UBC SCAIM Seminar: Alexandra Bünger

Isogeometric analysis (IGA) has been recently developed to build a bridge between geometric computer-aided design and PDE analysis. In IGA, spline functions used for geometric domain descriptions are also used to discretize PDE problems on such domains. However, due to the nature of the spline functions the resulting equation systems for, e.g., optimization problems may quickly become very costly to assemble or even infeasible to treat with conventional methods due to their size and reduced sparsity, especially in higher dimensions.

We developed a method to exploit the underlying tensor structure of the IGA discretization with low-rank tensor train approximations to separate the problem dimensions and reduce its complexity. This low-rank formulation gives rise to a fast and memory efficient assembly for the system matrices of arbitrary geometries. The PDE problem can be interpreted as a low-rank tensor train formulation, which can be efficiently solved in a low-rank format without assembling the full matrices. We illustrate this for challenging time-dependent PDE-constrained problems.