基于残差的自适应GMsFEM的高斯过程替代指标
Gaussian-process surrogate indicators for residual-based adaptive GMsFEM
- The Chinese University of Hong Kong（香港中文大学）
- The University of Alabama（阿拉巴马大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出高斯过程替代指标加速基于残差的自适应GMsFEM，在不改变多尺度求解下，通过KRR等价预测器实现2.0-2.1倍在线加速，并保持误差-自由度趋势相当。
中文摘要 AI 辅助
针对高对比度椭圆问题的基于残差的自适应GMsFEM方法在每个粗邻域上反复评估局部加权$H^{-1}$指标，使得在重复查询场景中指标评估成为一项重复性成本。我们引入了一种非侵入式的高斯过程（GP）替代模型，用于Dörfler标记中使用的指标分数。操作预测器为GP后验均值，在所述约定下代数等价于核岭回归（KRR）估计器；它使用压缩的局部解和谱特征，而不改变多尺度求解、局部谱构造或基扩充。一个非均匀扰动标记结果量化了点态分数误差如何影响由替代模型选择的邻域捕获的精确指标质量，而一个条件有界差异KRR路径为这类分数界限确定了充分假设。在受控的保留分布内测试中，替代模型引导的方法给出了与经典$H^{-1}$残差离线自适应性相当的误差-自由度趋势，并将在线指标组件评估速度提高了2.0-2.1倍，不包括离线数据生成和GP训练。
英文摘要
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.
