This work develops a numerical solver based on the combination of isogeometric analysis (IGA) and the tensor train (TT) decomposition for the approximation of partial differential equations (PDEs) on parameter-dependent geometries. First, the discrete Galerkin operator as well as the solution for a fixed geometry configuration are represented as tensors and the TT format is employed to reduce their computational complexity. Parametric dependencies are included by considering the parameters that control the geometry configuration as additional dimensions next to the physical space coordinates. The parameters are easily incorporated within the TT-IGA solution framework by introducing a tensor product basis expansion in the parameter space. The discrete Galerkin operators are accordingly extended to accommodate the parameter dependence, thus obtaining a single system that includes the parameter dependency. The system is solved directly in the TT format and a low-rank representation of the parameter-dependent solution is obtained. The proposed TT-IGA solver is applied to several test cases which showcase its high computational efficiency and tremendous compression ratios achieved for representing the parameter-dependent IGA operators and solutions.

这项工作开发了一种基于等几何分析 (IGA) 和张量序列 (TT) 分解相结合的数值求解器，用于在参数相关几何上近似偏微分方程 (PDE)。首先，离散 Galerkin 算子以及固定几何配置的解表示为张量，并采用 TT 格式来降低其计算复杂度。通过将控制几何配置的参数视为物理空间坐标旁边的附加维度，可以包含参数相关性。通过引入一个参数空间中的张量积基展开。离散 Galerkin 算子相应地被扩展以适应参数依赖性，从而获得包括参数依赖性的单个系统。系统直接以 TT 格式求解，并获得参数相关解的低秩表示。所提出的 TT-IGA 求解器应用于多个测试用例，展示了其高计算效率和巨大的压缩比，用于表示依赖于参数的 IGA 算子和解决方案。