A low-rank tensor method for PDE-constrained optimization with isogeometric analysis

Isogeometric analysis (IgA) has become one of the most popular methods for the discretization of PDEs, motivated by the use of nonuniform rational B-splines for geometric representations in industry and science. A crucial challenge lies in the solution of the discretized equations, which we discuss in this paper 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 as proposed in [A. Mantzaflaris, B. Juttler, B. N. Khoromskij, and U. Langer, Comput. Methods Appl. Mech. Engrg., 316 (2017), pp. 1062-1085] have become a promising approach. We present a framework for the assembly of these matrices in low-rank form using tensor train approximations. Furthermore, our framework allows for the exploitation of the resulting low-rank structure of the mass and stiffness matrices, and it can be used to solve a PDE-constrained optimization problem without assembling the actual system matrices and carries the low-rank format over to the solution. We use the block alternating minimal energy method to efficiently solve the corresponding KKT system of the optimization problem. We show several numerical experiments with threedimensional geometries to demonstrate that the low-rank assembly and solution drastically reduce the memory demands and computing times, depending on the approximation ranks of the domain.