A Tensor Train-Based Isogeometric Solver for Large-Scale 3D Poisson Problems on Complex Geometries

Abstract

We introduce a three-dimensional (3D) fully tensor train (TT)-assembled isogeometric analysis (IGA) framework, TT-IGA, for solving partial differential equations (PDEs) on complex geometries. Our method reformulates IGA discrete operators into TT format, enabling efficient compression and computation while retaining geometric flexibility and accuracy. Unlike previous low-rank approaches that typically rely on structured domains, our framework accommodates general 3D geometries through low-rank TT representations of both the geometry mapping and the PDE discretization. We demonstrate the effectiveness of the proposed TT-IGA framework on the 3D Poisson equation, achieving substantial reductions in memory usage and computational cost without compromising solution quality.