Monte Carlo Fundamentals
Variance Reduction
Why Variance Reduction Is Essential
The fundamental challenge of Monte Carlo simulation is that statistical uncertainty decreases only as 1/√N -- cutting your error in half requires four times the computational effort. For many practical problems, particularly deep-penetration shielding and rare-event detection, analog (unbiased) Monte Carlo is prohibitively expensive. Variance reduction techniques change how particles are sampled to concentrate computational effort where it matters most, without biasing the expected result.
Where R is the relative error and T is the computation time. Higher FOM means greater efficiency.
The Figure of Merit is the single most important metric for evaluating variance reduction. A good technique reduces R (uncertainty) faster than it increases T (cost per particle). If FOM stays constant or increases when you apply variance reduction, the technique is working. If FOM drops, you are making things worse -- the technique is either poorly tuned or inappropriate for the problem.
All variance reduction methods obey a fundamental constraint: they change the sampling probability of particle events while adjusting particle weights to compensate. The expected value of any tally remains unbiased, but the variance of the estimate changes. The goal is to make the variance smaller for a given computational budget.
The Analog Baseline
In an analog simulation, every particle has weight 1. Interactions are sampled at their natural probabilities. If only 1 in 10⁸ source neutrons reaches a detector behind thick shielding, you need billions of histories to score even a few detector hits, resulting in enormous statistical uncertainty for reasonable run times.
Variance reduction addresses this by making more particles reach the detector (each carrying a reduced weight), so the average contribution remains correct but thevariance of the estimate shrinks dramatically.
Implicit Capture (Survival Biasing)
The simplest and most universally applied variance reduction technique is implicit capture (also called survival biasing or absorption weighting). Instead of killing a particle when it undergoes absorption, the particle survives every collision but has its weight reduced by the non-absorption probability.
The particle weight is multiplied by the ratio of scattering to total cross-section at each collision.
This keeps every particle in play longer, generating more tally contributions per history. In highly absorbing materials, analog transport kills most particles at their first collision, wasting the computational effort of tracking them to that point. Implicit capture avoids this waste entirely. This technique is so beneficial that it is enabled by default in all major Monte Carlo codes (MCNP, OpenMC, Serpent).
Geometry Splitting and Russian Roulette
Splitting and Russian roulette are complementary techniques that control particle population as particles move through the geometry. They are based on assigning an importance value to each geometric region.
Splitting
When a particle moves from a low-importance region to a high-importance region, it is split into multiple particles, each carrying a fraction of the original weight.
If the importance ratio is 4:1, one particle becomes four particles, each with 1/4 the original weight. More particles explore the important region, improving statistics.
Russian Roulette
When a particle moves from a high-importance region to a low-importance region, it plays "Russian roulette" -- it is either killed (with high probability) or survives with increased weight.
This prevents wasting computation on particles that have moved away from the region of interest. Survivors carry enough weight to preserve the unbiased estimate.
A particle entering a region with importance I₂ from a region with importance I₁ is split into n particles (if I₂ > I₁) or rouletted with survival probability I₁/I₂ (if I₂ < I₁).
Practical Importance Assignment
For shielding problems, a good starting point is to increase importance by a factor of 2-4 for each mean free path of penetration toward the detector. This roughly compensates for the exponential attenuation of the flux through the shield.
Avoid importance ratios greater than about 5-10 between adjacent regions. Large ratios cause extreme splitting (many low-weight particles) that can degrade performance. Gradual importance gradients are almost always better than sharp jumps.
Weight Windows: The Modern Standard
Weight windows generalize and improve upon simple geometry splitting/roulette by defining acceptable weight bounds in each region of phase space (position, energy, and optionally direction). Particles with weights above the upper bound are split; particles below the lower bound are rouletted. Particles within the window pass through without modification.
Lower bound is inversely proportional to the adjoint flux (importance). The ratio R between upper and lower bounds is typically 2-5.
The key advantage of weight windows over simple importance-based splitting is that they control particle weights in phase space, not just geometry. A particle can be split or rouletted based on its energy as well as its location, allowing much finer control over the simulation.
Weight windows also prevent the "weight explosion" problem where variance reduction creates a few particles with extremely large weights. By enforcing both upper and lower bounds, weight windows keep the particle population well-behaved, leading to more reliable statistics.
Weight Window Tuning
- Start conservatively: Use wide windows (large R ratio) initially, then tighten as you gain confidence
- Monitor the FOM: If FOM drops after applying weight windows, the parameters need adjustment
- Check particle population: Look for regions with too many or too few particles -- this indicates poorly tuned bounds
- Iterate: Run a short calculation, examine the weight distribution, adjust bounds, repeat
CADIS and FW-CADIS: Automated Variance Reduction
The Consistent Adjoint Driven Importance Sampling (CADIS) method automates the generation of variance reduction parameters by solving a deterministic adjoint transport problem. This removes the guesswork from variance reduction and provides near-optimal parameters for many practical problems.
CADIS for Single-Tally Problems
CADIS works by first running a fast deterministic (Sn) adjoint calculation with a source placed at the detector location. The adjoint flux φ† represents the importance of particles at each point for contributing to the detector response. CADIS then uses φ† to generate both optimized weight windows and a biased source distribution:
- Weight windows: Set inversely proportional to φ† -- particles in high-importance regions get tight, low weight bounds (encouraging more particles there)
- Biased source: The source is biased proportionally to q(r,E) × φ†(r,E), so more source particles start where they will contribute most to the detector
FW-CADIS for Global Problems
Forward-Weighted CADIS extends the method to problems where you need good statistics everywhere, not just at a single detector. Examples include full-geometry dose maps, mesh tally calculations, and shutdown dose rate problems.
FW-CADIS first runs a forward deterministic calculation to estimate the flux everywhere. It then constructs an adjoint source that is inversely weighted by the forward flux, so that regions with naturally low flux get extra importance. The result is relatively uniform statistical uncertainty across the entire problem domain -- exactly what you need for global tallies.
CADIS/FW-CADIS Availability
These methods are implemented in the ADVANTG code (which generates weight windows for MCNP) and in the MAVRIC sequence of the SCALE code system. OpenMC can import weight windows generated by external tools. Serpent does not natively implement CADIS but supports user-defined importance maps. The CADIS methodology is one of the most significant practical advances in Monte Carlo efficiency over the past two decades.
Other Variance Reduction Techniques
Forced Collisions
In thin cells where particles usually pass through without interacting, forced collisions guarantee that a (reduced-weight) collision occurs. The uncollided particle continues with appropriately adjusted weight. This is valuable for thin detector volumes or activation foils.
DXTRAN Spheres (MCNP)
DXTRAN is an MCNP-specific technique that deterministically transports a fraction of each collision's outgoing weight to a spherical shell around a detector. It is extremely effective for small point-like detectors in large geometries, but requires careful setup -- poorly placed DXTRAN spheres can waste computation or introduce large weight fluctuations.
Source Biasing
Instead of sampling source particles uniformly, you can bias the source distribution to emit more particles toward the region of interest. Source energy biasing emits more particles at energies important for the tally. Source spatial biasing concentrates starting positions where they contribute most. Both require corresponding weight adjustments to maintain an unbiased result.
Exponential Transform
This technique modifies the effective cross-section along the particle's flight direction to stretch path lengths toward a preferred direction (usually toward the detector). It is most effective for one-dimensional streaming problems but can be counterproductive for complex 3D geometries where the optimal streaming direction varies spatially.
Practical Variance Reduction Strategy
Applying variance reduction effectively requires a systematic approach. Jumping directly to aggressive biasing without first understanding the problem usually leads to wasted effort or, worse, biased-looking results with artificially small uncertainties.
Step-by-Step Recipe
- Run a short analog calculation first. Get baseline FOM and understand the particle transport behavior. Where do particles go? Where do they die? How many reach the tally?
- Identify the bottleneck. Is the problem deep penetration (particles attenuated before reaching the detector)? Rare events (low-probability interactions)? Geometric streaming (particles missing a small target)?
- Choose the simplest appropriate technique. Start with coarse importance-based splitting/roulette. If that is insufficient, upgrade to energy-dependent weight windows. Only use CADIS/FW-CADIS or DXTRAN for genuinely difficult problems.
- Apply conservatively and verify. Use moderate splitting ratios (2-4). Check that the mean result is consistent with the analog run. Verify that the FOM has increased.
- Iterate and refine. Tighten weight windows gradually. Monitor statistical tests. Stop when the FOM plateaus or when the required accuracy is achieved.
Warning Signs of Bad Variance Reduction
- Decreasing FOM: The technique is making things worse, not better
- Large weight fluctuations: A few particles dominate the tally -- look for very high or very low weights
- Failed statistical tests: MCNP's 10 statistical checks should all pass; failures indicate unreliable results
- Mean value shift: If the mean changes significantly when you apply variance reduction, something is wrong with the implementation (possible bug, not a biasing issue per se)
- Very small reported uncertainty: Paradoxically, extremely small uncertainties from a short run can indicate that a few high-weight particles are dominating and the true uncertainty is much larger