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: Collocation with piecewise polynomial functions is developed as a method for solving two-point boundary value problems in this paper, and convergence is shown for a general class of linear problems and a rather broad class of nonlinear problems.
Abstract: Collocation with piecewise polynomial functions is developed as a method for solving two-point boundary value problems. Convergence is shown for a general class of linear problems and a rather broad class of nonlinear problems. Some computational examples are presented to illustrate the wide applicability and efficiency of the procedure.
309 citations
TL;DR: It is proved that one cannot approximate a general function f∈Eβ(Rd) using neural networks that are less complex than those produced by the construction, which partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions.
307 citations
TL;DR: A segmentation algorithm based on sequential optimization which produces a hierarchical decomposition of the picture that can be viewed as a tree, where the nodes correspond to picture segments and where links between nodes indicate set inclusions.
Abstract: A segmentation algorithm based on sequential optimization which produces a hierarchical decomposition of the picture is presented. The decomposition is data driven with no restriction on segment shapes. It can be viewed as a tree, where the nodes correspond to picture segments and where links between nodes indicate set inclusions. Picture segmentation is first regarded as a problem of piecewise picture approximation, which consists of finding the partition with the minimum approximation error. Then, picture segmentation is presented as an hypothesis-testing process which merges only segments that belong to the same region. A hierarchical decomposition constraint is used in both cases, which results in the same stepwise optimization algorithm. At each iteration, the two most similar segments are merged by optimizing a stepwise criterion. The algorithm is used to segment a remote-sensing picture, and illustrate the hierarchical structure of the picture. >
305 citations
TL;DR: Based on a piecewise Lyapunov function combined with some novel convexifying techniques, the solutions to the problem of robust H∞ output-feedback control for a class of nonlinear systems under unreliable communication links are formulated in the form of linear matrix inequalities.
Abstract: This paper investigates the problem of robust H∞ output-feedback control for a class of nonlinear systems under unreliable communication links. The nonlinear plant is represented by a Takagi-Sugeno (T-S) uncertain fuzzy model, and the communication links between the plant and controller are assumed to be imperfect, i.e., data-packet dropouts occur intermittently, which is often the case in a network environment. Stochastic variables that satisfy the Bernoulli random-binary distribution are adopted to characterize the data-missing phenomenon, and the attention is focused on the design of a piecewise static-output-feedback (SOF) controller such that the closed-loop system is stochastically stable with a guaranteed H∞ performance. Based on a piecewise Lyapunov function combined with some novel convexifying techniques, the solutions to the problem are formulated in the form of linear matrix inequalities (LMIs). Finally, simulation examples are also provided to illustrate the effectiveness of the proposed approaches.
303 citations
TL;DR: In this article, it was shown that the error in an interior domain 2 can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error of a weaker norm over a slightly larger domain which measures the effects from outside of the domain Q.
Abstract: Interior a priori error estimates in Sobolev norms are derived from interior RitzGalerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain 2 can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain Q. Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given. 0. Introduction. There are presently many methods which are available for computing approximate solutions of elliptic boundary value problems which may be classified as Ritz-Galerkin type methods. Many of these methods differ from each other in some respects (for example, in how they treat the boundary conditions) but have much in common in that they have what may be called "interior Ritz-Galerkin equations" which are the same. Here we shall be concerned with finding interior estimates for the rate of convergence for such a class of methods which are consequences of these interior equations. Let us briefly describe, in a special case, the type of question we wish to consider. Let &2 be a bounded domain in RN with boundary M2 and consider, for simplicity, the problem of finding an approximate solution of a boundary value problem (0.1) \u =f in Q2, (0.2) Au= g on U2, where A is some boundary operator. Suppose now that we are given a one-parameter family of finite-dimensional subspaces Sh (0 < h < 1) of an appropriate Hilbert space in which u lies and that, for each h, we have computed an approximate solution Uh c Sh to u using some Ritz-Galerkin type method. Here we have in mind, for example, methods such as the "engineer's" finite element method [8], [22], the Aubin-Babuska penalty method [2], [4], the methods of Nitsche [12], [13] or the Received October 15, 1973. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. Copyright i 1974, American Mathematical Society
302 citations