The present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Padé approximation techniques introduced in Bonizzoni et al. (ESAIM Math. Model. Numer. Anal., 52(4), 1261–1284 2018, Math. Comput. 89, 1229–1257 2020). Algorithmical aspects concerning the construction of rational LS-Padé approximants are described. In particular, we show that the computation of the Padé denominator can be carried out efficiently by solving an eigenvalue-eigenvector problem involving a Gramian matrix. The LS-Padé techniques are employed to approximate the frequency response map associated with two parametric time-harmonic acoustic wave problems, namely a transmission-reflection problem and a scattering problem. In both cases, we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Padé approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation methods.

中文翻译：

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参数和随机Helmholtz映射的最小二乘Padé逼近

本工作涉及基于Bonizzoni等人中介绍的单点最小二乘（LS）Padé逼近技术的合理模型降阶方法。（ESAIM数学模型。NUMER。元素分析，52（4），1261年至1284年2018年，数学式COMPUT。89，1229–1257 2020）。描述了有关有理LS-Padé近似构造的算法方面。尤其是，我们表明，通过解决涉及Gramian矩阵的特征值-特征向量问题，可以有效地执行Padé分母的计算。LS-Padé技术用于近似与两个参数时谐声波问题（即透射反射问题和散射问题）相关的频率响应图。在这两种情况下，我们都建立了频率响应图的亚纯性。还考虑了具有随机波数的亥姆霍兹方程。特别是，对于解的Lipschitz泛函及其相应的概率测度，我们建立了从LS-Padé逼近真实值的测度的弱收敛。