This paper is concerned with model order reduction of parametric Partial Differential Equations (PDEs) using tree-based library approximations. Classical approaches are formulated for PDEs on Hilbert spaces and involve one single linear space to approximate the set of PDE solutions. Here, we develop reduced models relying on a collection of linear or nonlinear approximation spaces called a library, and which can also be formulated on general metric spaces. To build the spaces of the library, we rely on greedy algorithms involving different splitting strategies which lead to a hierarchical tree-based representation. We illustrate through numerical examples that the proposed strategies have a much wider range of applicability in terms of the parametric PDEs that can successfully be addressed. While the classical approach is very efficient for elliptic problems with strong coercivity, we show that the tree-based library approaches can deal with diffusion problems with weak coercivity, convection-diffusion problems, and with transport-dominated PDEs posed on general metric spaces such as the $L^2$-Wasserstein space.

翻译：本文关注参数化偏微分方程（PDEs）的模型降阶，采用基于树的库逼近方法。经典方法针对希尔伯特空间上的PDEs构建，并利用单个线性空间来逼近PDE解集。本文开发了基于线性或非线性逼近空间集合（称为库）的降阶模型，同时可在一般度量空间上建立。为构建库中的空间，我们采用基于不同分裂策略的贪婪算法，从而得到层次化的树形表示。通过数值算例表明，所提策略在可成功处理的参数化PDEs类型上具有更广泛的适用性。经典方法对强椭圆问题高效，而本文证明基于树的库方法能处理弱椭圆扩散问题、对流扩散问题，以及定义在一般度量空间（如$L^2$-Wasserstein空间）上的传输主导型PDEs。