In this paper, we address discrete linear systems in the frequency domain, where both frequency and random parameters are considered. Sampling such a system many times is computationally challenging if the size is too large, which is often the case for a finite element discretization of partial differential equations. We propose an adaptive method, steered by dual error indicators, which combines rational Arnoldi model order reduction and sparse grid interpolation with hierarchical Leja nodes. At each Leja node, a reduced order model (ROM) is constructed such that the ROM-error in the frequency domain is balanced with the sparse grid error in the random parameter domain. Both the ROM basis and the sparse grid set are enlarged in a hierarchical manner to achieve a prescribed accuracy in statistical moments. Moreover, parameter sensitivities over the frequency range can be easily extracted from the combined reduced order-surrogate model. In the numerical tests considered in the paper, the method employs sampling sets, which are reduced by at least an order of magnitude, compared to Monte Carlo simulation. Additionally, for an example from vibroacoustics, building a ROM reduces the system size by roughly a factor of 100, while still providing an acceptable accuracy.

中文翻译：

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用于频域动力系统不确定性量化的自适应稀疏网格有理 Arnoldi 方法

在本文中，我们讨论频域中的离散线性系统，其中考虑了频率和随机参数。如果规模太大，对这样的系统进行多次采样在计算上具有挑战性，这通常是偏微分方程的有限元离散化的情况。我们提出了一种由双误差指标引导的自适应方法，该方法将合理的 Arnoldi 模型阶数减少和稀疏网格插值与分层 Leja 节点相结合。在每个 Leja 节点，构建降阶模型 (ROM)，使得频域中的 ROM 误差与随机参数域中的稀疏网格误差相平衡。ROM 基和稀疏网格集都以分层方式扩大，以达到规定的统计矩精度。而且，可以从组合的降阶代理模型中轻松提取频率范围内的参数敏感性。在论文中考虑的数值测试中，该方法采用了采样集，与蒙特卡罗模拟相比，采样集至少减少了一个数量级。此外，以振动声学为例，构建 ROM 可将系统尺寸缩小大约 100 倍，同时仍能提供可接受的精度。