  Isogeometric analysis (IGA) has become one of the most popular methods for
the discretization of partial differential equations motivated by the use of
NURBS for geometric representations in industry and science. A crucial
challenge lies in the solution of the discretized equations, which we discuss
in this talk with a particular focus on PDE-constrained optimization
discretized using IGA. The discretization results in a system of large mass and
stiffness matrices, which are typically very costly to assemble. To reduce the
computation time and storage requirements, low-rank tensor methods have become
a promising tool. We present a framework for the assembly of these matrices in
low-rank form as the sum of a small number of Kronecker products. For assembly
of the smaller matrices only univariate integration is required. The resulting
low rank Kronecker product structure of the mass and stiffness matrices can be
used to solve a PDE-constrained optimization problem without assembling the
actual system matrices. We present a framework which preserves and exploits the
low-rank Kronecker product format for both the matrices and the solution. We
use the block AMEn method to efficiently solve the corresponding KKT system of
the optimization problem. We show several numerical experiments with 3D
geometries to demonstrate that the low-rank assembly and solution drastically
reduces the memory demands and computing times, depending on the approximation
ranks of the domain.
