Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
A generalized discrepancy and quadrature error bound
Mathematics of Computation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Weighted tensor product algorithms for linear multivariate problems
Journal of Complexity
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces
SIAM Journal on Numerical Analysis
Strong tractability of multivariate integration using quasi-Monte Carlo algorithms
Mathematics of Computation
The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension
Mathematics of Computation
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
Journal of Complexity
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?
SIAM Journal on Scientific Computing
Good Lattice Rules in Weighted Korobov Spaces with General Weights
Numerische Mathematik
Finite-order weights imply tractability of linear multivariate problems
Journal of Approximation Theory
Diaphony, discrepancy, spectral test and worst-case error
Mathematics and Computers in Simulation
Randomly shifted lattice rules on the unit cube for unbounded integrands in high dimensions
Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
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
Tractability of quasilinear problems I: General results
Journal of Approximation Theory
Searching for extensible Korobov rules
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
Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing
INFORMS Journal on Computing
Finite-order weights imply tractability of linear multivariate problems
Journal of Approximation Theory
Randomly shifted lattice rules on the unit cube for unbounded integrands in high dimensions
Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
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Multivariate integration of high dimension s occurs in many applications. In many such applications, for example in finance, integrands can be well approximated by sums of functions of just a few variables. In this situation the superposition (or effective) dimension is small, and we can model the problem with finite-order weights, where the weights describe the relative importance of each distinct group of variables up to a given order (where the order is the number of variables in a group), and ignore all groups of variables of higher order.In this paper we consider multivariate integration for the anchored and unanchored (nonperiodic) Sobolev spaces equipped with finite-order weights. Our main interest is tractability and strong tractability of QMC algorithms in the worst-case setting. That is, we want to find how the minimal number of function values needed to reduce the initial error by a factor ε depends on s and ε-1. If there is no dependence on s, and only polynomial dependence on ε-1, we have strong tractability, whereas with polynomial dependence on both s and ε-1 we have tractability.We show that for the anchored Sobolev space we have strong tractability for arbitrary finite-order weights, whereas for the unanchored Sobolev space we have tractability for all bounded finite-order weights. In both cases, the dependence on ε-1 is quadratic. We can improve the dependence on ε-1 at the expense of polynomial dependence on s. For finite-order weights, we may achieve almost linear dependence on ε-1 with a polynomial dependence on s whose degree is proportional to the order of the weights.We show that these tractability bounds can be achieved by shifted lattice rules with generators computed by the component-by-component (CBC) algorithm. The computed lattice rules depend on the weights. Similar bounds can also be achieved by well-known low discrepancy sequences such as Halton, Sobol and Niederreiter sequences which do not depend on the weights. We prove that these classical low discrepancy sequences lead to error bounds with almost linear dependence on n-1 and polynomial dependence on d. We present explicit worst-case error bounds for shifted lattice rules and for the Niederreiter sequence. Better tractability and error bounds are possible for finite-order weights, and even for general weights if they satisfy certain conditions. We present conditions on general weights that guarantee tractability and strong tractability of multivariate integration.