Vortrag
Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs
Vortrag von Prof. Dr. Markus Bachmayr
Datum: 23.05.18 Zeit: 16.15 - 17.45 Raum: Y27H25
Folding grid value vectors into high-order tensors, combined with low-rank representation in the tensor train format, has been shown to lead to 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. Straightforward choices of the underlying basis, such as piecewise multilinear finite elements on uniform tensor product grids, lead to the well-known basis ill-conditioning of discretized operators. We demonstrate that for low-rank representations, the use of tensor structure additionally leads to representation ill-conditioning, a new effect specific to computations in tensor networks. We construct an explicit tensor-structured representation of a BPX preconditioner with ranks independent of the number of discretization levels, which combined with a carefully chosen representation of its product with the stiffness matrix turns out to remove both basis and representation ill-conditioning. Numerical tests, including problems with highly oscillatory coefficients, show that one arrives at reliable and efficient solvers which remain numerically stable for mesh sizes near machine precision.