Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Quasi-random sequences and their discrepancies
SIAM Journal on Scientific Computing
Quasirandom points and global function fields
FFA '95 Proceedings of the third international conference on Finite fields and applications
A generalized discrepancy and quadrature error bound
Mathematics of Computation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Complexity and information
A constructive approach to strong tractability using Quasi-Monte Carlo algorithms
Journal of Complexity
Strong tractability of multivariate integration using quasi-Monte Carlo algorithms
Mathematics of Computation
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
The existence of good extensible rank-1 lattices
Journal of Complexity
Finite-order weights imply tractability of multivariate integration
Journal of Complexity
Exact cubature for a class of functions of maximum effective dimension
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Low discrepancy sequences in high dimensions: How well are their projections distributed?
Journal of Computational and Applied Mathematics
Multi-level Monte Carlo algorithms for infinite-dimensional integration on RN
Journal of Complexity
Journal of Complexity
Deterministic multi-level algorithms for infinite-dimensional integration on RN
Journal of Complexity
Liberating the dimension for function approximation: Standard information
Journal of Complexity
Liberating the dimension for L2-approximation
Journal of Complexity
Liberating the dimension for L2-approximation
Journal of Complexity
On tractability of linear tensor product problems for ∞-variate classes of functions
Journal of Complexity
On weighted Hilbert spaces and integration of functions of infinitely many variables
Journal of Complexity
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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 s is bounded by Cε-p for some exponent p and some positive constant C. The ε-exponent of tractability is defined as the smallest power of ε-1 in these bounds. It is shown by using Monte Carlo quadrature that the ε-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 ε-exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.