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
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Mathematics of Computation
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Mathematics of Computation
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Computers in Physics
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
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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
Variance Reduction via Lattice Rules
Management Science
Finite-order weights imply tractability of multivariate 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
Smoothness and dimension reduction in Quasi-Monte Carlo methods
Mathematical and Computer Modelling: An International Journal
Low-Discrepancy Sequences and Global Function Fields with Many Rational Places
Finite Fields and Their Applications
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Proceedings of the 2008 IEEE/ACM International Conference on Computer-Aided Design
Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing
INFORMS Journal on Computing
Acceleration of market value-at-risk estimation
Proceedings of the 2nd Workshop on High Performance Computational Finance
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Evolutionary optimization of low-discrepancy sequences
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Accelerating Value-at-Risk estimation on highly parallel architectures
Concurrency and Computation: Practice & Experience
Advances in variation-aware modeling, verification, and testing of analog ICs
DATE '12 Proceedings of the Conference on Design, Automation and Test in Europe
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Quasi-Monte Carlo (QMC) methods have been successfully used to compute high-dimensional integrals arising in many applications, especially in finance. To understand the success and the potential limitation of QMC, this paper focuses on quality measures of point sets in high dimensions. We introduce the order-@?, superposition and truncation discrepancies, which measure the quality of selected projections of a point set on lower-dimensional spaces. These measures are more informative than the classical ones. We study their relationships with the integration errors and study the tractability issues. We present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the Sobol' nets with that of the sets of Latin hypercube sampling and random points. Numerical results show that in high dimensions the superiority of the Sobol' nets mainly derives from the one-dimensional projections and the projections associated with the earlier dimensions; for order-2 and higher-order projections all these point sets have similar behavior (on the average). In weighted cases with fast decaying weights, the Sobol' nets have a better performance than the other two point sets. The investigation enables us to better understand the properties of QMC and throws new light on when and why QMC can have a better (or no better) performance than Monte Carlo for multivariate integration in high dimensions.