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Folding grid value vectors of size $2^L$ into $L$th order tensors of modesizes $2\times \cdots\times 2$, combined with low-rank representation in thetensor train format, has been shown to lead to highly efficient approximationsfor various classes of functions. These include solutions of elliptic PDEs onnonsmooth domains or with oscillatory data. This tensor-structured approach isattractive because it leads to highly compressed, adaptive approximations basedon simple discretizations. Standard choices of the underlying basis, such aspiecewise multilinear finite elements on uniform tensor product grids, entailthe well-known matrix ill-conditioning of discrete operators. We demonstratethat for low-rank representations, the use of tensor structure itselfadditionally introduces representation ill-conditioning, a new effect specificto computations in tensor networks. We analyze the tensor structure of a BPXpreconditioner for second-order linear elliptic operators and construct anexplicit tensor-structured representation of the preconditioner, with ranksindependent of the number $L$ of discretization levels. The straightforwardapplication of the preconditioner yields discrete operators whose matrixconditioning is uniform with respect to the discretization parameter, but indecompositions that suffer from representation ill-conditioning. Byadditionally eliminating certain redundancies in the representations of thepreconditioned discrete operators, we obtain reduced-rank decompositions thatare free of both matrix and representation ill-conditioning. For an iterativesolver based on soft thresholding of low-rank tensors, we obtain convergenceand complexity estimates and demonstrate its reliability and efficiency fordiscretizations with up to $2^{50}$ nodes in each dimension.

Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs