M2P 2023

Reduced order models for time-dependent problems using the Laplace transform

  • Reyes, Ricardo (EPFL)
  • Hesthaven, Jan (EPFL)
  • Vallaghe, Sylvain (Akselos)
  • Knezevic, David (Akselos)

Please login to view abstract download link

We propose a new method to incorporate temporal evolution into general reduced-order parametrised models. Instead of following the traditional approach of time step sampling, we use frequency-domain solutions as the data input. The method consists in offline-online stages as standard reduced basis methods. In the offline part first we transfer the problem from the time domain to the frequency domain using the Laplace transform. Here, we sample solutions for a discrete set of frequencies and use the singular valued decomposition to construct a reduced basis. In the online part, we use this reduced basis for the time-dependent problem, as we would normally do with a basis obtained with any time-sampled or parametric data set. Although the method can be seen as a proper orthogonal decomposition extended to the frequency domain, there are two important considerations: we have to select appropriate sampling points so all the needed frequencies are captured, and we construct the basis using only the real part of the complex-valued solutions. Here we want to present some of the harmonic analysis needed to explain the method — which include inverse Laplace transforms and H2 Hardy spaces, and an example solving a multi component elastodynamic problem implemented in the Akselos software.