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Parallel Quasirandom Walks on the Boundary

  • Aneta Karaivanova , Michael Mascagni and Nikolai A. Simonov
Published/Copyright: May 9, 2008
Monte Carlo Methods and Applications
From the journal Volume 10 Issue 3-4

The Monte Carlo method called “random walks on boundary” has been successfully used for solving boundary-value problems. This method has significant advantages when compared with random walks on spheres, balls or on discrete grids when an exterior Dirichlet or Neumann problem is solved, or when we are interested in computing the solution to a problem at an arbitrary number of points using a single random walk.

In this paper we will investigate ways:

  • • to increase the convergence rate of this method by using quasirandom sequences instead of pseudorandom numbers for the construction of the boundary walks,

  • • to find an efficient parallel implementation of this method on a cluster using MPI.

In our parallel implementation we use disjoint contiguous blocks of quasirandom numbers extracted from a given quasirandom sequence for each processor. In this case, the increased convergence rate does not come at the cost of less trustworthy answers. We also present some numerical examples confirming both the increased rate of convergence and the good parallel efficiency of the method.

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

© de Gruyter 2004

Articles in the same Issue

  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.2004.10.3-4.311

Articles in the same Issue

  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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