007) of the (a priori divergent) series given by Ruelle's conjecture. In fact, we show that t ?t is differentiable within Radon measures. Linear response is violated if and only if ft is transversal to the topological class of f.
110 citations