Gaussian-process surrogate indicators for residual-based adaptive GMsFEM
Abstract
Residual-based adaptive GMsFEM for high-contrast elliptic problems repeatedly evaluates local weighted H−1 indicators on every coarse neighborhood, making indicator evaluation a recurring cost in repeated-query settings. We introduce a non-intrusive Gaussian-process (GP) surrogate for the indicator scores used in Dörfler marking. The operational predictor is the GP posterior mean, algebraically equivalent to a kernel ridge regression (KRR) estimator under the stated convention; it uses compressed local solution and spectral features without changing the multiscale solve, local spectral construction, or basis enrichment. A nonuniform perturbed-marking result quantifies how pointwise score errors affect the exact indicator mass captured by surrogate-selected neighborhoods, while a conditional bounded-discrepancy KRR pathway identifies sufficient assumptions for such score bounds. In controlled held-out in-distribution tests, the surrogate-guided method gives error-versus-DoF trends comparable with classical H−1-residual offline adaptivity and evaluates the online indicator component 2.0-2.1 times faster, excluding offline data generation and GP training.
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BibTeX
@misc{liu2026gaussianprocess,
title = {{Gaussian-process surrogate indicators for residual-based adaptive GMsFEM}},
author = {Siqing Liu and Eric Chung and Yiran Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26843},
eprint = {2609.26843},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26843},
}
RIS
TY - UNPB
TI - Gaussian-process surrogate indicators for residual-based adaptive GMsFEM
AU - Siqing Liu
AU - Eric Chung
AU - Yiran Wang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26843
UR - https://arxiv.org/abs/2609.26843
AB - Residual-based adaptive GMsFEM for high-contrast elliptic problems repeatedly evaluates local weighted H^−1 indicators on every coarse neighborhood, making indicator evaluation a recurring cost in repeated-query settings. We introduce a non-intrusive Gaussian-process (GP) surrogate for the indicator scores used in Dörfler marking. The operational predictor is the GP posterior mean, algebraically equivalent to a kernel ridge regression (KRR) estimator under the stated convention; it uses compressed local solution and spectral features without changing the multiscale solve, local spectral construction, or basis enrichment. A nonuniform perturbed-marking result quantifies how pointwise score errors affect the exact indicator mass captured by surrogate-selected neighborhoods, while a conditional bounded-discrepancy KRR pathway identifies sufficient assumptions for such score bounds. In controlled held-out in-distribution tests, the surrogate-guided method gives error-versus-DoF trends comparable with classical H^−1-residual offline adaptivity and evaluates the online indicator component 2.0-2.1 times faster, excluding offline data generation and GP training.
ER -