scispace - formally typeset
Search or ask a question
Topic

Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.


Papers
More filters
Journal ArticleDOI
TL;DR: A computation scheme based directly on the dynamic programming formulation is proposed, a time-sharing computer program is discussed, and the results of an example problem are presented.
Abstract: A method for determining where to locate the inspection stations in a multistage production process with imperfect inspection is presented. Dynamic programming is used to establish that the optimal expected total cost function at every stage is piecewise linear and concave. While the optimal policy at every stage usually consists of one “inspect” region and one “do not inspect” region, this policy structure is found not to hold in general. A computation scheme based directly on the dynamic programming formulation is proposed, a time-sharing computer program is discussed, and the results of an example problem are presented.

82 citations

Journal ArticleDOI
TL;DR: The diffusion coefficient obtained shows a fractal structure as a function of the slope of the map, which may be typical for a wide class of maps, such as two-dimensional sawtooth maps.
Abstract: We consider chains of one-dimensional, piecewise linear, chaotic maps with uniform slope. We study the diffusive behavior of an initially nonuniform distribution of points as a function of the slope of the map by solving the Frobenius-Perron equation. For Markov partition values of the slope, we relate the diffusion coefficient to eigenvalues of the topological transition matrix. The diffusion coefficient obtained shows a fractal structure as a function of the slope of the map. This result may be typical for a wide class of maps, such as two-dimensional sawtooth maps.

82 citations

Journal ArticleDOI
TL;DR: A survey of the known approximation properties of the outputs of neural networks with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines is presented in this paper.
Abstract: Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing theoretical explanation for this success is still lacking. Since these applications revolve around approximating an unknown function from data observations, part of the answer must involve the ability of NNs to produce accurate approximations. This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion, i.e. error versus the number of parameters used to create the approximant. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation, and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward. The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of f into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parametrized nonlinear manifold. It is shown that this manifold has certain space-filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates the challenge to the numerical method of finding best or good parameter choices when trying to approximate.

82 citations

Journal ArticleDOI
TL;DR: This paper gets quantitative estimates for the convergence to equilibrium, in terms of the W1 Wasserstein coupling distance, for the TCP window size process and also for its embedded chain.

81 citations

Journal ArticleDOI
TL;DR: An algorithm is presented that computes feedback controls for a kinematic point robot in an arbitrary dimensional space with piecewise linear boundary and requires minimal preprocessing of the environment and is extremely fast during execution.
Abstract: This paper presents a novel approach to computing feedback laws in the presence of obstacles. Instead of computing a trajectory between a pair of initial and goal states, our algorithms compute a vector field over the entire state space; all trajectories obtained from following this vector field are guaranteed to asymptotically reach the goal state. As a result, the vector field globally solves the navigation problem and provides robustness to disturbances in sensing and control. The vector field's integral curves (system trajectories) are guaranteed to avoid obstacles and are C∞ smooth. We construct a vector field with these properties by partitioning the space into simple cells, defining local vector fields for each cell, and smoothly interpolating between them to obtain a global vector field. We present an algorithm that computes these feedback controls for a kinematic point robot in an arbitrary dimensional space with piecewise linear boundary; the algorithm requires minimal preprocessing of the environment and is extremely fast during execution. For many practical applications in two-dimensional environments, full computation can be done in milliseconds. We also present an algorithm for computing feedback laws over cylindrical algebraic decompositions, thereby solving a smooth feedback version of the generalized piano movers' problem.

81 citations


Network Information
Related Topics (5)
Nonlinear system
208.1K papers, 4M citations
89% related
Linear system
59.5K papers, 1.4M citations
88% related
Optimization problem
96.4K papers, 2.1M citations
87% related
Robustness (computer science)
94.7K papers, 1.6M citations
86% related
Differential equation
88K papers, 2M citations
86% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023179
2022377
2021312
2020353
2019329
2018297