Infinite-dimensional integration and the multivariate decomposition method

Abstract

We further develop the Multivariate Decomposition Method (MDM) for the Lebesgue integration of functions of infinitely many variables x 1 , x 2 , x 3 , . . . with respect to a corresponding product of a one dimensional probability measure. The method is designed for functions that admit a dominantly convergent decomposition f = u f u , where u runs over all finite subsets of positive integers, and for each u = {i 1 , . . . , i k } the function f u depends only on x i1 , . . . , x i k .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.For approximating the integral, the MDM assumes that point values of f u 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 f u belongs to a normed space F u , and that bounds B u on f u Fu are known. This contrasts with the assumption in many papers that weights γ u , appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights γ u were determined by minimizing an error bound depending on the B u , the γ u and the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper only the bounds B u are assumed known. We give two examples in which we specialize the MDM: in the first case F u is the |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. arXiv:1501.05445v3 [math.NA]

Efficient Implementations of the Multivariate Decomposition Method for Approximating Infinite-Variate Integrals

“…where T > 0 is a "threshold" parameter that depends on the overall error demand ε > 0 and possibly on all of w(u). For example, w(u) can be related to the weight parameters from a weighted function space setting (as in [16,17,13,4]), or it can be related to the bounds on the norm of f u (as in [7]). In this section we will treat T and w(u) as input parameters (ignoring the mathematical details of where they come from), and focus on the efficient implementation of the active set given these parameters.…”

“…is a second sequence of nonnegative real numbers controlling the "order dependent aspect", with the restriction on Ω that its growth is controlled by ω , i.e., Ω +1 ω +1 ≤ Ω for all ∈ N. This assumption is satisfied in all practical cases that we are aware of. Further, in the theoretical framework for the MDM (see, e.g., [7]), a sufficient condition for the infinite-dimensional integral to be well-defined is for the parameters w(u) to be summable, u⊂N w(u) < ∞, which will not hold unless the condition Ω +1 ω +1 ≤ Ω holds (at least asymptotically in ).…”

“…Each variant of our MDM algorithm involves three stages, as outlined in the pseudocodes; a summary is given as follows: In Section 6 we derive a computable expression for estimating an infinite series that may appear in the definition of the active set, which is another novel and significant contribution of this paper. Finally in Section 7 we combine all ingredients and follow the mathematical setting of [7] to construct the active set and choose the quadrature rules. We then apply the MDM algorithm to an example integrand that mimics the characteristics of the integrands arising from some parametrized PDE problems (see, e.g., [8]).…”

“…, }) ≤ T . The assumptions on the structure of w(u) and properties 1-5 above ensure that this stopping criteria is valid, and hence that Pseudocode 1 does indeed construct the active set (7). In particular, property 3 implies w(u) ≤ w({1, 2, .…”

“…Note that in practice calculating the number of Smolyak levels m u (or the number of QMC points in the next subsection) normally requires knowledge of the entire active set U, see, e.g., [7,Section 4.3], hence we compute them when constructing U ext .…”