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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Papers
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TL;DR: Adaptive Finite Element Methods (AFEM) are numerical proce- dures that approximate the solution to a partial differential equation (PDE) by piecewise polynomials on adaptively generated triangulations as mentioned in this paper.
Abstract: Adaptive Finite Element Methods (AFEM) are numerical proce- dures that approximate the solution to a partial differential equation (PDE) by piecewise polynomials on adaptively generated triangulations. Only re- cently has any analysis of the convergence of these methods (10, 13) or their rates of convergence (2) become available. In the latter paper it is shown that a certain AFEM for solving Laplace's equation on a polygonal domain ⊂ R 2 based on newest vertex bisection has an optimal rate of convergence in the following sense. If, for some s > 0 and for each n = 1,2, . . ., the solu- tion u can be approximated in the energy norm to order O(n s ) by piecewise linear functions on a partition P obtained from n newest vertex bisections, then the adaptively generated solution will also use O(n) subdivisions (and floating point computations) and have the same rate of convergence. The question arises whether the class of functions A s with this approximation rate can be described by classical measures of smoothness. The purpose of the present paper is to describe such approximation classes A s by Besov smoothness.
89 citations
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01 Jan 2020TL;DR: It is demonstrated that, given the ability to sample from the locally reachable subset of the configuration space with positive probability, the first asymptotically optimal algorithm for motion planning problems with piecewise-analytic differential constraints can construct random geometric graphs that contain optimal plans with probability one in the limit of infinite samples.
Abstract: We present the first asymptotically optimal algorithm for motion planning problems with piecewise-analytic differential constraints, like manipulation or rearrangement planning. This class of problems is characterized by the presence of differential constraints that are local in nature: a robot can only move an object once the object has been grasped. These constraints are not analytic and thus cannot be addressed by standard differentially constrained planning algorithms. We demonstrate that, given the ability to sample from the locally reachable subset of the configuration space with positive probability, we can construct random geometric graphs that contain optimal plans with probability one in the limit of infinite samples. This approach does not require a hand-coded symbolic abstraction. We demonstrate our approach in simulation on a simple manipulation planning problem, and show it generates lower-cost plans than a sequential task and motion planner.
89 citations
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TL;DR: In this paper, the authors give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions, based on a theorem of Hofbauer.
Abstract: Generalizing a theorem ofHofbauer (1979), we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. Using these results we prove:
(1)
IfT is anS-unimodal map with an attracting invariant Cantor set, then ∫log|T′|dμ=0 for the unique invariant measure μ on the Cantor set.
(2)
IfT is piecewise invertible, iff is the Radon-Nikodym derivative ofT with respect to a σ-finite measurem, if logf has bounded distortion underT, and if μ is an ergodicT-invariant measure satisfying a certain lower estimate for its entropy, then μ≪m iffhμ (T)=Σlogf dμ.
89 citations
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TL;DR: This paper presents local and global sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains, and n-dimensional Diracs and convex polytopes, using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions).
Abstract: In this paper, we consider the problem of sampling signals that are nonband-limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI). We know that the classical ("band-limited sine") sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band-limited. However, the recent results on FRI sampling suggest that it is possible to sample and perfectly reconstruct such nonband-limited signals using a rich class of kernels. In this paper, we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples. Using these moments and the annihilating filter method (Prony's method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional-derivatives-based) and global (complex-moments-based, Radon-transform-based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains (e.g., circles, ellipses, and cardioids), 2D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This work has been explored in a promising way in super-resolution algorithms and distributed compression, and might find its applications in photogrammetry, computer graphics, and machine vision.
89 citations
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TL;DR: A delay-dependent stochastic stability condition is derived for discrete-time singular Markov jump systems with time-varying delay and time- varying transition probabilities, formulated by linear matrix inequalities and thus can be checked easily.
Abstract: This paper concerns the stochastic stability analysis for discrete-time singular Markov jump systems with time-varying delay and time-varying transition probabilities. The time-varying transition probabilities in the underlying systems are assumed to be finite piecewise-constant. Based on the delay partitioning technique, a delay-dependent stochastic stability condition is derived for these systems, which is formulated by linear matrix inequalities and thus can be checked easily. Some special cases are also considered. Finally, two numerical examples are provided to demonstrate the application and less conservativeness of the developed approaches.
89 citations