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Grzegorz W. Wasilkowski

Publications and source records attributed to Grzegorz W. Wasilkowski.

4 recordsLinked to original sources
arXiv2018

Efficient implementations of the Multivariate Decomposition Method for approximating infinite-variate integrals

In this paper we focus on efficient implementations of the Multivariate Decomposition Method (MDM) for approximating integrals of $\infty$-variate functions. Such $\infty$-variate integrals occur for example as expectations in uncertainty quantification. Starting with the anchored decomposition $f = \sum_{\mathfrak{u}\subset\mathbb{N}} f_\mathfrak{u}$, where the sum is over all finite subsets of $\mathbb{N}$ and each $f_\mathfrak{u}$ depends only on the variables $x_j$ with $j\in\mathfrak{u}$, our MDM algorithm approximates the integral of $f$ by first truncating the sum to some `active set' and then approximating the integral of the remaining functions $f_\mathfrak{u}$ term-by-term using Smolyak or (randomized) quasi-Monte Carlo (QMC) quadratures. The anchored decomposition allows us to compute $f_\mathfrak{u}$ explicitly by function evaluations of $f$. Given the specification of the active set and theoretically derived parameters of the quadrature rules, we exploit structures in both the formula for computing $f_\mathfrak{u}$ and the quadrature rules to develop computationally efficient strategies to implement the MDM in various scenarios. In particular, we avoid repeated function evaluations at the same point. We provide numerical results for a test function to demonstrate the effectiveness of the algorithm.

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arXiv2016

Equivalence of Weighted Anchored and ANOVA Spaces of Functions with Mixed Smoothness of Order one in $L_p$

We consider $γ$-weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in a weighted $L_p$ norm with $1 \leq p \leq \infty$. The domain of the functions is $D^d$, where $D \subseteq \mathbb{R}$ is a bounded or unbounded interval. We provide conditions on the weights $γ$ that guarantee that anchored and ANOVA spaces are equal (as sets of functions) and have equivalent norms with equivalence constants uniformly or polynomially bounded in $d$. Moreover, we discuss applications of these results to integration and approximation of functions on $D^d$.

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arXiv2016

Infinite-dimensional integration and the multivariate decomposition method

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.

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