Adaptive Low-Rank Methods for Problems on Sobolev Spaces with Error Control in $L_2$
For researchers solving high-dimensional PDEs with low-rank methods, this work provides a more efficient error control approach by switching to a weaker norm, though it is incremental as it builds on prior energy-norm schemes.
The paper develops adaptive low-rank tensor methods for high-dimensional elliptic PDEs that control the error in the $L_2$ norm rather than the energy norm, achieving substantial simplifications and reduced computational work for a given error tolerance.
Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such approximations, not with respect to a fixed finite dimensional discrete background problem, but with respect to the exact solution of the continuous problem. While the energy norm offers a natural error measure corresponding to the underlying operator considered as an isomorphism from the energy space onto its dual, this norm requires a careful treatment in its interplay with the tensor structure of the problem. In this paper we build on our previous work on energy norm-convergent subspace-based tensor schemes contriving, however, a modified formulation which now enforces convergence only in $L_2$. In order to still be able to exploit the mapping properties of elliptic operators, a crucial ingredient of our approach is the development and analysis of a suitable asymmetric preconditioning scheme. We provide estimates for the computational complexity of the resulting method in terms of the solution error and study the practical performance of the scheme in numerical experiments. In both regards, we find that controlling solution errors in this weaker norm leads to substantial simplifications and to a reduction of the actual numerical work required for a certain error tolerance.