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: A novel unifying algorithmic framework, dynamo (dynamic optimisation platform) is introduced designed to provide the quantum-technology community with a convenient matlab-based toolset for optimal control, and gives researchers in optimal-control techniques a framework for benchmarking and comparing new proposed algorithms to the state-of-the-art.
Abstract: For paving the way to novel applications in quantum simulation, computation, and technology, increasingly large quantum systems have to be steered with high precision. It is a typical task amenable to numerical optimal control to turn the time course of pulses, i.e., piecewise constant control amplitudes, iteratively into an optimized shape. Here, we present a comparative study of optimal-control algorithms for a wide range of finite-dimensional applications. We focus on the most commonly used algorithms: GRAPE methods which update all controls concurrently, and Krotov-type methods which do so sequentially. Guidelines for their use are given and open research questions are pointed out. Moreover, we introduce a unifying algorithmic framework, DYNAMO (dynamic optimization platform), designed to provide the quantum-technology community with a convenient matlab-based tool set for optimal control. In addition, it gives researchers in optimal-control techniques a framework for benchmarking and comparing newly proposed algorithms with the state of the art. It allows a mix-and-match approach with various types of gradients, update and step-size methods as well as subspace choices. Open-source code including examples is made available at http://qlib.info.
237 citations
01 Jun 1999
TL;DR: The novelty of the approach lies in the use of inter-image homographies to validate and best estimate the plane, and in the minimal initialization requirements-only a single 3D line with a textured neighbourhood is required to generate a plane hypothesis.
Abstract: A new method is described for automatically reconstructing 3D planar faces from multiple images of a scene. The novelty of the approach lies in the use of inter-image homographies to validate and best estimate the plane, and in the minimal initialization requirements-only a single 3D line with a textured neighbourhood is required to generate a plane hypothesis. The planar facets enable line grouping and also the construction of parts of the wireframe which were missed due to the inevitable shortcomings of feature detection and matching. The method allows a piecewise planar model of a scene to be built completely automatically, with no user intervention at any stage, given only the images and camera projection matrices as input. The robustness and reliability of the method are illustrated on several examples, from both aerial and interior views.
237 citations
01 Jul 1991
TL;DR: New techniques for interactive piecewise flattening of parametric 3-D surfaces, leading to a non-distorted, hence realistic, texture mapping, based on results from differential geometry.
Abstract: This paper introduces new techniques for interactive piecewise flattening of parametric 3-D surfaces, leading to a non-distorted, hence realistic, texture mapping. Cuts are allowed on the mapped texture and we make a compromise between discontinuities and distortions. These techniques are based on results from differential geometry, more precisely on the notion of "geodesic curvature": isoparametric curves of the surface are mapped, in a constructive way, onto curves in the texture plane with preservation of geodesic curvature at each point. As an application, we give a concrete example which is a first step towards an efficient and robust CAD tool for shoe modeling.
236 citations
TL;DR: Given a continuous-time control law that guarantees global uniform asymptotic tracking of the desired trajectory, this algorithm provides an event-based controller that not only guarantees uniform ultimate boundedness of the tracking error, but also ensures non-accumulation of inter-execution times.
Abstract: In this paper we study an event based control algorithm for trajectory tracking in nonlinear systems. The desired trajectory is modelled as the solution of a reference system with an exogenous input and it is assumed that the desired trajectory and the exogenous input to the reference system are uniformly bounded. Given a continuous-time control law that guarantees global uniform asymptotic tracking of the desired trajectory, our algorithm provides an event based controller that not only guarantees uniform ultimate boundedness of the tracking error, but also ensures non-accumulation of inter-execution times. In the case that the derivative of the exogenous input to the reference system is also uniformly bounded, an arbitrarily small ultimate bound can be designed. If the exogenous input to the reference system is piecewise continuous and not differentiable everywhere then the achievable ultimate bound is constrained and the result is local, though with a known region of attraction. The main ideas in the paper are illustrated through simulations of trajectory tracking by a nonlinear system.
235 citations
TL;DR: It is shown that under some circumstances, finite-time blow-up of solutions is possible and in other circumstances, the solutions will decay to a spatially constant solution (collapse).
Abstract: We investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks. We show that under some circumstances, finite-time blow-up of solutions is possible. In other circumstances, the solutions will decay to a spatially constant solution (collapse). We also give some intuitive arguments which demonstrate the possibility of the existence of aggregation (piecewise constant) solutions.
235 citations