Construction of Quasi-Monte Carlo Rules for Multivariate Integration in Spaces of Permutation-Invariant Functions

Abstract

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens, Suryanarayana, and Weimar [10] (Adv. Comput. Math. (2016), 42(1):55-84), the authors derived an upper estimate for the nth minimal worst case error for such problems, and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-1 lattice rule that obtains a rate of convergence arbitrarily close to O(n −α ), where α > 1/2 denotes the smoothness of our function space and n is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets which can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension d is significantly improved.

Cubature Formulas for Multisymmetric Functions and Applications to Stochastic Partial Differential Equations

“…The limiting factor when the algorithm is applied in the way we propose is memory consumption. For large n, the number k of columns in the matrix A P R n˚ˆk is usually very large (see (17)), and executing the simplex algorithm with a large constraint matrix uses sizable amounts of memory. Observe that in the critical cases we have k " n˚, and n˚is an upper bound for the number of nonzero entries in a solution.…”

“…Recently, the idea to exploit permutation-invariance conditions as another kind of a priori knowledge has been enunciated [25,26], and it has been shown that the complexity of such integration problems can be significantly reduced if the permutationinvariance conditions are exploited in the construction of quasi-Monte Carlo rules [16]. In [17], a component-by-component construction scheme for a quasi-Monte Carlo method utilizing permutation-invariance properties has been proposed. This work also features a semiconstructive scheme for cubature formulas that is built on the idea of variance reduction and also makes use of permutation invariance.…”

“…The idea to exploit permutation-invariance properties to facilitate multivariate high-dimensional numerical integration is still young, and the work performed so far [25,26,16,17] has been focused on a mathematical setting tailored to the Schrödinger equation, and as a consequence employs different notions of permutation invariance. Our definition of G-invariance includes the symmetry properties introduced in [25,26], but not the respective antisymmetry properties.…”

Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions

“…Here we analyze unshifted and shifted rank-1 lattice rules for the approximation of the integral (1) of I d -permutation-invariant functions from the Korobov-type spaces F d (r α,β ) defined in Section 2. First of all we give an exact error formula which holds for general cubature rules Q d,n of the form (8). The proof can be found in the appendix (Section 5).…”

“…Lemma 4.1. For d, n ∈ N let Q d,n denote a general cubature rule given by (8). Then its worst case error on the I d -permutation-invariant subspace of F d (r α,β ) satisfies e wor (Q d,n ;…”