Infinite-dimensional integration and the multivariate decomposition method

For approximating the integral, the MDM assumes that point values of fu are available for important subsets u, at some known cost. In this paper, we introduce a new setting, in which it is assumed that each fu belongs to a normed space Fu, and that bounds Bu on ‖fu‖Fu are known. This contrasts with the assumption in many papers that weights γu, appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights γu were determined by minimizing an error bound depending on the Bu, the γu and the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper, only the bounds Bu are assumed to be known. We give two examples in which we specialize the MDM: in the first case, Fu is the |u|-fold tensor product of an anchored reproducing kernel Hilbert space; in the second case, it is a particular non-Hilbert space for integration over an unbounded domain.