We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction-diffusion PDE (RD-PDE) systems. We show that solutions of surrogate models built by classical Proper Orthogonal Decomposition (POD) exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback, first of all, we propose a POD-DEIM technique with a correction term that includes missing information in the reduced models. To improve the computational efficiency, we propose an adaptive version of this algorithm in time that accounts for the peculiar dynamics of the RD-PDE in presence of Turing instability. We show the effectiveness of the proposed methods in terms of accuracy and computational cost for a selection of RD systems, i.e. FitzHugh-Nagumo, Schnakenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns.

中文翻译：

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反应扩散 PDE 系统中图灵模式近似的自适应 POD-DEIM 校正

我们研究了模型降阶 (MOR) 技术在图灵模式数值逼近中的合适应用，图灵模式是反应扩散 PDE (RD-PDE) 系统的稳态解。我们表明，由经典本征正交分解 (POD) 构建的替代模型的解在缩减空间的维度上表现出不稳定的错误行为。为了克服这个缺点，首先，我们提出了一种带有校正项的 POD-DEIM 技术，该校正项包括简化模型中的缺失信息。为了提高计算效率，我们及时提出了该算法的自适应版本，该版本考虑了存在图灵不稳定性时 RD-PDE 的特殊动力学。我们在选择 RD 系统的准确性和计算成本方面展示了所提出方法的有效性，即