Tensors, Sparsity, and High-dimensional PDEs

Problems in high spatial dimensions are typically subject to the "curse of dimensionality" which roughly means that the computational work needed to realize a desired target accuracy increases exponentially in the spatial dimension. As a consequence, on the one hand, standard methods developed for low dimensional regimes become prohibitively expensive. On the other hand, classical regularity notions do no longer provide meaningful complexity criteria. Possible remedies would be unveiling a hidden low dimensionality of the objects of interest, typically expressed in terms of "sparsity". This talk addresses tensor sparsity of solutions to high-dimensional PDEs aiming at approximating a solution within a given accuracy tolerance by possible short expansions in solution dependent tensors.

The two central themes are: (i) under which assumptions on the data are the solutions in a certain sense tensor sparse which can be viewed as a "regularity theorem", (ii) how to stably generate tensor approximations with provably near-minimal ranks. We present first results concerning (i) and (ii) for a simple model problem in an infinite-dimensional framework drawing on and highlighting, in particular, several pivotal contributions by W. Hackbusch in this field.