Variance Reduction Techniques¶

Analog Monte Carlo converges as \(O(1/\sqrt{N})\), which can be prohibitively slow for deep-penetration or rare-event problems. MC/DC provides several variance reduction (VR) techniques that reduce the statistical uncertainty per particle history without introducing bias.

All techniques below are activated through the mcdc.simulation interface.

Implicit Capture¶

Also called survival biasing or absorption suppression. Instead of terminating a particle at every capture event, the particle’s weight is reduced by the non-absorption probability after each collision:

\[w' = w \; \frac{\Sigma_s + \Sigma_f}{\Sigma_t}\]

The particle continues with the reduced weight, ensuring that every history contributes to tallies for longer. This is especially effective in highly absorbing media.

Usage:

mcdc.simulation.implicit_capture()

Note

Implicit capture is enabled by default in most MC/DC examples. Without a companion technique (weight roulette or weight windows) to eliminate very-low-weight particles, a memory overhead can build up over time.

Weight Roulette¶

When a particle’s weight drops below a threshold \(w_{\text{thresh}}\), Russian roulette is applied:

With probability \(p = w / w_{\text{target}}\), the particle survives and its weight is set to \(w_{\text{target}}\).
With probability \(1 - p\), the particle is killed.

This prevents an ever-growing population of low-weight particles while preserving the expected weight (unbiased).

Usage:

mcdc.simulation.global_weight_roulette(weight_threshold=0.25, weight_target=1.0)

weight_threshold and weight_target should be chosen so that \(w_{\text{thresh}} < w_{\text{target}}\); a common ratio is \(w_{\text{thresh}} / w_{\text{target}} \approx 0.25\).

Weighted Emission¶

In fission problems, the number of secondary neutrons is \(\lfloor\nu + \xi\rfloor\) in analog mode. Weighted emission instead emits a fixed number of secondaries (weight_target worth of weight), adjusting their weights so that the total expected weight is preserved:

\[w_{\text{child}} = \frac{w \cdot \nu}{n_{\text{emit}}}\]

This reduces the variance of the fission source weight distribution.

Usage:

mcdc.simulation.weighted_emission(active=True, weight_target=1.0)

Population Control¶

In time-dependent (transient) problems, the neutron population can grow or decay exponentially, making it difficult to maintain a well-sampled phase space. Population control adjusts the particle bank at each time census by splitting high-weight particles and rouletting low-weight ones, targeting a uniform weight close to \(w_{\text{target}}\).

Usage:

mcdc.simulation.population_control()

Population control is typically combined with a time census (set_time_census) that checkpoints the particle population at specified time boundaries.

Weight Windows¶

Weight windows define position-dependent (and optionally energy- and time-dependent) target weights and bounds. They combine splitting and roulette to focus computational effort in regions of high importance.

MC/DC supports both user-defined and automatically generated weight windows. See Weight Windows for a full description of the available strategies (WW_USER and WW_PREVIOUS) and modification schemes (WW_MIN and WW_WOLLABER).

Combining Techniques¶

VR techniques are designed to be composable. A typical production setup might use:

mcdc.simulation.implicit_capture()
mcdc.simulation.global_weight_roulette(weight_threshold=0.25, weight_target=1.0)

For time-dependent fission problems:

mcdc.simulation.implicit_capture()
mcdc.simulation.weighted_emission(active=True, weight_target=1.0)
mcdc.simulation.population_control()

The order of activation does not matter — MC/DC applies them in the correct transport-physics order internally.

For quasi-Monte Carlo acceleration of the source iteration, see iQMC.

References¶

T. E. Booth. “A Sample Problem for Variance Reduction in MCNP.” LA-10363-MS, LANL (1985).
A. B. Wollaber. “Advanced Monte Carlo Methods for Radiation Transport.” PhD diss., Univ. of Michigan (2016).