SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1874-1900, August 2024.
Abstract. We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyze Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique, which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients, e.g., integration by quasi–Monte Carlo methods.

中文翻译：

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参数椭圆特征值问题的解析和Gevrey类正则性及其应用


《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1874-1900 页，2024 年 8 月。

抽象的。我们研究一类具有齐次基本边界条件的参数椭圆特征值问题，其中系数（以及解）可能取决于参数。为了有效地近似评估整个参数空间上第一特征对的参数敏感性，我们提出并分析了关于参数的解的 Gevrey 类和解析规律。这是通过一种新颖的证明技术实现的，我们在本文中介绍并演示了该技术。我们的正则性结果对于参数椭圆特征值问题的各种数值方案的收敛具有直接影响，特别是对于由具有随机系数的椭圆微分算子产生的具有无限多个参数的椭圆特征值问题，例如，通过准蒙特卡罗方法进行积分。