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Computation of empirical eigenfunctions and order reduction for control of time-dependent parabolic PDEs

机译：经验本征函数的计算和降阶，以控制时间相关的抛物线型偏微分方程

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This article presents a methodology for the computation of empirical eigenfunctions and the construction of accurate low-dimensional approximations for control of nonlinear and time-dependent parabolic partial differential equations (PDE) systems. Initially, a representative (with respect to different initial conditions and input perturbations) ensemble of solutions of the time-varying PDE system is constructed by computing and solving a high-order discretization of the PDE. Then, the Karhunen-Loeve expansion is directly applied to the ensemble of solutions to derive a small set of empirical eigenfunctions (dominant spatial patterns) that capture almost all the energy of the ensemble of solutions. The empirical eigenfunctions are subsequently used as basis functions within a Galerkin's model reduction framework to derive low-order ordinary differential equation (ODE) systems that accurately describe the dominant dynamics of the PDE system. The method is applied to a diffusion-reaction process with nonlinearities, spatially-varying coefficients and time-dependent spatial domain, and is shown to lead to the construction of accurate low-order models and the synthesis of low-order controllers. The robustness of the predictions of the low-order models with respect to variations in the model parameters and different initial conditions, as well as the comparison of their performance with respect to low-order models which were constructed by using off-the-self basis function sets are successfully shown through computer simulations.
机译：本文提出了一种计算经验特征函数的方法，并为控制非线性和时变抛物型偏微分方程（PDE）系统提供了精确的低维近似方法。最初，通过计算和求解PDE的高阶离散化，构建时变PDE系统解决方案的代表（针对不同的初始条件和输入扰动）整体。然后，将Karhunen-Loeve展开直接应用到解的集合中，以导出一小组经验特征函数（主要的空间模式），这些特征函数几乎捕获了解的集合中的所有能量。经验特征函数随后被用作Galerkin模型简化框架内的基础函数，以导出可精确描述PDE系统支配动力学的低阶常微分方程（ODE）系统。该方法被应用于具有非线性，空间变化系数和时域依赖于空间域的扩散反应过程，并被证明可以建立精确的低阶模型并合成低阶控制器。低阶模型对模型参数变化和不同初始条件的预测的鲁棒性，以及相对于通过非常规方法构建的低阶模型的性能比较通过计算机仿真成功显示了功能集。

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