Randomized approximation numbers on Besov classes with mixed smoothness

2020 ◽  
Vol 18 (04) ◽  
pp. 2050023
Author(s):  
Liqin Duan ◽  
Peixin Ye
Keyword(s):  
Linear Approximation ◽  
Approximation Numbers ◽  
Randomized Methods ◽  
Mixed Smoothness

We study the Kolmogorov and the linear approximation numbers of the Besov classes [Formula: see text] with mixed smoothness in the norm of [Formula: see text] in the randomized setting. We first establish two discretization theorems. Then based on them, we determine the exact asymptotic orders of the Kolmogorov and the linear approximation numbers for certain values of the parameters [Formula: see text]. Our results show that the linear randomized methods lead to considerably better rates than those of the deterministic ones for [Formula: see text].

scholarly journals Function Values Are Enough for $$L_2$$-Approximation

2020 ◽  
Author(s):  
David Krieg ◽  
Mario Ullrich
Keyword(s):  
Hilbert Space ◽  
Fourier Coefficients ◽  
Approximation Number ◽  
Approximation Numbers ◽  
Approximation Of Functions ◽  
Worst Case ◽  
Linear Information ◽  
Grid Algorithm ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

AbstractWe study the $$L_2$$ L 2 -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number $$e_n$$ e n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number $$a_n$$ a n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that $$\begin{aligned} e_n \,\lesssim \, \sqrt{\frac{1}{k_n} \sum _{j\ge k_n} a_j^2}, \end{aligned}$$ e n ≲ 1 k n ∑ j ≥ k n a j 2 , where $$k_n \asymp n/\log (n)$$ k n ≍ n / log ( n ) . This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces $$H^s_\mathrm{mix}(\mathbb {T}^d)$$ H mix s ( T d ) with dominating mixed smoothness $$s>1/2$$ s > 1 / 2 and dimension $$d\in \mathbb {N}$$ d ∈ N , and we obtain $$\begin{aligned} e_n \,\lesssim \, n^{-s} \log ^{sd}(n). \end{aligned}$$ e n ≲ n - s log sd ( n ) . For $$d>2s+1$$ d > 2 s + 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.

Turbo decoding algorithm based on linear approximation of correction function

2013 ◽  
Vol 32 (8) ◽  
pp. 2113-2115
Author(s):  
Zheng LI ◽  
Chun-lin SONG ◽  
Yun-jie ZHAO ◽  
Zhu-jia WU
Keyword(s):  
Linear Approximation ◽  
Correction Function ◽  
Turbo Decoding ◽  
Decoding Algorithm
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