Interpolatory tensorial reduced order models for parametric dynamical systems

• Published in: Computer methods in applied mechanics and engineering 2022-07, Vol.397, p.115122, Article 115122
• Main Authors: Mamonov, Alexander V., Olshanskii, Maxim A.
• Format: Article
• Language: English
• Subjects: Accuracy; Cost analysis; Dynamical systems; Forecasting; Low-rank tensors; Mathematical models; Model order reduction; Numerical integration; Parameters; Parametric PDEs; Proper orthogonal decomposition; Reduced order models; Representations; Tensors
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2022.115122

The paper introduces a reduced order model (ROM) for numerical integration of a dynamical system which depends on multiple parameters. The ROM is a projection of the dynamical system on a low dimensional space that is both problem-dependent and parameter-specific. The ROM exploits compressed tensor formats to find a low rank representation for a sample of high-fidelity snapshots of the system state. This tensorial representation provides ROM with an orthogonal basis in a universal space of all snapshots and encodes information about the state variation in parameter domain. During the online phase and for any incoming parameter, this information is used to find a reduced basis that spans a parameter-specific subspace in the universal space. The computational cost of the online phase then depends only on tensor compression ranks, but not on space or time resolution of high-fidelity computations. Moreover, certain compressed tensor formats enable to avoid the adverse effect of parameter space dimension on the online costs (known as the curse of dimension). The analysis of the approach includes an estimate for the representation power of the acquired ROM basis. We illustrate the performance and prediction properties of the ROM with several numerical experiments, where tensorial ROM’s complexity and accuracy is compared to those of conventional POD-ROM. •New projection-based ROM is introduced for multi-parameter dynamical system.•Tensorial ROM finds problem-dependent and parameter-specific reduced basis.•CP-, HOSVD- and TT-tensor decomposition are considered for the purpose of finding ROM basis.•Costs of online computations depend only on compression ranks, but not on dimensions of highfidelity model.•The introduced tensorial ROM can be seen as extension of POD-ROM for multi-parameter systems.