Journal of Computational Physics
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Computing the discrepancy with applications to supersampling patterns
ACM Transactions on Graphics (TOG)
Efficient algorithms for computing the L2-discrepancy
Mathematics of Computation
Computing discrepancies of Smolyak quadrature rules
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
Application of Threshold-Accepting to the Evaluation of the Discrepancy of a Set of Points
SIAM Journal on Numerical Analysis
A generalized discrepancy and quadrature error bound
Mathematics of Computation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
On the L2-discrepancy for anchored boxes
Journal of Complexity
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Some open problems concerning the star-discrepancy
Journal of Complexity
Covering numbers, vapnik-červonenkis classes and bounds for the star-discrepancy
Journal of Complexity
Construction Algorithms for Digital Nets with Low Weighted Star Discrepancy
SIAM Journal on Numerical Analysis
Tractability properties of the weighted star discrepancy
Journal of Complexity
Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy
Journal of Complexity
Constructing Sobol Sequences with Better Two-Dimensional Projections
SIAM Journal on Scientific Computing
Monte Carlo and Quasi-Monte Carlo Methods 2008
Monte Carlo and Quasi-Monte Carlo Methods 2008
Algorithmic construction of low-discrepancy point sets via dependent randomized rounding
Journal of Complexity
Hardness of discrepancy computation and ε-net verification in high dimension
Journal of Complexity
A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting
SIAM Journal on Numerical Analysis
A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting
SIAM Journal on Numerical Analysis
Constructing low star discrepancy point sets with genetic algorithms
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Uniform point sets and the collision test
Journal of Computational and Applied Mathematics
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We present a new algorithm for estimating the star discrepancy of arbitrary point sets. Similar to the algorithm for discrepancy approximation of Winker and Fang [SIAM J. Numer. Anal., 34 (1997), pp. 2028-2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a nonuniform sampling strategy, which is more suited for higher-dimensional inputs and additionally takes into account the topological characteristics of given point sets, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into critical test boxes. These critical test boxes provably yield higher discrepancy values and contain the box that exhibits the maximum value of the local discrepancy. We provide comprehensive experiments to test the new algorithm. Our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked (i.e., where the exact discrepancy of the point set can be computed in feasible time). Most importantly, in higher dimensions the new method behaves clearly better than all previously known methods.