Universität Jena, Mathematisches Institut, Lehrstuhl Analysis

Forschungsseminar "Funktionenräume"

Dienstag, 24. Oktober 2000, 15.00 Uhr
Konferenzzimmer (R. 3319), Ernst-Abbe-Platz 1-4


V.N. Temlyakov

(South Carolina, Columbia)

"Greedy Algorithms in Nonlinear Approximation."



Abstract

Our main interest in this talk is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. While the scope of the talk is mostly theoretical, we should note that this form of approximation appears in many numerical applications such as adaptive PDE solvers, compression of images and signals, statistical classification, and so on.
The standard problem in this regard is the problem of $m$-term approximation where one fixes a basis and looks to approximate a target function by a linear combination of $m$ terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We will discuss the quantitative aspects of this type of approximation. We will also discuss stable algorithms for finding good or near best approximations using $m$ terms. These algorithms are representatives of a family of greedy algorithms.
More recently, there has emerged another more complicated form of nonlinear approximation which we call highly nonlinear approximation. It takes many forms but has the basic ingredient that a basis is replaced by a larger system of functions that is usually redundant. Some types of approximation that fall into this general category are mathematical frames, adaptive pursuit (or greedy algorithms) and adaptive basis selection. Redundancy on the one hand offers much promise for greater efficiency in terms of approximation rate, but on the other hand gives rise to highly nontrivial theoretical and practical problems. We will discuss some of these theoretical problems in the talk.


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Dr. D.D. Haroske; 2000-10-12