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We present a new general class of methods for the computation of high-dimensional integrals. The quadrature schemes result by truncation and discretization of the anchored-ANOVA decomposition. They are designed to exploit low effective dimensions and include sparse grid methods as special case. To derive bounds for the resulting modelling and discretization errors, we introduce effective dimensions...
We show that the lower-order terms in the ANOVA decomposition of a function f(x)≔max(ϕ(x),0) for x∈[0,1]d, with ϕ a smooth function, may be smoother than f itself. Specifically, f in general belongs only to Wd,∞1, i.e., f has one essentially bounded derivative with respect to any component of x, whereas, for each u⊆{1,…,d}, the ANOVA term fu (which depends only on the variables xj with j∈u) belongs...
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