Folding grid value vectors of size \(2^L\) into Lth-order tensors of mode size \(2\times \cdots \times 2\), combined with low-rank representation in the tensor train format, has been shown to result in highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Standard choices of the underlying bases, such as piecewise multilinear finite elements on uniform tensor product grids, entail the well-known matrix ill-conditioning of discrete operators. We demonstrate that, for low-rank representations, the use of tensor structure itself additionally introduces representation ill-conditioning, a new effect specific to computations in tensor networks. We analyze the tensor structure of a BPX preconditioner for a second-order linear elliptic operator and construct an explicit tensor-structured representation of the preconditioner, with ranks independent of the number L of discretization levels. The straightforward application of the preconditioner yields discrete operators whose matrix conditioning is uniform with respect to the discretization parameter, but in decompositions that suffer from representation ill-conditioning. By additionally eliminating certain redundancies in the representations of the preconditioned discrete operators, we obtain reduced-rank decompositions that are free of both matrix and representation ill-conditioning. For an iterative solver based on soft thresholding of low-rank tensors, we obtain convergence and complexity estimates and demonstrate its reliability and efficiency for discretizations with up to \(2^{50}\) nodes in each dimension.

折叠尺寸的网格值矢量\（2 ^大号\）到大号模式尺寸的阶张量\（2 \倍\ cdots \倍2 \） ，与在该伸张列车格式低等级表示相结合，已经表明从而对各种功能类别进行高效逼近。这些包括非光滑域上或带有振荡数据的椭圆形PDE的解。这种张量结构化的方法很有吸引力，因为它导致基于简单离散化的高度压缩，自适应近似。标准基础的选择，例如均匀张量积网格上的分段多线性有限元，需要众所周知的矩阵不适离散运算符。我们证明，对于低秩表示，张量结构本身的使用还引入了表示病态，这是特定于张量网络计算的新效果。我们分析了二阶线性椭圆算子的BPX预调节器的张量结构，并构造了一个独立于数L的秩的显式张量结构表示。离散化水平。预调节器的直接应用会产生离散算子，这些算子的矩阵条件关于离散化参数是统一的，但是在分解中表现不佳。通过在预条件离散算子的表示中另外消除某些冗余，我们获得了无矩阵分解和表示不适的降秩分解。对于基于低秩张量的软阈值的迭代求解器，我们获得了收敛性和复杂性估计，并证明了其离散化在每个维度中最多有（2 ^ {50} \）个节点的可靠性和效率。