From the Abstract: “Tensor-compressed numerical solution of elliptic multiscale-diffusion and high frequency scattering problems is considered.”
From the Introduction: “The outline of this paper is as follows: In Section 2, we introduce the model problems, their variational formulations and the standard finite element (FE) discretizations. The quantized tensor train (QTT) decomposition and related notions are recapitulated in Section 3.2. Section 3 provides a short summary of quantized FE approximations. Section 4 introduces the homogenization problem, the asymptotic analysis of its solution and provides scale-separated finite-dimensional approximations with exponential convergence rate bounds which are interesting in their own right. These bounds are subsequently used to prove logarithmic in accuracy QTT rank bounds for the quantized FE solution vectors. Section 5 is devoted to the same program for the high frequency Helmholtz equation, where we prove QTT rank bounds which are mildly depending on the wavenumber and, again, logarithmic in accuracy. For the practical implementation of quantized tensor train FE methods, analogous rank bounds for the stiffness and mass matrices are required, and we prove this in Section 6. Finally, we provide in Section 7 numerical experiments which show that the logarithmic in accuracy, and scale-robust tensor rank bounds of QTT FE approximations are achieved in practice, for the problem classes under consideration here.”