冻结Calderón：形状不确定性下Maxwell散射的鲁棒预条件

中文摘要 AI 辅助

Calderón预条件恢复了Maxwell电场积分方程边界元离散的网格无关条件数，但组装对偶重心算子可能主导计算成本。我们证明该预条件器可以在名义几何上一次组装，并在保持连通性的形状扰动中重复使用。RWG-BC自由度的精确组合对齐传递了对偶配对，仅留下对偶网格算子被冻结。在均匀离散稳定性和几何算子连续性假设下，双参数算子预条件分析产生了谱条件数界，该界在网格尺寸上一致，并在张量化后在随机伽辽金维度上一致。在光滑和非光滑散射体上的数值实验表明，冻结预条件器在显著形状变化下紧密跟踪新组装的Calderón预条件器。在NASA杏仁体上的蒙特卡洛活动，其中每个形状构建前筛选可接受性，因此没有实现被拒绝，将两种策略的差异小于实现之间的变化，并且那里的网格细化使冻结迭代次数保持不变。在我们的密集实现中，重用每个新实现产生约30倍的实测加速；基于已发表压缩组装时间的成本模型估计约为2倍。

英文摘要

Calderón preconditioning restores mesh-independent conditioning for boundary-element discretizations of the Maxwell electric field integral equation, but assembling the dual barycentric operator can dominate the computational cost. We show that this preconditioner can be assembled once at a nominal geometry and reused across connectivity-preserving shape perturbations. Exact combinatorial alignment of the RWG-BC degrees of freedom transports the duality pairing, leaving only the dual-mesh operator frozen. Under uniform discrete-stability and geometric operator-continuity assumptions, a bi-parametric operator-preconditioning analysis yields a spectral-condition-number bound uniform in the mesh size and, after tensorization, in the stochastic-Galerkin dimension. Numerical experiments on smooth and non-smooth scatterers show that the frozen preconditioner closely tracks freshly assembled Calderón preconditioners over substantial shape variations. A Monte Carlo campaign on a NASA almond, whose admissibility is screened before each shape is constructed so that no realization is rejected, separates the two strategies by less than the variation between realizations, and mesh refinement there leaves the frozen iteration count unchanged. Reuse yields roughly 30x measured speedup per new realization in our dense implementation; a cost-model estimate based on published compressed-assembly timings gives roughly 2x.

发表机构

• Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学工程与科学学院）
• Department of Mathematics and Digital Futures, KTH Royal Institute of Technology（瑞典皇家理工学院数学与数字未来系）
• Inria Chile（法国国家信息与自动化研究所智利分部）

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