We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that allow for low-rank variation in the state matrix. Usual approaches for parametric model reduction typically involve exploring the parameter space to identify representative parameter values and the associated models become the principal focus of model reduction methodology. These models are then combined in various ways in order to interpolate the response. The initial exploration of the parameter space can be a forbiddingly expensive task. A different approach is proposed here that requires neither parameter sampling nor parameter space exploration. Instead, we represent the system response function as a composition of four subsystem response functions that are non-parametric with a purely parameter-dependent function. One may apply any one of a number of standard (non-parametric) model reduction strategies to reduce the subsystems independently, and then conjoin these reduced models with the underlying parameterization to obtain the overall parameterized response. Our approach has elements in common with the parameter mapping approach of Baur et al. (PAMM 14(1), 19–22 2014) but offers greater flexibility and potentially greater control over accuracy. In particular, a data-driven variation of our approach is described that exercises this flexibility through the use of limited frequency-sampling of the underlying non-parametric models. The parametric structure of our system representation allows for a priori guarantees of system stability in the resulting reduced models across the full range of parameter values. Incorporation of system theoretic error bounds allows us to determine appropriate approximation orders for the non-parametric systems sufficient to yield uniformly high accuracy across the parameter range. We illustrate our approach on a class of structural damping optimization problems and on a benchmark model of thermal conduction in a semiconductor chip. The parametric structure of our reduced system representation lends itself very well to the development of optimization strategies making use of efficient cost function surrogates. We discuss this in some detail for damping parameter and location optimization for vibrating structures.

我们考虑具有仿射参数依赖性的线性动力学系统的参数族的减少，从而允许状态矩阵的低秩变化。用于参数化模型简化的常用方法通常涉及探索参数空间以识别代表性参数值，并且相关联的模型成为模型简化方法学的主要重点。然后以各种方式组合这些模型，以内插响应。对参数空间的初步探索可能是一项昂贵的任务。这里提出了一种不同的方法，该方法既不需要参数采样也不需要参数空间探索。相反，我们将系统响应函数表示为四个非参数子系统响应函数的组合具有纯参数相关的功能。可以应用多种标准（非参数）模型简化策略中的任何一种来独立地简化子系统，然后将这些简化的模型与基础参数化结合起来，以获得整体参数化响应。我们的方法具有与Baur等人的参数映射方法相同的元素。（PAMM 14（1），2014年19月22日），但提供了更大的灵活性，并可能更好地控制准确性。特别是，描述了我们方法的数据驱动型变体，该变体通过使用底层非参数模型的有限频率采样来行使这种灵活性。我们系统表示的参数结构允许在整个参数值范围内的简化模型中获得系统稳定性的先验保证。系统理论误差范围的合并使我们能够为非参数系统确定适当的近似阶次，足以在整个参数范围内产生均一的高精度。我们将说明针对一类结构阻尼优化问题和半导体芯片热传导基准模型的方法。我们减少的系统表示形式的参数结构非常适合于使用高效成本函数代理的优化策略的开发。我们将针对振动结构的阻尼参数和位置优化进行详细讨论。