SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A800-A827, January 2021.
The aim of this paper is to propose a robust numerical solver, which is capable of efficiently solving a three-dimensional elliptic problem in a data-sparse quantized tensor format. In particular, we use the combined Tucker and quantized tensor train format (TQTT), which allows us to use astronomically large grid sizes. However, due to the ill-conditioning of discretized differential operators, such fine grids lead to numerical instabilities. To obtain a robust solver, we utilize the well-known alternating direction implicit method and modify it to avoid multiplication by differential operators. So as to make the method efficient, we derive an explicit TQTT representation of the iteration matrix and quantized tensor train representations of the inverses of symmetric tridiagonal Toeplitz matrices as an auxiliary result. As an application, we consider accurate solution of elliptic problems with singular potentials arising in electronic Schrödinger's equation.

中文翻译：

---

三维椭圆PDE的量化张量格式的鲁棒交替方向隐式求解器

SIAM科学计算杂志，第43卷，第2期，第A800-A827页，2021年1月。
本文的目的是提出一种鲁棒的数值求解器，它能够以数据稀疏量化张量格式有效地解决三维椭圆问题。特别是，我们使用了塔克（Tucker）和量化张量序列格式（TQTT）的组合，这使我们能够使用天文上很大的网格尺寸。但是，由于离散微分算子的不适，这种精细的网格导致数值不稳定。为了获得鲁棒的求解器，我们利用了众所周知的交替方向隐式方法并对它进行了修改，以避免被微分算子相乘。为了使该方法高效，我们得出了迭代矩阵的显式TQTT表示和对称三对角Toeplitz矩阵的逆的量化张量图表示作为辅助结果。作为应用，