A low-rank tensor method for PDE-constrained optimization with isogeometric analysis
This work addresses the high computational cost of solving PDE-constrained optimization problems with IGA, offering a more efficient assembly and solution method for practitioners in computational science and engineering.
The paper presents a low-rank tensor framework for assembling mass and stiffness matrices in isogeometric analysis (IGA) for PDE-constrained optimization, reducing memory and computation time by exploiting Kronecker product structure. Numerical experiments on 3D geometries show drastic reductions in memory and computing times.
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.