Degree of Adaptive Approximation

DeVore, Ronald A.; Yu, Xiang

doi:10.2307/2008437

Cited by 16 publications

(32 citation statements)

References 5 publications

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“…This section is devoted to the proof of Theorem 3.5, that extends the approximation result of DeVore and Yu [14] (Sect. 3) to the approximation of R r -valued functions defined on {1, .…”

Section: Proof Of the Approximation Results Over Besov Bodiesmentioning

confidence: 63%

“…6.2), which is not the case anymore for r ≥ 2. We first describe the approximation algorithm adapted from [14]. Then, we give the main lines of the proof and also demonstrate the key result, which is a direct consequence of the approximation algorithm.…”

Section: Proof Of the Approximation Results Over Besov Bodiesmentioning

confidence: 99%

“…, n} by piecewise constant functions. For r = 1, that extension simply results from [14] (Cor. 3.2) and Proposition 6.5 (cf.…”

Section: Proof Of the Approximation Results Over Besov Bodiesmentioning

confidence: 99%

“…Thus, we will refer to our estimator as the d-estimator. The collection of linear spaces we consider has been chosen for its potential qualities of approximation, as suggested by approximation results for real-valued functions due to DeVore and Yu [14] and DeVore (cf. [6]).…”

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confidence: 99%

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Estimating a discrete distributionviahistogram selection

Akakpo

2011

ESAIM: PS

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Abstract. Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.Mathematics Subject Classification. 62G05, 62C20, 41A17.

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“…This section is devoted to the proof of Theorem 3.5, that extends the approximation result of DeVore and Yu [14] (Sect. 3) to the approximation of R r -valued functions defined on {1, .…”

Section: Proof Of the Approximation Results Over Besov Bodiesmentioning

confidence: 63%

Section: Proof Of the Approximation Results Over Besov Bodiesmentioning

confidence: 99%

“…, n} by piecewise constant functions. For r = 1, that extension simply results from [14] (Cor. 3.2) and Proposition 6.5 (cf.…”

Section: Proof Of the Approximation Results Over Besov Bodiesmentioning

confidence: 99%

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confidence: 99%

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Estimating a discrete distributionviahistogram selection

Akakpo

2011

ESAIM: PS

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“…The following result for i = 1 is a special case of the Jackson estimates from [3]. The case i = 0 is given in [15]. Such estimates for one-dimensional problems can be found in [13, p. 102].…”

Section: Mixed Norm Estimatesmentioning

confidence: 98%

Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

et al. 2003

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Abstract. We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form h α u W s,p (Ω) for positive and negative s and α, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N , the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results-previously known only for quasi-uniform meshes-to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.

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Sequential function approximation for the solution of differential equations

Meade

Kokkolaras

Zeldin

1997

Commun. Numer. Meth. Engng.

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SUMMARYA computational method for the solution of dierential equations is proposed. With this method an accurate approximation is built by incremental additions of optimal local basis functions. The parallel direct search software package (PDS), that supports parallel objective function evaluations, is used to solve the associated optimization problem eciently. The advantage of the method is that, although it resembles adaptive methods in computational mechanics, an a priori grid is not necessary. Moreover, the traditional matrix construction and evaluations are avoided. Computational cost is reduced while eciency is enhanced by the low-dimensional parallel-executed optimization and parallel function evaluations. In addition, the method should be applicable to a broad class of interpolation functions. Results and global convergence rates obtained for one-and two-dimensional boundary value problems are satisfactorily compared to those obtained by the conventional Galerkin ®nite element method. #

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Degree of Adaptive Approximation

Abstract: Abstract.We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation.

Cited by 16 publications

References 5 publications

Estimating a discrete distributionviahistogram selection

Estimating a discrete distributionviahistogram selection

Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

Sequential function approximation for the solution of differential equations

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