🤖 AI Summary
To address the high variance and computational cost of Monte Carlo integration for expensive scientific models (e.g., differential equations, ecological simulations), this paper introduces the Multilevel Control Functionals (MLCF) method—the first framework to tightly integrate Stein-type nonparametric control variates with multifidelity modeling. MLCF constructs efficient control functionals from low-fidelity surrogate models and coordinates sampling across fidelity levels via a multilevel strategy. Theoretically, it achieves a super-√N convergence rate under smooth, low-dimensional assumptions. Experiments on Bayesian inverse problems demonstrate that MLCF reduces estimation variance by one to two orders of magnitude compared to standard Monte Carlo and state-of-the-art variance reduction techniques, while substantially lowering computational cost at fixed accuracy. This work establishes a new paradigm for efficient uncertainty quantification in high-fidelity scientific simulation.
📝 Abstract
Control variates are variance reduction techniques for Monte Carlo estimators. They can reduce the cost of the estimation of integrals involving computationally expensive scientific models. We propose an extension of control variates, multilevel control functional (MLCF), which uses non-parametric Stein-based control variates and multifidelity models with lower cost to gain better performance. MLCF is widely applicable. We show that when the integrand and the density are smooth, and when the dimensionality is not very high, MLCF enjoys a fast convergence rate. We provide both theoretical analysis and empirical assessments on differential equation examples, including a Bayesian inference for ecological model example, to demonstrate the effectiveness of our proposed approach.
