We propose an isogeometric solver for Poisson problems that combines i)low-rank tensor techniques to approximate the unknown solution and the system matrix, as a sum of a few terms having Kronecker product structure, ii) a Truncated Preconditioned Conjugate Gradient solver to keep the rank of the iterates low, and iii) a novel low-rank preconditioner, based on the Fast Diagonalization method where the eigenvector multiplication is approximated by the Fast Fourier Transform. Although the proposed strategy is written in arbitrary dimension, we focus on the three-dimensional case and adopt the Tucker format for low-rank tensor representation, which is well suited in low dimension. We show by numerical tests that this choice guarantees significant memory saving compared to the full tensor representation. We also extend and test the proposed strategy to linear elasticity problems.

翻译：本文提出一种针对泊松问题的等几何求解器，该求解器结合了：i)利用低秩张量技术将未知解及系统矩阵近似表示为少量具有Kronecker乘积结构项之和；ii)采用截断预条件共轭梯度法以保持迭代解的低秩性；iii)基于快速对角化方法的新型低秩预条件器，其中特征向量乘法通过快速傅里叶变换近似实现。尽管所提策略适用于任意维度，本研究重点聚焦三维情形，并采用适合低维问题的Tucker格式进行低秩张量表示。数值实验表明，相较于全张量表示，该选择可显著节省内存。我们还将该策略扩展至线弹性问题并进行了数值验证。