Multi-Fidelity Monte Carlo

Multi-fidelity Monte Carlo estimates statistics by combining a few high-fidelity samples with many cheap low-fidelity ones as control variates.

Generalized control variates

Multi-fidelity Monte Carlo (MFMC) extends control variates to a hierarchy of models with unknown means. The expensive high-fidelity model is the target; cheaper models serve as control variates whose means are estimated from their own large sample sets. The combined estimator is unbiased for the high-fidelity mean but has far lower variance than high-fidelity sampling alone.

No grid hierarchy required

Kronos motion — monte carlo

Unlike multilevel Monte Carlo, MFMC does not need nested discretizations. The low-fidelity models can be reduced-physics models, coarse surrogates, or emulators of any kind, as long as they correlate with the high-fidelity output. This flexibility makes it broadly applicable.

Optimal allocation

Estimate correlations rho_k between each model and the high-fidelity target
Estimate per-model evaluation costs
Solve for the number of samples per model and the control coefficients that minimize estimator variance under a compute budget

When it pays off

The variance reduction grows with correlation and with the cost ratio between levels. Highly correlated, much cheaper low-fidelity models yield the largest gains; weakly correlated models add little and can be dropped by the allocation formula, which assigns them near-zero weight.

Practice

MFMC is well suited to design studies where a fast reduced model and a slow high-fidelity model describe the same system. For a fusion figure of merit, a 0-D scaling model can act as the control variate for full transport runs, so the expensive solver is invoked only enough times to correct the cheap estimate. Correlations must be re-estimated if the operating region changes, since a model correlation valid in one regime may not hold in another.