Posterior variance as an adaptive stopping criterion

Determine whether the posterior variance of the multilevel Gaussian-process surrogate can serve as a computable error estimator and stopping criterion that enables the adaptive multilevel algorithm to target a prescribed approximation tolerance.

Background

The paper develops a multilevel Gaussian-process surrogate whose posterior mean approximates a hierarchy-based response surface and whose posterior variance quantifies predictive uncertainty. The theoretical analysis establishes that the posterior mean error and integrated posterior standard deviation have matching convergence-rate bounds, while numerical experiments show that the integrated posterior standard deviation can track or overestimate the posterior-mean error in the tested regime.

The authors identify an unresolved practical question: whether this computable posterior-variance quantity can reliably function as an error estimator and thereby provide a stopping rule for the adaptive algorithm. Resolving this issue would allow adaptive refinement to terminate when a prescribed tolerance is reached without requiring access to the exact response or its true approximation error.