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Session III.5 - Information-Based Complexity

Wednesday, June 21, 17:30 ~ 18:00

$L^2$-approximation and numerical integration on Gaussian Spaces

Robin Rüßmann

RPTU in Kaiserslautern, Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We study $L^2$-approximation and integration on reproducing kernel Hilbert spaces $H(L_\sigma)$ of $d$ variables, where $d\in\mathbb{N}$ or $d=\infty$. Here, $L_\sigma$ is given as the tensor product of univariate Gaussian kernels, i.e., $L_\sigma(x,y) := \prod_{j=1}^d \exp(-\sigma_j^2 \cdot (x_j-y_j)^2)$. These spaces are closely related to Hermite spaces $H(K_\beta)$, where $K_\beta$ is again of tensor product form, but based on univariate Hermite kernels, i.e., $K_\beta(x,y):= \prod_{j=1}^{d}\sum_{\nu=0}^\infty \beta_j^{\nu}\cdot h_\nu(x_j)\cdot h_\nu(y_j)$, where $h_v$ is the Hermite polynomial of degree $\nu$. More precisely, for each space $H(L_\sigma)$ there exists a corresponding space $H(K_\beta)$ and an isometric isomorphism $Q$ between both spaces such that one function evaluation of $Qf$ needs only one function evaluation of $f$ and vice versa.

Via this correspondence, we are able to constructively transform any algorithm for $L^2$-approximation or integration on $H(K_\beta)$ into an algorithm for the same problem on $H(L_\sigma)$ and vice versa, preserving error and cost. In the case $d=\infty$, this allows us to investigate both problems on $H(L_\sigma)$ for the first time. In the case $d\in\mathbb{N}$, we are able to transfer some known results between the two function space settings.

Joint work with Michael Gnewuch (University of Osnabrück, Germany) and Klaus Ritter (RPTU in Kaiserslautern, Germany).

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