iQMC

The iterative Quasi-Monte Carlo (iQMC) method replaces the pseudo-random number sequences used in conventional Monte Carlo transport with low-discrepancy sequences (e.g., Halton or Sobol sequences). This reduces the variance convergence rate from the standard \(O(1/\sqrt{N})\) to as fast as \(O((\log N)^d / N)\) for \(d\)-dimensional problems, yielding significant efficiency gains for smooth solutions.

Algorithm

iQMC reformulates the transport problem as a fixed-point iteration and combines it with Krylov linear solvers such as GMRES. At each iteration:

  1. Low-discrepancy sample points are generated for the source particle phase-space coordinates.

  2. Particles are transported using the standard MC kernel, but with quasi-random initial conditions rather than pseudo-random ones.

  3. Scattering and fission source tallies are accumulated to update the right-hand side of the transport equation.

  4. A Krylov solver (e.g., GMRES) accelerates the convergence of the source iteration.

This process repeats until the scattering/fission source converges.

Spatial Error Mitigation

A known challenge with iQMC is spatial discretization error introduced by the tally mesh. MC/DC addresses this with linear discontinuous (LD) source tilting, which uses a piecewise-linear representation of the scattering and fission sources within each mesh cell rather than a flat (piecewise-constant) approximation. Additionally, effective scattering and fission rate tallies improve the consistency between the transport solve and the source update, reducing spatial bias.

Applications

iQMC is applicable to both fixed-source and k-eigenvalue problems. Output data from iQMC simulations is stored in the iqmc/tally/ group of the HDF5 output file.

For more details, see:

    1. Pasmann, I. Variansyah, C. T. Kelley, and R. G. McClarren. “Mitigating Spatial Error in iQMC with Linear Discontinuous Source Tilting and Effective Scattering and Fission Rate Tallies.” NSE (2024). Preprint DOI 10.48550/arXiv.2401.04029.

    1. Pasmann, I. Variansyah, C. T. Kelley, and R. G. McClarren. “A Quasi-Monte Carlo Method with Krylov Linear Solvers for Multigroup Neutron Transport Simulations.” NSE (2023). DOI 10.1080/00295639.2022.2143704.