Jan S. Hesthaven (EPFL)

Structure preserving reduced order models

B. Maboudi Afkham, J.S. Hesthaven, C. Pagliantini, and N. Ripamonti

The development of reduced order models for complex applications has the promise for rapid and accurate evaluation of the output of complex models under parameterized variation with applications in problems which require many evaluations, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed.
However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large scale applications, in particular for nonlinear and/or time-dependent problems.

In this talk we discuss the development of reduced methods which seek to conserve chosen invariants for nonlinear time-dependent problems. In the first part, we develop structure-preserving reduced basis methods for a broad class of Hamiltonian dynamical systems, endowed with a general Poisson manifold structure which encodes the physical properties, symmetries and conservation laws of the dynamics.

In second part, we focus on reduced order modeling of more general hyperbolic problems, discuss the importance of the skew-symmetric form of the governing equations, and the benefits of using the skew-symmetric form for the reduced order model. We demonstrate the methods through the numerical simulation of various fluid flows.