Global smoothness preservation and the variation-diminishing property

Cottin, Claudia; Gavrea, Ioan; Gonska, Heiner; Kacsó, Daniela

doi:10.1155/s1025583499000314

Cited by 11 publications

(13 citation statements)

References 4 publications

(11 reference statements)

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“…Non-negative linearly consistent approximants have long been studied in the literature [5]. These methods often present a number of attractive features, such as the related properties of monotonicity, the variation diminishing property (the approximation does not create extrema not present in the data), or smoothness preservation [16], of particular interest in the presence of shocks or sharp gradients. The nonoscillatory behavior for discontinuous data of the approximants presented here contrasts with the behavior of second-order triangular finite elements or second-order MLS approximants, as exemplified in Figure 1.…”

Section: Setupmentioning

confidence: 99%

Smooth, second order, non‐negative meshfree approximants selected by maximum entropy

Cyron

Arroyo

Ortiz

2009

Numerical Meth Engineering

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SUMMARYWe present a family of approximation schemes, which we refer to as second-order maximum-entropy (max-ent) approximation schemes, that extends the first-order local max-ent approximation schemes to second-order consistency. This method retains the fundamental properties of first-order max-ent schemes, namely the shape functions are smooth, non-negative, and satisfy a weak Kronecker-delta property at the boundary. This last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial. The evaluation of the shape functions is not explicit, but it is very efficient and robust. To our knowledge, the proposed method is the first higher-order scheme for function approximation from unstructured data in arbitrary dimensions with non-negative shape functions. As a consequence, the approximants exhibit variation diminishing properties, as well as an excellent behavior in structural vibrations problems as compared with the Lagrange finite elements, MLS-based meshfree methods and even B-Spline approximations, as shown through numerical experiments. When compared with usual MLS-based second-order meshfree methods, the shape functions presented here are much easier to integrate in a Galerkin approach, as illustrated by the standard benchmark problems. 1 Smooth, second order, non-negative meshfree approximants selected by maximum entropy, Cyron, ;Arroyo, M.; Ortiz, M.; International

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Section: Setupmentioning

confidence: 99%

Smooth, second order, non‐negative meshfree approximants selected by maximum entropy

Cyron

Arroyo

Ortiz

2009

Numerical Meth Engineering

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“…Recent examples include the natural element method shape functions [11] and subdivision schemes [12]. These methods often present a number of attractive features, such as the related properties of monotonicity, the variation diminishing property (the approximation is not more 'wiggly' than the data), or smoothness preservation [13], of particular interest in the presence of shocks. Furthermore, they lead to well behaved mass matrices.…”

Section: Introductionmentioning

confidence: 99%

Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methods

Arroyo

Ortiz

2005

Numerical Meth Engineering

338

654

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SUMMARYWe present a one-parameter family of approximation schemes, which we refer to as local maximumentropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum-entropy (max-ent) statistical inference. Local max-ent approximation schemes represent a compromise-in the sense of Pareto optimality-between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max-ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker-delta property at the boundary. Local max-ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max-ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max-ent approximation schemes is vastly superior to that of finite elements.

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“…Hence, the main purpose of this manuscript is to justify/modify our statement in [13] and show how (1.2) "can be derived from [4,8,9]" (note that [10] in our original statement is replaced by an earlier paper [8]) by constructing positive linear polynomial approximation operators that simultaneously preserve k-monotonicity for all k ≤ q and yield (1.2). Additionally, we make this paper self-contained and provide all proofs (except for some straightforward statements that can be verified directly and some classical properties of ultraspherical polynomials).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this section, we discuss several properties of the operator H n+2 that was introduced by Gavrea [9]. Everything here follows from [4,9], and we include this section in the current manuscript only for readers' convenience (we also somewhat clean up some of the proofs making them, in our opinion, more transparent by utilizing the notation (3.9) and Corollary 3.5). For any n ∈ N and a fixed (generating) polynomial P n (x) = n k=0 a k x k , Gavrea's operator H n+2 :…”

Section: Gavrea's Operatormentioning

confidence: 99%