Multifidelity methods improve estimates for complex high-dimensional data
Multifidelity Monte Carlo Estimator for High-Dimensional Quantities of Interest
Machine Learning
Summary
Making accurate predictions from complex models can be very expensive to run. The authors extended a method called multifidelity Monte Carlo that uses cheaper, simpler models to help reduce the work needed for high-dimensional results. They show how to smartly combine information from multiple model types to get better estimates. Their new approach works better for vector data (multiple numbers at once) and can choose simpler or more complex combinations depending on data availability. Tests on different problems confirm the approach reduces errors efficiently.
What this means in practice
- •For computational engineers: Estimate statistics from complex physics simulations more efficiently by combining high- and low-fidelity models with vector-based corrections.
- •For environmental modelers: Improve accuracy-cost tradeoffs in spatially detailed flow and transport models by using multifidelity estimators that handle multiple output variables.
Authors
Zhibo Dai, Hamdi A. Tchelepi, Daniel M. Tartakovsky
Abstract
Multifidelity Monte Carlo (MFMC) reduces the cost of estimating the statistics of quantities of interest (QoIs) computed via expensive high-fidelity model by using cheaper low-fidelity models as control variates. We extend MFMC to vector-valued QoIs by replacing the scalar control-variate coefficients with matrices and minimizing a weighted mean-squared error. The optimal full-matrix coefficients are available in closed form, and their variance reduction is summarized by an effective correlation that takes the place of the squared Pearson correlation in the closed-form MFMC design. Restricting the coefficients to block-diagonal, diagonal, and scalar classes gives a nested hierarchy of estimators with increasing variance reduction and increasing pilot-data demand, with closed-form coefficients for compatible weightings. For a scalar QoI obtained by linearly projecting the state, correcting the full state before projecting is never worse at the population level. Experiments on a Gaussian problem, a reduced-order model of an advection--diffusion problem, and a coarse-grid two-phase flow problem show that the best class depends on the pilot size relative to the number of covariance quantities it must estimate, and that reusing the pilot samples lowers the mean-squared error.