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Isogeometric analysis (IGA) has become one of the most popular methods forthe discretization of partial differential equations motivated by the use ofNURBS for geometric representations in industry and science. A crucialchallenge lies in the solution of the discretized equations, which we discussin this talk with a particular focus on PDE-constrained optimizationdiscretized using IGA. The discretization results in a system of large mass andstiffness matrices, which are typically very costly to assemble. To reduce thecomputation time and storage requirements, low-rank tensor methods have becomea promising tool. We present a framework for the assembly of these matrices inlow-rank form as the sum of a small number of Kronecker products. For assemblyof the smaller matrices only univariate integration is required. The resultinglow rank Kronecker product structure of the mass and stiffness matrices can beused to solve a PDE-constrained optimization problem without assembling theactual system matrices. We present a framework which preserves and exploits thelow-rank Kronecker product format for both the matrices and the solution. Weuse the block AMEn method to efficiently solve the corresponding KKT system ofthe optimization problem. We show several numerical experiments with 3Dgeometries to demonstrate that the low-rank assembly and solution drasticallyreduces the memory demands and computing times, depending on the approximationranks of the domain.

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