Wavelet compression and nonlinearn-widths

DeVore, Ronald A.; Kyriazis, George; Leviatan, D.; Тихомиров, В. М.

doi:10.1007/bf02071385

Cited by 48 publications

(62 citation statements)

References 7 publications

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“…Important examples include rational approximation, approximation by splines with free knots, and wavelet compression. See [7,12,15,25,27,28]. A general problem is the approximation of f ʦ X by an arbitrary expression of the form…”

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confidence: 99%

On the Power of Adaption

Novak

1996

Journal of Complexity

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Optimal error bounds for adaptive and nonadaptive numerical methods are compared. Since the class of adaptive methods is much larger, a well-chosen adaptive method might seem to be better than any nonadaptive method. Nevertheless there are several results saying that under natural assumptions adaptive methods are not better than nonadaptive ones. There are also other results, however, saying that adaptive methods can be significantly better than nonadaptive ones as well as bounds on how much better they can be. It turns out that the answer to the ''adaption problem'' depends very much on what is known a priori about the problem in question; even a seemingly small change of the assumptions can lead to a different answer. © 1996 Academic Press, Inc. THE ADAPTION PROBLEMOne of the more controversial issues in numerical analysis concerns adaptive algorithms. The use of such algorithms is widespread and many people believe that well-chosen adaptive algorithms are much better than nonadaptive methods in most situations. Such a belief is usually based on numerical experimentation. In this paper we survey what is known theoretically regarding the power of adaption. We will present some results which state that under natural assumptions adaptive methods are not better than nonadaptive ones. There are also other results, however, saying that adaptive methods can be significantly superior to nonadaptive ones. As we will see, the power of adaption is critically dependent on our a priori knowledge concerning the problem being studied; even a seemingly small change in the assumptions can lead to a different answer. ERICH NOVAKLet us begin with some well-known examples. The bisection method and the Newton method for zero finding of a function are adaptive, since they compute a sequence (x n ) n of knots that depends on the function. The Gauss formula for numerical integration is nonadaptive since its knots and weights do not depend on the function.A nonadaptive method provides an immediate decomposition for parallel computation. If adaptive information is superior to nonadaptive information, then an analysis of the tradeoff between using adaptive or nonadaptive information on a parallel computer should be carried out.To formulate the adaption problem precisely, we need some definitions and notations. Many problems of numerical analysis can be described as computing an approximation of the value S( f ) of an operatorHere we assume that X is a normed space of functions and G is also a normed space. The operator S describes the solution of a mathematical problem, for example the solution of a boundary value problem or an integral equation. Also, numerical integration (with G ϭ R) and the recovery of functions (with an imbedding S ϭ id: X Ǟ L p , where X ʚ L p ) can be stated in this way. In many cases the space X is infinite dimensional and therefore f ʦ X cannot directly be an input of a computation. We usually replace S with a discretization method given, for example, by a finite element method. Accordingly, numerical methods a...

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confidence: 99%

On the Power of Adaption

Novak

1996

Journal of Complexity

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“…Interesting ideas concerning non-linear n-widths, which are not based on continuous algorithms, have been recently introduced in [13] and [16]. For other notions of non-linear n-widths, see [18], [3]. Non-continuous algorithms of n-term approximation and the n-term approximation for classes of functions with bounded mixed derivatives or differences, have been considered in [10], [17].…”

Section: Introductionmentioning

confidence: 99%

“…The interested reader is referred to [3], [9] for brief surveys on the non-linear n-widths a n and $ n of the classical Sobolev and Besov classes.…”

Section: Introductionmentioning

confidence: 99%

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Continuous Algorithms in n-Term Approximation and Non-Linear Widths

Dũng¹

2000

Journal of Approximation Theory

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In the present paper we investigate optimal continuous algorithms in n-term approximation based on various non-linear n-widths, and n-term approximation by the dictionary V formed from the integer translates of the mixed dyadic scales of the tensor product multivariate de la Valle e Poussin kernel, for the unit ball of Sobolev and Besov spaces of functions with common mixed smoothness. The asymptotic orders of these quantities are given. For each space the asymptotic orders of nonlinear n-widths and n-term approximation coincide. Moreover, these asymptotic orders are achieved by a continuous algorithm of n-term approximation by V, which is explicitly constructed. Academic Press

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“…The restrictions on the parameters in the Besov space imply that this Besov space is embedded in L 1 . For 0=R d this kind of problem has been already addressed in various settings in [1,6,11,12,13].…”

Section: Introductionmentioning

confidence: 98%