On Uncertainty Quantification of Eigenvalues and Eigenspaces with Higher Multiplicity

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-02-07 DOI:10.1137/22m1529324

Jürgen Dölz, David Ebert

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Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 422-451, February 2024.
Abstract. We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fréchet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fréchet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results.

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论高倍性特征值和特征空间的不确定性量化

SIAM 数值分析期刊》第 62 卷第 1 期第 422-451 页，2024 年 2 月。摘要。我们考虑双线性形式中带有随机扰动的变分形式广义算子特征值问题。这种设置的动机是具有随机输入数据的偏微分方程的变分形式。所考虑的特征对可能具有更高但有限的多重性。我们研究了特征对的相关随机量，并讨论了为什么当倍率大于 1 时，只有特征空间的随机特性是有意义的，而单个特征对的随机特性却没有意义。为此，我们描述了特征对相对于扰动的弗雷谢特导数，并为更高倍率的特征对提供了新的线性描述。作为附带结果，我们证明了特征空间的局部解析性。基于特征对的弗雷谢特导数，我们讨论了针对多特征值的有意义的蒙特卡罗采样策略，并开发了一种不确定性量化扰动方法。为了说明理论结果，我们给出了一些数值示例。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.