Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAdaptive sampling recovery of functions with higher mixed regularity
147
0
0.0
(
0
)
Added by
Nadiia Derevianko
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We tensorize the Faber spline system from [14] to prove sequence space isomorphisms for multivariate function spaces with higher mixed regularity. The respective basis coefficients are local linear combinations of discrete function values similar as for the classical Faber Schauder system. This allows for a sparse representation of the function using a truncated series expansion by only storing discrete (finite) set of function values. The set of nodes where the function values are taken depends on the respective function in a non-linear way. Indeed, if we choose the basis functions adaptively it requires significantly less function values to represent the initial function up to accuracy $varepsilon>0$ (say in $L_infty$) compared to hyperbolic cross projections. In addition, due to the higher regularity of the Faber splines we overcome the (mixed) smoothness restriction $r<2$ and benefit from higher mixed regularity of the function. As a byproduct we present the solution of Problem 3.13 in the Triebel monograph [46] for the multivariate setting.
rate research
Read More
This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber-Schauder basis except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebels monograph (see References [47]) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylors remainder formula to the dual Chui-Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel-Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber-Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber-Schauder situation.
In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will be on the behavior of the best $m$-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm. It turns out that these quantities share a few fundamental abstract properties like their behavior under real interpolation, such that they can be treated simultaneously. We start with proving estimates on finite rank convolution operators with range in a step hyperbolic cross. These results imply bounds for the corresponding function space embeddings by a well-known decomposition technique. The decay of Kolmogorov numbers have direct implications for the problem of sampling recovery in $L_2$ in situations where recent results in the literature are not applicable since the corresponding approximation numbers are not square summable.
In this paper we study $L_2$-norm sampling discretization and sampling recovery of complex-valued functions in RKHS on $D subset R^d$ based on random function samples. We only assume the finite trace of the kernel (Hilbert-Schmidt embedding into $L_2$) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomially in $n$, the number of samples. Under the mild additional assumption of separability we observe improved rates of convergence related to the decay of the singular values. Our main tool is a spectral norm concentration inequality for infinite complex random matrices with independent rows complementing earlier results by Rudelson, Mendelson, Pajor, Oliveira and Rauhut.
The purpose of this paper is concerned with the approximate solution of split equality problems. We introduce two types of algorithms and a new self-adaptive stepsize without prior knowledge of operator norms. The corresponding strong convergence theorems are obtained under mild conditions. Finally, some numerical experiments demonstrate the efficiency of our results and compare them with the existing results.
We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into $L_2(D,varrho_D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. Here the measure $varrho_D$ is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weavers conjecture we mostly lose a $sqrt{log n}$ and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
