This paper presents an adaptive sampling algorithm tailored for the optimization of parametrized dynamical systems using projection-based model order reduction. Unlike classical sampling strategies, this framework does not aim for a small approximation error in the global sense but focuses on identifying and refining promising regions early on while reducing expensive full order model evaluations. The algorithm is tested on two models: a Timoshenko beam and a Kelvin cell, which ought to be optimized in terms of the system output in the frequency domain. For that, different norms of the transfer function are used as the objective function, while up to two geometrical parameters form the vector of design variables. The sampled full order models are reduced using the iterative rational Krylov algorithm and reprojected into a global basis. Subsequently, the models are parametrized by performing sparse Bayesian regression on matrix entry level of the reduced operators. Thompson sampling is carried out using the posterior distribution of the polynomial coefficients in order to account for uncertainties in the trained regression models. The strategy deployed for sample acquisition incorporates a gradient-based search on the parametrized reduced order model, which involves analytical gradients obtained via adjoint sensitivity analysis. By adding the found optimum to the sample set, the sample set is iteratively refined. Results demonstrate robust convergence towards the global optimum but highlight the computational cost introduced by the gradient-based optimization. The probabilistic extensions seamlessly integrate into existing matrix-interpolatory reduction frameworks and enable the analytical calculation of gradients under uncertainty.

翻译：本文提出一种自适应采样算法，专为基于投影的模型降阶方法在参数化动力系统优化中的应用而设计。与经典采样策略不同，该框架并不追求全局意义上的较小近似误差，而是侧重于早期识别并细化有潜力的区域，同时减少昂贵的高阶模型评估。该算法在两个模型上进行测试：Timoshenko梁和Kelvin胞元结构，两者均需在频域内针对系统输出进行优化。为此，采用传递函数的不同范数作为目标函数，同时使用最多两个几何参数构成设计变量向量。采样得到的高阶模型通过迭代有理Krylov算法进行降阶，并重新投影到全局基中。随后，通过对降阶算子矩阵元素进行稀疏贝叶斯回归实现模型参数化。为考虑训练回归模型中的不确定性，采用Thompson采样方法，利用多项式系数的后验分布进行采样。样本采集策略结合了基于参数化降阶模型的梯度搜索，其中梯度通过伴随灵敏度分析获得解析解。通过将寻得的最优点加入样本集，实现样本集的迭代优化。结果表明，该方法能稳健收敛至全局最优解，但也凸显了基于梯度的优化带来的计算成本。概率扩展可无缝集成到现有矩阵插值降阶框架中，实现在不确定性条件下的解析梯度计算。