Quantized Tensor FEM for Multiscale Problems: Diffusion Problems in Two and Three Dimensions
Homogenization in terms of multiscale limits transforms a multiscale problem with 𝑛+1n+1 asymptotically separated microscales posed on a physical domain 𝐷⊂ℝ𝑑D⊂Rd into a one-scale problem posed on a product domain of dimension (𝑛+1)𝑑(n+1)d by introducing 𝑛n so-called fast variables. This procedure allows one to convert 𝑛+1n+1 scales in 𝑑d physical dimensions into a single-scale structure in (𝑛+1)𝑑(n+1)ddimensions. We prove here that both the original, physical multiscale problem and the corresponding high-dimensional, one-scale limiting problem can be efficiently treated numerically with the recently developed quantized tensor-train finite-element method (QTT-FEM). The QTT-FE approximation consists in restricting approximation and computation to sequences of nested subspaces, each of which is a tensor product of two factors of low dimension (rank), within a vast, but generic “virtual” (background) discretization space. In practice, these subspaces are determined iteratively and data-adaptively at runtime bypassing any “offline precomputation.” For theoretical analysis, low-dimensional subspaces are constructed analytically to bound the tensor ranks against the error tolerance. We consider a model linear elliptic multiscale problem in several physical dimensions and show, theoretically and experimentally, that both (i) the solution of the associated high-dimensional one-scale problem and (ii) the approximation to the solution of the multiscale problem induced thereby admit efficient QTT-FE approximations. These problems can therefore be numerically solved in a scale-robust fashion by standard (low-order) PDE discretizations combined with state-of-the-art general-purpose solvers for tensor-structured linear systems. Specifically, we prove the existence of QTT-FE approximations with an upper bound on the tensor ranks growing no faster than algebraically with respect to log𝜖−1logϵ−1 and independent of the scale parameters, where 𝜖ϵ is the target accuracy for the approximation of the solution and of its gradient. In numerical experiments, we verify the theoretical rank bounds and computationally investigate the dependence of the complexity of the solutions on the number 𝑛n of microscales.