By Gregory E. Fasshauer, Larry L. Schumaker
ISBN-10: 3319064037
ISBN-13: 9783319064031
ISBN-10: 3319064045
ISBN-13: 9783319064048
These complaints have been ready in reference to the 14th foreign convention on Approximation conception, which used to be held April 7-10, 2013 in San Antonio, Texas. The convention used to be the fourteenth in a sequence of conferences in Approximation concept held at a number of destinations within the usa. The integrated invited and contributed papers hide different parts of approximation concept with a distinct emphasis at the most modern and energetic parts corresponding to compressed sensing, isogeometric research, anisotropic areas, radial foundation services and splines. Classical and summary approximation is usually incorporated. The e-book can be of curiosity to mathematicians, engineers\ and machine scientists operating in approximation idea, computer-aided geometric layout, numerical research and comparable software areas.
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Extra info for Approximation Theory XIV: San Antonio 2013
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L. -L. Bouchot We investigate in particular some variations of the Hard Thresholding Pursuit (HTP) algorithm [7], an iterative thresholding-based method, and its graded approach, a recent variation that does not require prior knowledge of the sparsity [2]. We analyze the reconstruction abilities of these algorithms in both idealized and realistic settings. In particular we introduce a generalization that improve the speed performance of (GHTP). The idealized setting is characterized by the fact that the signal to be recovered x is exactly s-sparse and that the measurements occur in an error-free manner.
The condition number κM of M is given by ⎤ κM := Δd (MT M) = ≥M≥2 ≥M−1 ≥2 . Δ1 (MT M) Definition 2 Given a regular expanding matrix M ∈ Zd × d , a fundamental interpolant IM ∈ L1 (Td ) fulfills the ellipsoidal (periodic) Strang-Fix conditions of order s > 0 for q ≥ 1 and an Ω ∈ R+ , if there exists a nonnegative sequence T d b = {bz }z ∈ Zd → R+ 0 , such that for all h ∈ G (M ), z ∈ Z \{0} we have −s ≥M−T h≥s2 , 1. |1 − mch (IM )| ∇ b0 κM −s −T s 2. |mch + MT z (IM )| ∇ bz κM ≥M≥−Ω 2 ≥M h≥2 with πSF := {θΩM (z)bz }z ∈ Zd ψq (Zd ) < ∅.
While these conditions might be unrealistic in some cases, it is shown that with high probability, our algorithms select a correct set of indices at each iteration, as long as the active support is smaller than the actual support of the vector to be recovered, with a proviso on the shape of the vector. Our theoretical findings are illustrated by numerical examples. , we analyze the reconstruction of sparse signals x ∈ C N based only on a few number of (linear) measurements y ∈ Cm where m ≥ N . It is known from the compressive sensing literature that recovery of s-sparse signals x is ensured when the sensing (or measurement) matrix is random (Gaussian or sub-Gaussian for instance) and when the number of measurements scales linearly with the sparsity of the signal up to a log factor.
Approximation Theory XIV: San Antonio 2013 by Gregory E. Fasshauer, Larry L. Schumaker
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