Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
On the numerical integration of Walsh series by number-theoretic methods
Mathematics of Computation
On the numerical integration of high-dimensional Walsh-series by quasi-Monte Carlo methods
Mathematics and Computers in Simulation - Special issue: Numerical probabilities
The mean square discrepancy of randomized nets
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Monte Carlo Variance of Scrambled Net Quadrature
SIAM Journal on Numerical Analysis
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
The asymptotic efficiency of randomized nets for quadrature
Mathematics of Computation
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces
SIAM Journal on Numerical Analysis
The Mean Square Discrepancy of Scrambled (t,s)-Sequences
SIAM Journal on Numerical Analysis
The discrepancy and gain coefficients of scrambled digital nets
Journal of Complexity
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
Variance with alternative scramblings of digital nets
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Optimal quadrature for Haar wavelet spaces
Mathematics of Computation
On the convergence rate of the component-by-component construction of good lattice rules
Journal of Complexity
Journal of Complexity
On the root mean square weighted L2 discrepancy of scrambled nets
Journal of Complexity
Diaphony, discrepancy, spectral test and worst-case error
Mathematics and Computers in Simulation
Journal of Complexity
On the mean square weighted L2 discrepancy of randomized digital nets in prime base
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Journal of Complexity
On the approximation of smooth functions using generalized digital nets
Journal of Complexity
Construction algorithms for higher order polynomial lattice rules
Journal of Complexity
Constructions of general polynomial lattice rules based on the weighted star discrepancy
Finite Fields and Their Applications
Constructions of (t ,m,s)-nets and (t,s)-sequences
Finite Fields and Their Applications
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We introduce a weighted reproducing kernel Hilbert space which is based on Walsh functions. The worst-case error for integration in this space is studied, especially with regard to (t, m, s)-nets. It is found that there exists a digital (t, m, s)-net, which achieves a strong tractability worst-case error bound under certain condition on the weights.We also investigate the worst-case error of integration in weighted Sobolev spaces. As the main tool we define a digital shift invariant kernel associated to the kernel of the weighted Sobolev space. This allows us to study the mean square worst-case error of randomly digitally shifted digital (t, m, s)- nets. As this digital shift invariant kernel is almost the same as the kernel for the Hilbert space based on Walsh functions, we can derive results for the weighted Sobolev space based on the analysis of the Walsh function space. We show that there exists a (t, m, s)-net which achieves the best possible convergence order for integration in weighted Sobolev spaces and are strongly tractable under the same condition on the weights as for lattice rules.