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Operator Equations, Multiscale Concepts and Complexity
 LECTURES IN APPLIED MATHEMATICS
, 1996
"... In this paper, we review several recent developments centering upon the application of multiscale basis methods for the numerical solution of operator equations with special emphasis on complexity questions. In particular, issues like preconditioning, matrix compression, construction of special wav ..."
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In this paper, we review several recent developments centering upon the application of multiscale basis methods for the numerical solution of operator equations with special emphasis on complexity questions. In particular, issues like preconditioning, matrix compression, construction of special wavelet bases and adapted error estimators are addressed.
Nonlinear Approximation and the Space BV(R²)
 AMER. J. MATH
"... Given a function f 2 L 2 (Q), Q := [0; 1)² and a real number t ? 0, let U(f; t) := inf g2BV(Q) kf \Gamma gk 2 L 2 (I) + t VQ (g); where the infimum is taken over all functions g 2 BV of bounded variation on I. This and related extremal problems arise in several areas of mathematics such as interpo ..."
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Cited by 8 (1 self)
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Given a function f 2 L 2 (Q), Q := [0; 1)² and a real number t ? 0, let U(f; t) := inf g2BV(Q) kf \Gamma gk 2 L 2 (I) + t VQ (g); where the infimum is taken over all functions g 2 BV of bounded variation on I. This and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation, as well as in digital image processing. Techniques for finding minimizers g for U(f; t) based on variational calculus and nonlinear partial differential equations have been put forward by several authors ([DMS], [LOR], [MS], [CL]). The main disadvantage of these approaches is that they are numerically intensive. On the other hand, it is wellknown that more elementary methods based on wavelet shrinkage solve related extremal problems, for example, the above problem with BV replaced by the Besov space B 1 1 (L 1 (I)) (see e.g. [CDLL]). However, since BV has no simple description in terms of wavelet coefficients, it is not clear that minimi...
Approximation of functions, in
 Proc. Symp. Applied Mathematics
, 1986
"... using scattered shifts of a multivariate ..."
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Applied and Computational Aspects of Nonlinear Wavelet Approximation
, 2001
"... Nonlinear approximation has recently found computational applications such as data compression, statistical estimation or adaptive schemes for partial differential or integral equations, especially through the development of waveletbased methods. The goal of this paper is to provide with a short su ..."
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Nonlinear approximation has recently found computational applications such as data compression, statistical estimation or adaptive schemes for partial differential or integral equations, especially through the development of waveletbased methods. The goal of this paper is to provide with a short survey of nonlinear wavelet approximation in the perspective of these applications, as well as to stress some remaining open questions. 1. Introduction Numerous problems of approximation theory have in common the following general setting: we are given a family of subspaces (SN ) N0 of a normed space X, and for f 2 X, we consider the best approximation error oe N (f) := inf g2SN kf \Gamma gkX : (1) Typically, N represents the number of parameters needed to describe an element in SN , and in most cases of interest, oe N (f) goes to zero as this number tends to infinity. For a given f , we can then study the rate of approximation, i.e. the range of r 0 for which there exists C ? 0 such th...
Adaptive and anisotropic piecewise polynomial approximation, chapter 4 of the book Multiscale, Nonlinear and Adaptive Approximation
, 2009
"... We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given function f which is approximated. We focus our discussion on (i) ..."
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We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given function f which is approximated. We focus our discussion on (i) the properties that describe an optimal partition for f, (ii) the smoothness properties of f that govern the rate of convergence of the approximation in the Lpnorms, and (iii) fast refinement algorithms that generate near optimal partitions. While these results constitute a fairly established theory in the univariate case and in the multivariate case when dealing with elements of isotropic shape, the approximation theory for adaptive and anisotropic elements is still building up. We put a particular emphasis on some recent results obtained in this direction. 1
The Brouwer Lecture 2005 : Statistical estimation with model selection
, 2006
"... The purpose of this paper is to explain the interest and importance of (approximate) models and model selection in Statistics. Starting from the very elementary example of histograms we present a general notion of finite dimensional model for statistical estimation and we explain what type of risk b ..."
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The purpose of this paper is to explain the interest and importance of (approximate) models and model selection in Statistics. Starting from the very elementary example of histograms we present a general notion of finite dimensional model for statistical estimation and we explain what type of risk bounds can be expected from the use of one such model. We then give the performance of suitable model selection procedures from a family of such models. We illustrate our point of view by two main examples: the choice of a partition for designing a histogram from an nsample and the problem of variable selection in the context of Gaussian regression.
Nonlinear approximation and the space BV (R 2
 184 R R Coifman and D L Donoho. Translation invariant denoising. In Wavelets and Statistics : Springer Lecture Notes in Statistics 103
, 1999
"... Abstract. Given a function f 2 L2(Q), Q: = [0, 1) 2 and a real number t> 0, let U ( f, t):= inf g2BV (Q) kf, gk 2 L 2(I) + t VQ (g), where the infimum is taken over all functions g 2 BV of bounded variation on I. This and related extremal problems arise in several areas of mathematics such as int ..."
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Abstract. Given a function f 2 L2(Q), Q: = [0, 1) 2 and a real number t> 0, let U ( f, t):= inf g2BV (Q) kf, gk 2 L 2(I) + t VQ (g), where the infimum is taken over all functions g 2 BV of bounded variation on I. This and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation, as well as in digital image processing. Techniques for finding minimizers g for U ( f, t) based on variational calculus and nonlinear partial differential equations have been put forward by several authors [DMS], [RO], [MS], [CL]. The main disadvantage of these approaches is that they are numerically intensive. On the other hand, it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems, for example, the above problem with BV replaced by the Besov space B 1 1 (L1(I)) (see e.g. [CDLL]). However, since BV has no simple description in terms of wavelet coefficients, it is not clear that minimizers for U ( f, t) can be realized in this way. We shall show in this paper that simple methods based on Haar thresholding provide near minimizers for U ( f, t). Our analysis of this extremal problem brings forward many interesting relations between Haar decompositions and the space BV. 1. Introduction. Nonlinear
Adaptive Approximation of Curves ∗
, 2004
"... We propose adaptive multiscale refinement algorithms for approximating and encoding curves by polygonal curves. We establish rates of approximation of these algorithms in the Hausdorff metric. For example, we show that under the mere assumption that the original curve has finite length then the firs ..."
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We propose adaptive multiscale refinement algorithms for approximating and encoding curves by polygonal curves. We establish rates of approximation of these algorithms in the Hausdorff metric. For example, we show that under the mere assumption that the original curve has finite length then the first of these algorithms gives a rate of convergence O(1/n) where n is the number of vertices of the approximating polygonal curve. Similar results giving second order approximation are proven under weak assumptions on the curvature such as Lp integrability, p> 1. Note that for nonadaptive algorithms, to obtain the same order of approximation would require that the curvature is bounded. Key Words: Polygonal approximation of planar curves, adaptive multiscale refinement, maximal functions, convergence rates, encoding
Detecting changepoints in a discrete distribution via model selection
, 2008
"... Abstract: This paper is concerned with the detection of multiple changepoints in the joint distribution of independent categorical variables. The procedures introduced rely on model selection and are based on a penalized leastsquares criterion. Their performance is assessed from a nonasymptotic poi ..."
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Abstract: This paper is concerned with the detection of multiple changepoints in the joint distribution of independent categorical variables. The procedures introduced rely on model selection and are based on a penalized leastsquares criterion. Their performance is assessed from a nonasymptotic point of view. Using a special collection of models, a preliminary estimator is built. According to an existing model selection theorem, it satisfies an oracletype inequality. Moreover, thanks to an approximation result demonstrated in this paper, it is also proved to be adaptive in the minimax sense. In order to eliminate some irrelevant changepoints selected by that first estimator, a twostage procedure is proposed, that also enjoys some adaptivity property. Besides, the first estimator can be computed with a complexity only linear in the size of the data. A heuristic method allows to implement the second procedure quite satisfactorily with the same computational complexity.