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Coin Tossing Algorithms for Integral Equations and Tractability

  • Erich Novak and Harald Pfeiffer
Published/Copyright: May 9, 2008
Monte Carlo Methods and Applications
From the journal Volume 10 Issue 3-4

Integral equations with Lipschitz kernels and right-hand sides are intractable for deterministic methods, the complexity increases exponentially in the dimension d. This is true even if we only want to compute a single function value of the solution. For this latter problem we study coin tossing algorithms (or restricted Monte Carlo methods), where only random bits are allowed. We construct a restricted Monte Carlo method with error ε that uses roughly ε−2 function values and only d log2 ε random bits. The number of arithmetic operations is of the order ε−2 + d log2 ε. Hence, the cost of our algorithm increases only mildly with the dimension d, we obtain the upper bound C · (ε−2 + d log2 ε) for the complexity. In particular, the problem is tractable for coin tossing algorithms.

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.2004.10.3-4.491

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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