The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Abstract:

Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation. Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor $\varepsilon$ is bounded by $C \varepsilon ^{-p}$ for some exponent $p$ and some positive constant $C$. The $\varepsilon$-exponent of tractability is defined as the smallest power of $\varepsilon ^{-1}$ in these bounds. It is shown by using Monte Carlo quadrature that the $\varepsilon$-exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the $\varepsilon$-exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.