a b s t r a c tWe 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 for the anchored-ANOVA decomposition. We show that the new methods can be applied in locally adaptive and dimension-adaptive ways and demonstrate their efficiency by numerical experiments with high-dimensional integrals from finance.

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We propose a monotone multigrid method based on a B-spline basis of arbitrary smoothness for the efficient numerical solution of elliptic variational inequalities on closed convex sets. In order to maintain monotonicity (upper bound) and quasi-optimality (lower bound) of the coarse grid corrections, we propose coarse grid approximations of the obstacle function which are based on B-spline expansion coefficients. To illustrate the potential of the scheme, the method is applied to the pricing of American options in the Black-Scholes framework.

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