In this paper we consider integration and L2-approximation for functions over Rs from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and L2-approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold ε tends to 0 and the problem dimension s grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.
In this paper we consider integration and L2-approximation for functions over Rs from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and L2-approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold ε tends to 0 and the problem dimension s grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.
This paper is devoted to the study of tractability of the $ L_2 $-approximation and integration from weighted Korobov spaces of increasing smoothness in the worst-case setting. The considered algorithms use information from the class $ \Lambda^{\rm all} $, including all continuous linear functionals, and from the class $ \Lambda^{\rm std} $, including function evaluations. Necessary and sufficient conditions on the weights of the function space for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniform weak tractability, weak tractability and $ (\sigma, \tau) $-weak tractability, are provided. Our results give a comprehensive picture of the weight conditions for all standard notions of algebraic tractability. It may be helpful to study the tractability of nonhomogeneous tensor product spaces.
We study the multivariate L2-approximation problem APPd defined over a Banach space in the average case setting. The space is equipped with a zero-mean Gaussian measure with a weighted covariance kernel, which depends on parameter sequences α={αj}j∈N and β={βj}j∈N with 1<α1≤α2≤⋯ and 1≥β1≥β2≥⋯>0. In this paper, two interesting weighted covariance kernels are considered, which model the importance of the covariance kernels. Under the absolute error criterion or the normalized error criterion, we discuss (s,t)-weak tractability of the L2-approximation problem APP={APPd}d∈N from a Banach space whose zero-mean Gaussian measure has the above two weighted covariance kernels for some positive numbers s and t in the average case setting. Here, (s,t)-weak tractability means how the information complexity behaves as a function of dt and ε−s for large dimension d and small threshold ε. In particular, for all s>0 and t∈(0,1), we find the matching sufficient and necessary condition on the parameter sequences α={αj}j∈N and β={βj}j∈N to obtain (s,t)-weak tractability under the absolute error criterion or the normalized error criterion in the average case setting. We describe (s,t)-weak tractability by the matching sufficient and necessary condition, which reflects symmetry.
We study integration and L2-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space RN. We establish upper and lower bounds for the minimal worst case errors under general assumptions; these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods.
We study d-variate approximation for a weighted unanchored Sobolev space having smoothness m ≥ 1. Folk wisdom would lead us to believe that this problem should become easier as its smoothness increases. This is true if we are only concerned with asymptotic analysis: the nth minimal error is of order n^-(m-δ) for any δ > 0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? We prove that no matter how the weights are chosen, the waiting period is at least m^d, even if the error demand ε is arbitrarily close to 1. Hence, for m ≥ 2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So, we will be unable to vanquish the curse of dimensionality unless m = 1 , i.e., unless the smoothness is minimal. We then show that our problem can be tractable if m = 1. That is, we can find an ε-approximation using polynomially-many (in d and ε^-1) information operations, even if only function values are permitted. When m = 1, it is even possible for the problem to be strongly tractable, i.e., we can find an ε-approximation using polynomially-many (in ε^-1) information operations, independent of d. These positive results hold when the weights of the Sobolev space decay sufficiently quickly or are bounded finite-order weights, i.e., the d-variate functions we wish to approximate can be decomposed as sums of functions depending on at most ω variables, where ω is independent of d.
Tractability of L2-approximation in hybrid function spaces
We consider multivariate L2-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error eL2−app,Λ(N,d) of all algorithms that use N information evaluations from the class Λ in the d-dimensional case. The two classes Λ considered in this paper are the class Λall consisting of all linear functionals and the class Λstd consisting only of function evaluations. The focus lies on the dependence of eL2−app,Λ(N,d) on the dimension d. The main results are conditions for weak, polynomial, and strong polynomial tractability.
Tractability of L2-approximation in hybrid function spaces
We consider multivariate L2-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error eL2−app,Λ(N,d) of all algorithms that use N information evaluations from the class Λ in the d-dimensional case. The two classes Λ considered in this paper are the class Λall consisting of all linear functionals and the class Λstd consisting only of function evaluations. The focus lies on the dependence of eL2−app,Λ(N,d) on the dimension d. The main results are conditions for weak, polynomial, and strong polynomial tractability.