Novel hierarchical tensor product methods currently emerge as an important tool in numerical analysis and scienti.c computing. One reason is that these methods often enable one to attack high-dimensional problems successfully, another that they allow very compact representations of large data sets. These representations are in some sense optimal and by construction at least as good as approximations by classical function systems like polynomials, trigonometric polynomials, or wavelets. Moreover, the new tensor-product methods are by construction able to detect and to take advantage of self-similarities in the data sets. They should therefore be ideally suited to represent solutions of partial differential equations that exhibit certain types of multiscale behavior.
The aim of this project is both to develop methods and algorithms that utilize these properties and to check their applicability to concrete problems as they arise in the collobarative research centre. We plan to attack this task from two sides. On the one hand we will try to decompose solutions that are known from experiments, e.g., on Earthquake fault behavior, or large scale computations, such as turbulent flow fields. The question here is whether the new tensor product methods can support the development of improved understanding of the multiscale behavior and whether they are an improved starting point in the development of compact storage schemes for solutions of such problems relative to linear ansatz spaces.
On the other hand, we plan to apply such tensor product approximations in the framework of Galerkin methods, aiming at the reinterpretation of existing schemes and at the development of new approaches to the ef.cient approximation of partial differential equations involving multiple spatial scales. The basis functions in this setting are not taken from a given library, but are themselves generated and adapted in the course of the solution process.
One mid-to long-term goal of the project that combines the results from the two tracks of research described above is the construction of a self-consistent closure for Large Eddy Simulations (LES) of turbulent flows that explicitly exploits the tensorproduct approach’s capability of capturing self-similar structures. If this proves successful, we plan to transfer the developed concepts also to Earthquake modelling in joint work with partner project B01.
Kornhuber, R. and Peterseim, D. and Yserentant, H. (2018) An analysis of a class of variational multiscale methods based on subspace decomposition. Mathematics of Computation, 87 (314). pp. 2765-2774. ISSN 1088-6842 (online)
von Larcher, T. and Klein, R. (2017) On identification of self-similar characteristics using the Tensor Train decomposition method with application to channel turbulence flow. Theoretical and Computational Fluid Dynamics . pp. 1-22. ISSN 0935-4964 (Print) 1432-2250 (Online) (Submitted)
Kornhuber, R. and Podlesny, J. and Yserentant, H. (2017) Direct and Iterative Methods for Numerical Homogenization. In: Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, XXIII (116). SpringerLink, pp. 217-225. ISBN 978-3-319-52389-7
von Larcher, T. and Klein, R. (2017) Approximating turbulent and non-turbulent events with the Tensor Train decomposition method. In: Turbulence in the Complex Conditions. Springer. (Submitted)
Kornhuber, R. and Yserentant, H. (2016) Numerical Homogenization of Elliptic Multiscale Problems by Subspace Decomposition. Multiscale Model. Simul., 14 (3). pp. 1017-1036. ISSN print: 1540-3459; online: 1540-3467
Horenko, I. and Gerber, S. (2015) Improving clustering by imposing network information. Science Advances, 1 (7). ISSN 2375-2548