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：针对泊松问题，本文提出一种结合以下技术的等几何求解器：i) 利用低秩张量技术将未知解及系统矩阵近似表示为若干具有Kronecker乘积结构项的求和形式；ii) 采用截断预处理共轭梯度法保持迭代解的秩较低；iii) 提出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。尽管所提策略可适用于任意维度，但本文着重三维情形并采用适用于低维问题的Tucker格式进行低秩张量表示。数值实验表明，与全张量表示相比，该选择可显著节约内存。我们还将所提策略扩展至线弹性问题并进行了数值验证。