Model reduction using L¹‐norm minimization as an application to nonlinear hyperbolic problems

Abstract

Summary We are interested in the model reduction techniques for hyperbolic problems, particularly in fluids. This paper, which is a continuation of an earlier paper of Abgrall et al, proposes a dictionary approach coupled with an L1 minimization approach. We develop the method and analyze it in simplified 1‐dimensional cases. We show in this case that error bounds with the full model can be obtained provided that a suitable minimization approach is chosen. The capability of the algorithm is then shown on nonlinear scalar problems, 1‐dimensional unsteady fluid problems, and 2‐dimensional steady compressible problems. A short discussion on the cost of the method is also included in this paper.

Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism

“…The coefficient ν is the only physical/geometrical parameter of this problem. The range of ν is [2,6]. The snapshots for POD basis generation are collected from the high-fidelity simulations with 9 values of ν that are uniformly sampled.…”

“…However, the POD-Galerkin lacks a priori guarantees of stability, accuracy and convergence [31,33,58] for general time-dependent nonlinear problems. Extensive efforts have been made to improve the stability and accuracy of the POD-Galerkin method, such as the structure-preservation [43], supremizer enrichment [4,59],basis adaption [9,53], L 1 -norm minimization [2], and least-squares Petrov-Galerkin (LSPG) technique [10].…”

The Adjoint Petrov–Galerkin method for non-linear model reduction

“…15) into a linear partial differential equation. Equation 15 can be written equivalently as the following partial differential equation…”

“…The development of stable and accurate reduced-order modeling techniques for complex non-linear systems is the motivation for the current work.A significant body of research aimed at producing accurate and stable ROMs for complex non-linear problems exists in the literature. These efforts include, but are not limited to, "energy-based" inner products [9,11], symmetry transformations [12], basis adaptation [13,14], L 1 -norm minimization [15], projection subspace rotations [16], and least-squares residual minimization approaches [17,18,19,20,21,22,23,24]. The Least-Squares Petrov-Galerkin (LSPG) [22] method comprises a particularly popular leastsquares residual minimization approach and has been proven to be an effective tool for non-linear model reduction.…”

6 Model reduction in computational aerodynamics

“…Of particular interest is the L 1 -norm, which is a more natural norm for hyperbolic equations. Based on this observation, Abgrall and Crisovan [1] propose an ROM which identifies the solution through L 1 -minimization and apply it to parameterized transonic Euler flow over an airfoil.…”