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We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).

We study approximation properties of sequences of centered random elements X d , d ? N , with values in separable Hilbert spaces. We focus on sequences of tensor product-type random elements, which have covariance operators of corresponding tensor product form. The average case approximation complexity n X d ( e ) is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate X d with relative 2 -average error not exceeding a given threshold e ? ( 0 , 1 ) . The growth of n X d ( e ) as a function of e - 1 and d determines whether a sequence of corresponding approximation problems for X d , d ? N , is tractable or not. Different types of tractability were studied in the paper by Lifshits et?al. (J. Complexity, 2012), where for each type the necessary and sufficient conditions were found in terms of the eigenvalues of the marginal covariance operators. We revise the criterion of quasi-polynomial tractability and provide a simplified version. We illustrate our result by applying it to random elements corresponding to tensor products of squared exponential kernels. We also extend a recent result of Xu (2014) concerning weighted Korobov kernels.

We study approximation properties of sequences of centered random elements $X_d$, $d\in\mathbb N$, with values in separable Hilbert spaces. We focus on sequences of tensor product-type and, in particular, degree-type random elements, which have covariance operators of corresponding tensor form. The average case approximation complexity $n^{X_d}(\varepsilon)$ is defined as the minimal number of continuous linear functionals that is needed to approximate $X_d$ with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. In the paper we investigate $n^{X_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Namely, we find criteria of (un)boundedness for $n^{X_d}(\varepsilon)$ on $d$ and of tending $n^{X_d}(\varepsilon)\to\infty$, $d\to\infty$, for any fixed $\varepsilon\in(0,1)$. In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics \begin{eqnarray*} \ln n^{X_d}(\varepsilon)= a_d+q(\varepsilon)b_d+o(b_d),\quad d\to\infty, \end{eqnarray*} at continuity points of a non-decreasing function $q\colon (0,1)\to\mathbb R$. Here $(a_d)_{d\in\mathbb N}$ is a sequence and $(b_d)_{d\in\mathbb N}$ is a positive sequence such that $b_d\to\infty$, $d\to\infty$. Under rather weak assumptions, we show that for tensor product-type random elements only special quantiles of self-decomposable or, in particular, stable (for tensor degrees) probability distributions appear as functions $q$ in the asymptotics. We apply our results to the tensor products of the Euler integrated processes with a given variation of smoothness parameters and to the tensor degrees of random elements with regularly varying eigenvalues of covariance operator., Comment: 39 pages, 0 figures

We study EC- ( s , t ) -weak tractability of multivariate linear problems in the average case setting. This paper extends earlier work in the worst case setting. The parameters s ≥ 0 and t ≥ 0 allow us to study the information complexity n ( e , d ) of a d -variate problem with respect to different powers of ln e − 1 , corresponding to the bits of accuracy, and d . We consider the absolute and normalized error criteria. In particular, a multivariate problem is EC- ( s , t ) -weakly tractable iff lim d + e − 1 → ∞ ln n ( e , d ) ∕ [ d t + ln s e − 1 ] = 0 . We deal with general linear problems and linear tensor product problems. We show necessary and sufficient conditions for EC- ( s , t ) -weak tractability. In the case of general linear problem these conditions are matching. For linear tensor product problems, we also show matching conditions with the exception of some cases where s > 1 , in general.

Hinrichs (2009) [3] recently studied multivariate integration defined over reproducing kernel Hilbert spaces in the randomized setting and for the normalized error criterion. In particular, he showed that such problems are strongly polynomially tractable if the reproducing kernels are pointwise nonnegative and integrable. More specifically, let n^r^a^n(@e,INT"d) be the minimal number of randomized function samples that is needed to compute [email protected] for the d-variate case of multivariate integration. Hinrichs proved that n^r^a^n(@e,INT"d)@[email protected][email protected]([email protected])^[email protected]?for [email protected]@?(0,1) and [email protected]?N. In this paper we prove that the exponent 2 of @e^-^1 is sharp for tensor product Hilbert spaces whose univariate reproducing kernel is decomposable and univariate integration is not trivial for the two parts of the decomposition. More specifically we have n^r^a^n(@e,INT"d)>[email protected]?18([email protected])^[email protected]?for [email protected]@?(0,1) and d>[email protected]^-^[email protected]^-^1, where @[email protected]?[1/2,1) depends on the particular space. We stress that these estimates hold independently of the smoothness of functions in a Hilbert space. Hence, even for spaces of very smooth functions the exponent of strong polynomial tractability must be 2. Our lower bounds hold not only for multivariate integration but for all linear tensor product functionals defined over a Hilbert space with a decomposable reproducing kernel and with a non-trivial univariate functional for the two spaces corresponding to decomposable parts. We also present lower bounds for reproducing kernels that are not decomposable but have a decomposable part. However, in this case it is not clear if the lower bounds are sharp.

We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight γj moderates the behavior of functions with respect to the jth variable. For γj≡1, we obtain the classical Sobolev spaces whereas for decreasing γj's the weighted Sobolev spaces consist of functions with diminishing dependence on the jth variables. We study the minimal errors of quadratures that use n function values for the unit ball of the space Fd. It is known that if the smoothness parameter of the Sobolev space is one, then the minimal error is the same as the discrepancy. Let n(ε, Fd) be the smallest n for which we reduce the initial error, i.e., the error with n=0, by a factor ε. The main problem studied in this paper is to determine whether the problem is intractable, i.e., whether n(ε, Fd) grows faster than polynomially in ε−1 or d. In particular, we prove intractability of integration (and discrepancy) if limsupd∑dj=1γj/lnd=∞. Previously, such results were known only for restricted classes of quadratures. For γj≡1, the, following bounds hold for discrepancy •with boundary conditions1.0628d(1+o(1))⩽n(ε, Fd)⩽1.5dε−2,asd→∞,•without boundary conditions1.0463d(1+o(1))⩽n(ε, Fd)⩽1.125dε−2,asd→∞. These results are obtained by analyzing arbitrary linear tensor product functionals Id defined over weighted tensor product reproducing kernel Hilbert spaces Fd of functions of d variables. We introduce the notion of a decomposable kernel. For reproducing kernels that have a decomposable part we prove intractability of all Id with non-zero subproblems with respect to the decomposable part of the kernel, as long as the weights satisfy the condition mentioned above.