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Efficient implementations of the Multivariate Decomposition Method for approximating infinite-variate integrals. (arXiv:1712.06782v1 [math.NA])
来源于:arXiv
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 查看全文>>