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On the Scrambled Halton Sequence

  • Michael Mascagni und Hongmei Chi
Veröffentlicht/Copyright: 9. Mai 2008
Monte Carlo Methods and Applications
Aus der Zeitschrift Band 10 Heft 3-4

The Halton sequence is one of the standard (along with (t, s)-sequences and lattice points) low-discrepancy sequences, and thus is widely used in quasi-Monte Carlo applications. One of its important advantages is that the Halton sequence is easy to implement due to its definition via the radical inverse function. However, the original Halton sequence suffers from correlations between radical inverse functions with different bases used for different dimensions. These correlations result in poorly distributed two-dimensional projections. A standard solution to this is to use a randomized (scrambled) version of the Halton sequence. Here, we analyze the correlations in the standard Halton sequence, and based on this analysis propose a new and simpler modified scrambling algorithm. We also provide a number theoretic criterion to choose the optimal scrambling from among a large family of random scramblings. Based on this criterion, we have found the optimal scrambling for up to 60 dimensions for the Halton sequence. This derandomized Halton sequence is then numerically tested and shown empirically to be far superior to the original sequence.

Published Online: 2008-05-09
Published in Print: 2004-12

© de Gruyter 2004

Artikel in diesem Heft

  1. Foreword
  2. An outline of quasi-probability*: Why quasi-Monte-Carlo methods are statistically valid and how their errors can be estimated statistically
  3. QMC techniques for CAT bond pricing*
  4. Parallel Quasi-Monte Carlo Methods for Linear Algebra Problems
  5. Algorithm of statistical simulation of dynamic systems with distributed change of structure*
  6. Frequency Analysis of Semiconductor Devices Using Full-Band Cellular Monte Carlo Simulations
  7. The ⊝-Maruyama scheme for stochastic functional differential equations with distributed memory term*
  8. Subdiffusion and Superdiffusion in Lagrangian Stochastic Models of Oceanic Transport
  9. Approximations of functional integrals with respect to measures generated by solutions of stochastic differential equations
  10. Normalization of the Spectral Test in High Dimensions
  11. A spectral Monte Carlo method for the Poisson equation
  12. Convergence rate for spherical processes with shifted centres*
  13. On the Power of Quantum Algorithms for Vector Valued Mean Computation
  14. Parallel Quasirandom Walks on the Boundary
  15. Trajectory Splitting by Restricted Replication
  16. Upper Bounds for Bermudan Style Derivatives
  17. Stochastic Eulerian model for the flow simulation in porous media. Unconfined aquifers*
  18. Solution of the Space-dependent Wigner Equation Using a Particle Model
  19. Subgrid Modeling of Filtration in a Porous Medium with Multiscale Log-Stable permeability
  20. Comparison of Quasi-Monte Carlo-Based Methods for Simulation of Markov Chains
  21. Monte Carlo methods for fissured porous media: a gridless approach*
  22. Smoothed Transformed Density Rejection*
  23. Adaptive adjoint Monte Carlo simulation for the uncertainty
  24. A Nuclear Measurement Technique of Water Penetration in Concrete Barriers
  25. System availability and reliability analysis by direct Monte Carlo with biasing
  26. On the Scrambled Halton Sequence
  27. Monte-Carlo simulation of the chord length distribution function across convex bodies, non-convex bodies and random media
  28. Discrepancy of sequences generated by dynamical system
  29. Operator-Split Method for Variance Reduction in Stochastic Solutions of the Wigner Equation
  30. Two variants of a stochastic Euler method for homogeneous balance differential equations*
  31. Full band Monte Carlo simulation - beyond the semiclassical approach
  32. Coin Tossing Algorithms for Integral Equations and Tractability
  33. Optimal Korobov Coefficients for Good Lattice Points in Quasi Monte Carlo Algorithms
  34. A theoretical view on transforming low-discrepancy sequences from a cube to a simplex
  35. Weighted simulation of steady-state transport within the standard Monte Carlo paradigm
  36. Dynamic probabilistic method of numerical modeling of multidimensional hydrometeorological fields
  37. Quasirandom Sequences in Branching Random Walks*
  38. Discrete random walk on large spherical grids generated by spherical means for PDEs*
  39. Reusing paths in radiosity and global illumination
  40. Measures of Uniform Distribution in Wavelet Based Image Compression
  41. Random Walk Algorithms for Estimating Effective Properties of Digitized Porous Media*
  42. Security of Pseudo-random Generator and Monte Carlo Method
  43. Randomization of Quasi-Monte Carlo Methods for Error Estimation: Survey and Normal Approximation*
  44. Monte Carlo Simulation of Narrow-Width SOI Devices: Incorporation of the Short Range Coulomb Interaction
  45. Dagger-sampling variance reduction in Monte Carlo reliability analysis
https://doi.org/10.1515/mcma.2004.10.3-4.435

Artikel in diesem Heft

  1. Foreword
  2. An outline of quasi-probability*: Why quasi-Monte-Carlo methods are statistically valid and how their errors can be estimated statistically
  3. QMC techniques for CAT bond pricing*
  4. Parallel Quasi-Monte Carlo Methods for Linear Algebra Problems
  5. Algorithm of statistical simulation of dynamic systems with distributed change of structure*
  6. Frequency Analysis of Semiconductor Devices Using Full-Band Cellular Monte Carlo Simulations
  7. The ⊝-Maruyama scheme for stochastic functional differential equations with distributed memory term*
  8. Subdiffusion and Superdiffusion in Lagrangian Stochastic Models of Oceanic Transport
  9. Approximations of functional integrals with respect to measures generated by solutions of stochastic differential equations
  10. Normalization of the Spectral Test in High Dimensions
  11. A spectral Monte Carlo method for the Poisson equation
  12. Convergence rate for spherical processes with shifted centres*
  13. On the Power of Quantum Algorithms for Vector Valued Mean Computation
  14. Parallel Quasirandom Walks on the Boundary
  15. Trajectory Splitting by Restricted Replication
  16. Upper Bounds for Bermudan Style Derivatives
  17. Stochastic Eulerian model for the flow simulation in porous media. Unconfined aquifers*
  18. Solution of the Space-dependent Wigner Equation Using a Particle Model
  19. Subgrid Modeling of Filtration in a Porous Medium with Multiscale Log-Stable permeability
  20. Comparison of Quasi-Monte Carlo-Based Methods for Simulation of Markov Chains
  21. Monte Carlo methods for fissured porous media: a gridless approach*
  22. Smoothed Transformed Density Rejection*
  23. Adaptive adjoint Monte Carlo simulation for the uncertainty
  24. A Nuclear Measurement Technique of Water Penetration in Concrete Barriers
  25. System availability and reliability analysis by direct Monte Carlo with biasing
  26. On the Scrambled Halton Sequence
  27. Monte-Carlo simulation of the chord length distribution function across convex bodies, non-convex bodies and random media
  28. Discrepancy of sequences generated by dynamical system
  29. Operator-Split Method for Variance Reduction in Stochastic Solutions of the Wigner Equation
  30. Two variants of a stochastic Euler method for homogeneous balance differential equations*
  31. Full band Monte Carlo simulation - beyond the semiclassical approach
  32. Coin Tossing Algorithms for Integral Equations and Tractability
  33. Optimal Korobov Coefficients for Good Lattice Points in Quasi Monte Carlo Algorithms
  34. A theoretical view on transforming low-discrepancy sequences from a cube to a simplex
  35. Weighted simulation of steady-state transport within the standard Monte Carlo paradigm
  36. Dynamic probabilistic method of numerical modeling of multidimensional hydrometeorological fields
  37. Quasirandom Sequences in Branching Random Walks*
  38. Discrete random walk on large spherical grids generated by spherical means for PDEs*
  39. Reusing paths in radiosity and global illumination
  40. Measures of Uniform Distribution in Wavelet Based Image Compression
  41. Random Walk Algorithms for Estimating Effective Properties of Digitized Porous Media*
  42. Security of Pseudo-random Generator and Monte Carlo Method
  43. Randomization of Quasi-Monte Carlo Methods for Error Estimation: Survey and Normal Approximation*
  44. Monte Carlo Simulation of Narrow-Width SOI Devices: Incorporation of the Short Range Coulomb Interaction
  45. Dagger-sampling variance reduction in Monte Carlo reliability analysis
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