A Local Convergence Proof for the Minvar Algorithm for Computing Continuous Piecewise Linear Approximations

Groff, Richard E.; Khargonekar, Pramod P.; Koditschek, Daniel E.

doi:10.1137/s0036142902402213

Cited by 6 publications

(6 citation statements)

References 48 publications

(52 reference statements)

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“…The representation and approximation problems of piecewise linear functions have been studied in a variety of fields such as numerical analysis, system modelling, circuit synthesis, nonlinear identification and control, as examples of [1][2][3][4][5][6][7] and so on. In [8] we proposed a derivation of piecewise linear differential equations from nonlinear systems, with multiplication terms by using identities of products.…”

Section: Introductionmentioning

confidence: 99%

A constructive enclosure approximation of a continuous function of many variables by piecewise linear functions

Okazaki

Nakano

2016

NOLTA

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Abstract:In this paper, for contributing to solve the problems of one-sided, or two-sided approximation, a constructive enclosure approximation (having representable lower and upper bounds) of a continuous function of many variables by piecewise linear functions is provided. A theorem of a representation like Kolmogorov's superposition theorem of continuous functions of many variables is provided, and a theorem of a constructive piecewise linear enclosure of a continuous function of many variables is also provided. By using an implementation of the theorems on Maple, the examples of piecewise linear enclosures of a polynomial and a transcendental function of three variables are obtained.

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Section: Introductionmentioning

confidence: 99%

A constructive enclosure approximation of a continuous function of many variables by piecewise linear functions

Okazaki

Nakano

2016

NOLTA

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“…Although the concept of PWA systems was initially researched in the late 1940's, the first optimal algorithm to approximate nonlinearities with PWA functions appears, to the best of our knowledge, in 1970's by Cantoni [5] and Tomek [6]. Many different attempts have been made to produce suitable PWA models in references [7], [1], [8], [9], and [4]. Reference [7] addresses PWA approximation of continuous functions using uniform simplicial partitions.…”

Section: Introductionmentioning

confidence: 99%

Intersection-based piecewise affine approximation of nonlinear systems

Zavieh

Rodrigues

2013

21st Mediterranean Conference on Control and Automation

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This paper presents a new algorithm for PWA approximation of nonlinear systems. Such an approximation is very important to enable a reduction in the complexity of models of nonlinear systems while keeping the global validity of the models. The paper builds on previous work on piecewise affine (PWA) approximation methods, in particular on the work done by Casselman and Rodrigues, known as the Set of Linearization Points (SLP) PWA approximation. The proposed extension method can be used to approximate any continuous function of one variable by a PWA function. The algorithm is based on the points at which the linearization lines intersect with each other. The method assumes that a desired approximation error and one linearization point are given. The algorithm, then performs several linearizations. It is shown that the new linearization points are optimal in the sense of decreasing the error between the exact function and the approximation. The main advantages of this methodology compared to previous approaches are the reduction of the number of pieces of the PWA function, the guarantee that the approximation is continuous, and that the derivative of the approximation and the derivative of the exact function are equal at all linearization points. A detailed collection of examples from different fields of study highlight the effectiveness and the flexibility of the proposed method. It is shown that the proposed method compares favorably with other methods.

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“…Any point in the domain that is not a vertex is a unique convex combination of the vertices and its value is calculated accordingly. This approach was recently used for estimating continuous piecewise linear functions from data in Groff et al (2003) and the Ph.D. dissertation of Groff (2003). In these works, the authors introduce a novel non-linear algorithm called Minvar for the parametric estimation of continuous piecewise linear functions from data.…”

Section: Introductionmentioning

confidence: 99%

Riesz estimators

Aliprantis¹,

Harris²,

Tourky³

2007

Journal of Econometrics

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We consider properties of estimators that can be written as vector lattice (Riesz space) operations. Using techniques widely used in economic theory and functional analysis, we study the approximation properties of these estimators paying special attention to additive models. We also provide two algorithms RIESZVAR(i-ii) for the consistent parametric estimation of continuous multivariate piecewise linear functions. r 2005 Elsevier B.V. All rights reserved. JEL classification: C13; C14

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A Local Convergence Proof for the Minvar Algorithm for Computing Continuous Piecewise Linear Approximations

Cited by 6 publications

References 48 publications

A constructive enclosure approximation of a continuous function of many variables by piecewise linear functions

A constructive enclosure approximation of a continuous function of many variables by piecewise linear functions

Intersection-based piecewise affine approximation of nonlinear systems

Riesz estimators

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