Finite-order weights imply tractability of multivariate integration
Journal of Complexity
Finite-order weights imply tractability of linear multivariate problems
Journal of Approximation Theory
Tractability of quasilinear problems I: General results
Journal of Approximation Theory
Note: A note on the complexity and tractability of the heat equation
Journal of Complexity
Low discrepancy sequences in high dimensions: How well are their projections distributed?
Journal of Computational and Applied Mathematics
Tractability properties of the weighted star discrepancy
Journal of Complexity
Comparison of Point Sets and Sequences for Quasi-Monte Carlo and for Random Number Generation
SETA '08 Proceedings of the 5th international conference on Sequences and Their Applications
Approximate zero-variance simulation
Proceedings of the 40th Conference on Winter Simulation
Shifted lattice rules based on a general weighted discrepancy for integrals over Euclidean space
Journal of Computational and Applied Mathematics
A practical view of randomized quasi-Monte Carlo: invited presentation, extended abstract
Proceedings of the Fourth International ICST Conference on Performance Evaluation Methodologies and Tools
Finite-order weights imply tractability of linear multivariate problems
Journal of Approximation Theory
Journal of Complexity
Multidimensional pseudo-spectral methods on lattice grids
Applied Numerical Mathematics
Variance bounds and existence results for randomly shifted lattice rules
Journal of Computational and Applied Mathematics
Enhancing Quasi-Monte Carlo Methods by Exploiting Additive Approximation for Problems in Finance
SIAM Journal on Scientific Computing
On weighted Hilbert spaces and integration of functions of infinitely many variables
Journal of Complexity
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We study the problem of multivariate integration and the construction of good lattice rules in weighted Korobov spaces with general weights. These spaces are not necessarily tensor products of spaces of univariate functions. Sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found. These conditions are also necessary if the weights are such that the reproducing kernel of the weighted Korobov space is pointwise non-negative. The existence of a lattice rule which achieves the nearly optimal convergence order is proven. A component-by-component (CBC) algorithm that constructs good lattice rules is presented. The resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights. We also study special weights such as finite-order and order-dependent weights. For these special weights, the cost of the CBC algorithm is polynomial. Numerical computations show that the lattice rules constructed by the CBC algorithm give much smaller worst-case errors than the mean worst-case errors over all quasi-Monte Carlo rules or over all lattice rules, and generally smaller worst-case errors than the best Korobov lattice rules in dimensions up to hundreds. Numerical results are provided to illustrate the efficiency of CBC lattice rules and Korobov lattice rules (with suitably chosen weights), in particular for high-dimensional finance problems.