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 in 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：我们提出了一种用于泊松问题的等几何求解器，该求解器结合了：i) 低秩张量技术，将未知解和系统矩阵近似表示为少量具有Kronecker积结构的项之和；ii) 截断预处理共轭梯度求解器，以保持迭代解的秩较低；以及iii) 一种基于快速对角化方法的新型低秩预条件子，其中特征向量乘法通过快速傅里叶变换近似。尽管所提出的策略适用于任意维度，但我们重点关注三维情况，并采用Tucker格式进行低秩张量表示，该格式在低维度下表现优异。数值实验表明，与全张量表示相比，该方案能够显著节省内存。我们还将所提策略扩展至线性弹性问题并进行了测试。