The ROMES method for statistical modeling of reduced-order-model error
This work addresses error estimation for reduced-order models in computational science, offering a novel probabilistic approach that enhances accuracy and provides uncertainty control, though it builds on existing methods like Gaussian processes and error bounds.
The paper tackles the problem of modeling errors in reduced-order models by using Gaussian-process regression to map error indicators to error distributions, resulting in improved prediction accuracy by an order of magnitude and near-optimal expected effectivity compared to typical error bounds.
This work presents a technique for statistically modeling errors introduced by reduced-order models. The method employs Gaussian-process regression to construct a mapping from a small number of computationally inexpensive `error indicators' to a distribution over the true error. The variance of this distribution can be interpreted as the (epistemic) uncertainty introduced by the reduced-order model. To model normed errors, the method employs existing rigorous error bounds and residual norms as indicators; numerical experiments show that the method leads to a near-optimal expected effectivity in contrast to typical error bounds. To model errors in general outputs, the method uses dual-weighted residuals---which are amenable to uncertainty control---as indicators. Experiments illustrate that correcting the reduced-order-model output with this surrogate can improve prediction accuracy by an order of magnitude; this contrasts with existing `multifidelity correction' approaches, which often fail for reduced-order models and suffer from the curse of dimensionality. The proposed error surrogates also lead to a notion of `probabilistic rigor', i.e., the surrogate bounds the error with specified probability.