A Survey of Optimal Recovery

Abstract

The problem of optimal recovery is that of approximating as effectively as possible a given map of any function known to belong to a certain class from limited, and possibly error-contaminated, information about it. In this selective survey we describe some general results and give many examples of optimal recovery. C. A. Micchelli et al. (eds.), Optimal Estimation in Approximation Theory

De-noising by soft-thresholding

“…We start with an index set I n of cardinality n, and we observe y I = I + z I ; I 2 I n ; [31,41]). Suppose we h a v e an index set I (not necessarily nite), an object ( I ) o f i n terest, and observations y I = I + u I ; I 2 I :…”

Оптимальное Восстановление Функций И Их Производных По Приближенной Информации О Спектре И Неравенства Для Производных

“…Впо-следствии вся эта проблематика интенсивно развивалась в разных направлениях (см. [2][3][4][5]). Подход к задачам восстановления, основанный на общих принципах теории экстремума, развивался в работах [6][7][8].…”

Termination Conditions for Approximating Linear Problems with Noisy Information

“…To this end, we recall the concept of the «th diameter of information. It is given by (2.6) dn(N,A~) = 2.sup{||S(«)||: h£F, \\h\\ < I, \L¡(h)\ < A,-, l<z<«} (see, e.g., Micchelli and Rivlin [5], Traub et al [6]). Furthermore, we define a spline (p-spline) algorithm 4>* = {4>*}n>o (see Trojan [7] and Kacewicz and Plaskota [3]) as follows.…”

“…We study in this paper a termination criterion based on the use of the diameter of information, a quantity studied by many authors; see for instance Babenko [1], Micchelli and Rivlin [5], Marchuk and Osipenko [4], Kacewicz and Plaskota [3], and Traub et al [6]. This criterion (referred to as the diameter termination criterion) does not depend on the unknown error of approximation and /, but only on information used and the class K. In spite of this advantage, the diameter termination criterion may seem too strong when compared to the theoretical stopping condition, since it is based on a stronger inequality (see the relation (2.7)).…”