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: The local dynamics of all possible two-folds in three dimensions are derived, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but nondeterministic dynamics.
Abstract: A vector field is piecewise smooth if its value jumps across a hypersurface, and a two-fold singularity is a point where the flow is tangent to the hypersurface from both sides. Two-folds are generic in piecewise smooth systems of three or more dimensions. We derive the local dynamics of all possible two-folds in three dimensions, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but nondeterministic dynamics. In cases where the flow passes through the two-fold, upon reaching the singularity it is unique in neither forward nor backward time, meaning the causal link between inward and outward dynamics is severed. In one scenario this occurs recurrently. The resulting flow makes repeated, but nonperiodic, excursions from the singularity, whose path and amplitude is not determined by previous excursions. We show that this behavior is robust and has many of the properties associated with chaos. Local geometry reveals that the chaotic behavior can be eliminated by varying a single parameter: the angular jump of the vector field across the two-fold.
88 citations
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TL;DR: This paper presents an algorithm for finding an arbitrarily close arc spline approximation of a smooth curve, a tangent continuous piecewise curve made of circular arcs and straight line segments.
88 citations
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TL;DR: A nonlinear multiresolution scheme within Harten's framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique, which shows promising results in terms of convergence, smoothness, and stability.
Abstract: A nonlinear multiresolution scheme within Harten's framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.
88 citations
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TL;DR: Quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size and the approximation order are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices.
Abstract: We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation.
88 citations
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TL;DR: The basic idea is that waiting long enough a stable model will converge towards its equilibrium and, therefore, the Lyapunov functions under consideration are not necessarily decreasing at every sample, but are allowed to decrease every k samples.
88 citations