Multilevel Monte Carlo

Multilevel Monte Carlo estimates statistics by combining many cheap coarse-grid samples with a few expensive fine-grid corrections.

The telescoping sum

Multilevel Monte Carlo (MLMC), introduced by Giles, writes the fine-level expectation as a telescoping sum: E[P_L] = E[P_0] + sum_{l=1}^{L} E[P_l - P_{l-1}]. Each difference is estimated by its own Monte Carlo sampler. The coarsest term uses many cheap samples; the correction terms, whose variance shrinks as levels refine, need few expensive samples.

Why it saves work

Kronos motion — monte carlo

The key is that the variance of the difference P_l - P_{l-1} decreases as the discretization refines, so few samples are needed at fine, expensive levels. Optimal sample allocation across levels balances cost against variance, often reducing total work by orders of magnitude compared with sampling only the finest level.

Sample allocation

Estimate per-level variance V_l and cost C_l from pilot samples
Allocate samples N_l proportional to sqrt(V_l / C_l)
Add levels until the discretization bias falls below tolerance

Requirements

MLMC needs a hierarchy of discretizations of the same problem and coupled samples, meaning the same random input drives both P_l and P_{l-1} in each difference so their correlation is high and the difference variance is small. This coupling is what makes the correction terms cheap.

Relation to multi-fidelity

MLMC uses a structured grid hierarchy with theoretical variance-decay rates, while multi-fidelity Monte Carlo allows arbitrary correlated models without a nested grid structure. Both are control-variate ideas applied across fidelity. For discretization-driven simulations with provable convergence, MLMC gives the sharpest guarantees; for heterogeneous model hierarchies, multi-fidelity Monte Carlo is more flexible.