## Applied Math Seminar: Stability-preserving projection-based model order reduction for compressible flows

Seminar | December 6 | 4-5 p.m. | 891 Evans Hall

Irina Tezaur, Sandia National Laboratories

Department of Mathematics

Projection-based reduced order modeling is a promising tool for bridging the gap between high-fidelity and real- time/multi-query applications such as uncertainty quantification (UQ), optimization and control design. A popular approach to building projection-based reduced order models (ROMs) for fluid problems is the proper orthogonal decomposition (POD)/Galerkin projection method. This method consists of two steps: (1) the computation of the POD basis from a set of snapshots of the solution field, followed by (2) the Galerkin projection of the governing equations onto this reduced basis in some inner product. POD is a mathematical procedure that constructs a reduced basis for an ensemble of snapshots collected from a high-fidelity simulation. This basis is optimal in the sense that it describes more energy on average of the ensemble than any other linear basis of the same reduced dimension.

Unfortunately, ROMs constructed via the POD/Galerkin method using the L2 inner product lack, in general, an a priori stability guarantee when applied to compressible flow problems. This leads to practical limitations of ROMs obtained using the POD/Galerkin method: a ROM aimed to capture a flow in a physically stable (i.e., bounded as time tends to infinity) regime might be stable for a given number of modes, but unstable (i.e., unbounded as time tends to infinity), and therefore inaccurate, for other choices of basis size [1, 2]. The situation can be exacerbated by basis truncation: the removal of POD modes having low energy. Although necessary for model reduction, truncation can destroy the balance between energy production and dissipation in a fluid ROM, thereby leading to ROMs that are inaccurate and/or unstable. This talk will describe several approaches for building stable projection-based reduced order models (ROMs) for compressible flows developed by the speaker during the past 10 years.

linlin@berkeley.edu