Leszek Plaskota

Peter Kritzer, Friedrich Pillichshammer, Leszek Plaskota et al.

In this paper, we study the approximation of $d$-dimensional $ρ$-weighted integrals over unbounded domains $\mathbb{R}_+^d$ or $\mathbb{R}^d$ using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded $L_p$ norm of mixed partial derivatives of first order, where $p\in[1,+\infty].$ The main results give sufficient conditions on the change of variables $ν$ which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on $ρ$ and $p$. The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.

Frances Y. Kuo, Dirk Nuyens, Leszek Plaskota et al.

We further develop the \emph{Multivariate Decomposition Method} (MDM) for the Lebesgue integration of functions of infinitely many variables $x_1,x_2,x_3,\ldots$ with respect to a corresponding product of a one dimensional probability measure. Although a number of concepts of infinite-dimensional integrals have been used in the literature, questions of uniqueness and compatibility have mostly not been studied. We show that, under appropriate convergence conditions, the Lebesgue integral equals the `anchored' integral, independently of the anchor. The MDM assumes that point values of $f_{\mathfrak{u}}$ are available for important subsets ${\mathfrak{u}}$, at some known cost. In this paper we introduce a new setting, in which it is assumed that each $f_{\mathfrak{u}}$ belongs to a normed space $F_{\mathfrak{u}}$, and that bounds $B_{\mathfrak{u}}$ on $\|f_{\mathfrak{u}}\|_{F_{\mathfrak{u}}}$ are known. This contrasts with the assumption in many papers that weights $γ_{\mathfrak{u}}$, appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights $γ_{\mathfrak{u}}$ were determined by minimizing an error bound depending on the $B_{\mathfrak{u}}$, the $γ_{\mathfrak{u}}$ \emph{and} the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper only the bounds $B_{\mathfrak{u}}$ are assumed known. We give two examples in which we specialize the MDM: in the first case $F_{\mathfrak{u}}$ is the $|{\mathfrak{u}}|$-fold tensor product of an anchored reproducing kernel Hilbert space, and in the second case it is a particular non-Hilbert space for integration over an unbounded domain.