Low-Rank Tensor Methods for High-Dimensional Problems

In this thesis, we propose and analyze novel numerical algorithms for solving three different high-dimensional problems involving tensors. The commonality of these problems is that the tensors can potentially be well approximated in low-rank formats. Identifying and exploiting this low-rank structure allows us to mitigate the curse of dimensionality.

The first problem considered in this thesis is the computation of functional low-rank approximations of multivariate functions defined on tensor product domains. We develop two novel algorithms to compute such approximations by combining tensorized Chebyshev interpolation with low-rank approximations of the coefficient tensor. For most functions, our numerical experiments demonstrate that our algorithms require a lower number of function evaluations and achieve the same approximation error compared to existing methods. In addition, we solve partial differential equations using a novel global spectral method that can potentially be combined with functional low-rank approximations.

The second problem of interest is the solution of multi-marginal optimal transport problems. After adding entropic regularization, these are equivalent to tensor scaling problems that can be solved using the Sinkhorn algorithm. In literature, it has been suggested to accelerate the Sinkhorn algorithm by exploiting either a graphical model structure or a low-rank tensor approximation. We propose to combine these two approaches to accelerate the solution of the tensor scaling problem even further.

The third problem of interest is the computation of the self-diffusion matrix of a tagged particle process defined on a grid. We propose a novel approach based on computing the matrix via solving a high-dimensional tensor-valued optimization problem. We observe numerically that our approach is much less subject to statistical noise compared to classical approaches based on estimating long-time averages of empirical means of deviations of some stochastic processes.