Numerieke analyse in ruimten van vaste-rang tensoren en andere Riemannse variëteiten

We consider the geometry of low-rank tensors in two different ways. First, we look at the Lie group action induced by changes of basis in each mode. We show that this action is transitive under suitable restrictions on the sets of - tensors with fixed canonical polyadic rank, - tensors with fixed tensor train rank, - tensors with fixed multilinear rank. This induces a homogeneous smooth manifold structure. We explore a few natural applications of this structure. Most importantly, it induces a so-called canonical Riemannian metric on the aforementioned sets. Geodesics are then induced by horizontal geodesics on the change-of-basis Lie group, which is a product of general linear groups. Using a known expression for geodesics on the general linear group, along with a clever choice of representative where the horizontal space has a particular rank structure, we derive an expression for geodesics on low-rank tensors. Importantly, these can then be evaluated efficiently using ideas from the theory of exponential integrators. We further classify the manifolds that have an important additional property, namely those that are reductive. This property implies that the manifold is Riemannian-homogeneous, meaning that changes of basis are isometries, and has implications for so-called Lie integrators. Secondly, we continue the work of Lars Swijsen on the Riemannian geometry of rank-1 tensors as an embedded submanifold of Euclidean space. By identifying the metric locally as a warped product, we are able to derive the exponential and logarithmic maps. We are moreover able to put this manifold in a more general context. There is a one-parameter family of metrics, obtained from tweaking a parameter in the warping function of the warped product. It includes the Euclidean, but also some other interesting geometries. In particular, some of these are geodesically connected, meaning that any pair of points can be connected by a geodesic. This property is important in, for example, consensus aggregation, where we want to find the rank-1 tensor that best represents the average of a set of many rank-1 tensors. We then consider the general problem of how to approximate a map into a Riemannian manifold. Many established approximation schemes follow a general pullback-approximate-pushforward template. We relate their accuracy to the geometry of the target manifold by showing that if one uses the exponential and logarithmic maps to pull back and push forward to and from the tangent space, then it is possible to bound the approximation error on the manifold in terms of the linear approximation on the tangent space and a lower bound on the sectional curvature. It turns out that for many manifolds that appear organically in numerical analysis, we have explicit expressions for the exponential and logarithmic maps, and there are often global lower curvature bounds as well. Numerical experiments and a natural application to Krylov subspace recycling are presented. Our analysis uses Toponogov's theorem, a classic result from Riemannian comparison theory about geodesic triangles. The idea is that one can construct such a triangle where one corner is the true function value and another corner is the approximated value. Having constructed the correct comparison triangle, the analysis is straightforward. If the exponential and logarithmic maps are not known for a manifold, one can use a so-called retraction, which is a map that agrees with the exponential to first order. We discuss how it is possible to use our framework to think about the approximation error when retracting. Typically, this will overestimate the error. Numerical experiments confirm that there is practically no difference in the errors between exponentiating and retracting. Everything numerical is implemented in Julia, and the source code is available. The low-rank tensor geometries and the rank-1 warped product geometries are all contributed to the popular library Manifolds.jl. There is also a separate package, ManiFactor.jl, published to demo the process of approximating maps into Riemannian manifolds via this pullback-approximate-pushforward technique