发表机构
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez; Department of Mathematics and Digital Futures, KTH Royal Institute of Technology; Inria Chile（阿道夫·伊巴涅斯大学工程与科学学院; 皇家理工学院数学与数字未来系; 智利国家信息与自动化研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了时谐麦克斯韦散射不确定性量化中边界积分算子的形状全纯性问题，通过逆变Piola变换等方法消除两大阻碍，证明算子族等可实现维度无关的最优稀疏多项式逼近。
AI 中文摘要
针对形状不确定的障碍物产生的时谐麦克斯韦散射的不确定性量化，所需的不止是散射场的全纯依赖性：对于边界元法而言，必须是边界积分算子族本身全纯依赖于形状参数。目前存在两大阻碍。电场积分方程的自然能量空间$\boldsymbol H^{-1/2}_{\text{div}_Γ}(Γ)$依赖于几何结构，而现有的针对弱奇异核的算子值形状全纯性理论建立在$L^2$空间上，无法覆盖该能量空间。\n 我们消除了这两个阻碍。一种曲面逆变Piola变换将依赖几何结构的麦克斯韦迹空间与一个固定的参考空间等同起来，并且在拉回的变分公式中，曲面雅可比行列式恰好抵消。随后，主要的分析要素是针对一致$C^{1,1}$曲面上复形变标量单层族的一致分数阶映射定理$H^{-1/2}\to H^{1/2}$，该定理通过将拉普拉斯主部实现为固定环境空间上复系数牛顿问题的迹而得到。由此，拉回算子关于$(\bm b,p,\boldsymbol\beta)$是全纯的，其中$\bm b∈\boldsymbol\beta^p(\boldsymbol\beta)$，$0<p<1$，并且在紧实参数集上逐点排除内部电共振可得到一致可逆性。\n 因此，勒让德系数是$\boldsymbol\beta^p$可和的，故而算子族、面电流和远场均能以与维度无关的最优$N$项速率实现稀疏多项式逼近。这些结论针对的是算子族本身在其能量空间算子范数下的性质，而不仅仅是针对单个解。
英文摘要
Time-harmonic Maxwell scattering by a perfectly conducting obstacle is governed by the electric field integral equation (EFIE), whose natural energy space is $\boldsymbol H^{-1/2}(\mathrm{div}_Γ,Γ)$. For countably parametrized surface deformations, both the operator and its energy space depend on the geometry. We establish operator-valued shape holomorphy of the EFIE operator on a fixed reference space and derive dimension-independent sparse approximation rates. The main analytical difficulty is a fractional mapping property absent from existing $L^2$-based operator-valued shape-holomorphy theory for weakly singular kernels: the Maxwell graph norm requires uniform holomorphy of the complex-deformed scalar single layer as an operator from $H^{-1/2}$ to $H^{1/2}$. We prove this one-order smoothing by realizing the Laplace principal part as the trace of a uniformly sectorial complex-coefficient divergence-form problem on a fixed real ambient domain. Combined with a surface contravariant Piola transformation, which identifies the geometry-dependent Maxwell trace spaces and cancels the surface Jacobians in the pulled-back EFIE exactly, this yields $(\mathbf b,p,\varepsilon)$-holomorphy of the EFIE operator family for $\ell^p$-summable shape deformations, $0<p<1$. Under pointwise exclusion of interior electric resonances, parameter-uniform invertibility is derived rather than assumed, and the surface current and far field inherit the same parametric regularity. Their Legendre coefficients are $\ell^p$-summable, yielding dimension-independent best $N$-term approximation rates. The operator-level results also transfer to Galerkin boundary element operators on a single reference mesh, with constants independent of the discretization dimension, and provide reusable surrogates for multiple incident fields and bounded linear observables.
