Abstract
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.
Original language | English |
---|---|
Pages (from-to) | A140-A161 |
Number of pages | 22 |
Journal | SIAM Journal on Scientific Computing |
Volume | 42 |
Issue number | 1 |
Early online date | 8 Jan 2020 |
DOIs | |
Publication status | Published - 2020 |
Funding
\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section November 16, 2018; accepted for publication (in revised form) October 2, 2019; published electronically January 8, 2020. https://doi.org/10.1137/18M1227238 Funding: The work of the first and third authors was partially supported by the German Science Foundation (DFG) through grant 1742243256 - TRR 96 and DAAD-MIUR Joint Mobility Program 2018-2020 through grant 5739665. \dagger Technische Universit\a"t Chemnitz, Department of Mathematics, Chair of Scientific Computing, 09107 Chemnitz, Germany ([email protected], martin.stoll@ mathematik.tu-chemnitz.de). \ddagger University of Bath, Claverton Down, BA2 7AY, Bath, United Kingdom ([email protected]).
Keywords
- Isogeometric analysis
- Low-rank decompositions
- Optimal control
- Tensor train format
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics