Optimize parameter selection in MLMC with preintegration

Determine the optimal choices of the multilevel parameters—maximum level L, per-level outer sample sizes N_ℓ, numbers of Laguerre quadrature nodes M_{Lag,ℓ}, and Newton iteration tolerances TOL_{Newton,ℓ}—that minimize the total computational cost ∑_{ℓ=0}^L N_ℓ m_ℓ (M_{Lag,ℓ} + log(TOL_{Newton,ℓ}^{-1})) subject to the mean squared error constraint E[(E[g(φ(ω))] − Q̄)^2] = TOL^2, where Q̄ is the MLMC estimator with numerical preintegration and m_ℓ is the inner sample size at level ℓ.

Background

The paper formulates a concrete constrained optimization problem to minimize the total cost of the proposed MLMC estimator with numerical preintegration, under a prescribed mean squared error tolerance. The cost per level incorporates both quadrature-point selection for the preintegration and the Newton iteration tolerance used to locate discontinuity points, reflecting the practical trade-offs in computation.

Although the authors provide heuristic parameter choices and derive overall complexity bounds, they explicitly state that they did not solve the posed optimization problem and leave its resolution—optimal allocation of N_ℓ, M_{Lag,ℓ}, and TOL_{Newton,ℓ} across levels—for future work.