Seminar by Mattia Manucci

Speaker

Mattia Manucci (Universität Stuttgart)

Title

Contour Integral Methods and Eigenvalues approximation for projection MOR

Date

  • October 24, 2023 16:00 CEST+0200 (Europe/Rome)

  • October 24, 2023 10:00 EDT-0400 (US/Eastern)

  • October 24, 2023 09:00 CDT-0500 (US/Central)

  • October 24, 2023 07:00 PDT-0700 (US/Pacific)

Abstract

We present a numerical method for efficiently and reliably solving parametrized time-dependent partial differential equations (PPDEs). An important consideration for the addressed problems is that not all dynamics are significant; rather, only the solution at a given final time \(t = T\) or within a time window \([T\Lambda^{-1},\;T]\), where \(\Lambda > 1\), of moderate size.

In the first part of the talk, methods based on the Laplace transform and its numerical inversion are introduced. These methods, also known as contour integral methods, are well-suited for solving ordinary differential equations (ODEs) that arise from the discretization of linear parabolic PDEs. A notable advantage of this approach is that, unlike time-stepping methods such as Runge-Kutta integrators, the Laplace transform enables direct computation of the solution at a specific instant. This can be achieved by approximating the contour integral associated with the inverse Laplace transform using a suitable quadrature formula. Furthermore, the proposed approach automatically yields an approximation of the solution with a prescribed level of accuracy.

The second part of the talk discusses a projection model order reduction (MOR) method that employs the contour integral methods as a time integrator. With respect to some classical MOR methodology, like the one based on the classical proper orthogonal decomposition (POD), this determines a significant improvement in the reduction phase since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method.

Finally, the last part is dedicated to the numerically efficient and reliable approximation of eigenvalues and singular values associated with parametrically dependent matrices. Specifically, the focus lies on approximating the smallest eigenvalue and/or singular value, as this is crucial for the efficiency of greedy strategies in constructing reduced spaces within projection MOR.

Recording

Watch the recording on our YouTube channel.