In this manuscript we present the tensor-train reduced basis method, a novel projection-based reduced-order model for the efficient solution of parameterized partial differential equations. Despite their popularity and considerable computational advantages with respect to their full order counterparts, reduced-order models are typically characterized by a considerable offline computational cost. The proposed approach addresses this issue by efficiently representing high dimensional finite element quantities with the tensor train format. This method entails numerous benefits, namely, the smaller number of operations required to compute the reduced subspaces, the cheaper hyper-reduction strategy employed to reduce the complexity of the PDE residual and Jacobian, and the decreased dimensionality of the projection subspaces for a fixed accuracy. We provide a posteriori estimates that demonstrate the accuracy of the proposed method, we test its computational performance for the heat equation and transient linear elasticity on three-dimensional Cartesian geometries.

翻译：本文提出了一种基于投影的新型降阶模型——张量列降基方法，用于高效求解参数化偏微分方程。尽管降阶模型相较于全阶模型具有显著的计算优势且应用广泛，但其通常伴随着可观的离线计算成本。本方法通过张量列格式高效表示高维有限元量，有效解决了这一问题。该方法具有多重优势：计算降维子空间所需的操作次数更少，降低偏微分方程残差与雅可比矩阵复杂度的超降阶策略成本更低，以及在固定精度下投影子空间的维度更低。我们给出了验证方法精度的后验估计，并在三维笛卡尔几何结构上针对热传导方程和瞬态线性弹性问题测试了其计算性能。