Efficient Hyperreduction via Model Reduction Implicit Feature Tracking with an Accelerated Greedy Approach

This work introduces a new approach to reduce the computational cost of solving partial differential equations (PDEs) with convection-dominated solutions containing discontinuities (shocks): efficient hyperreduction via model reduction implicit feature tracking with an accelerated greedy approach. Traditional model reduction techniques use an affine subspace to reduce the dimensionality of the solution manifold and, as a result, yield limited reduction and require extensive training due to the slowly decaying Kolmogorov $n$-width of convection-dominated problems.

The proposed approach circumvents the slowly decaying $n$-width limitation by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with a space of bijections of the underlying domain. Central to the proposed approach is the reduced order model implicit feature tracking (ROM-IFT) procedure, which is a residual minimization problem over the reduced nonlinear manifold that simultaneously determines the reduced coordinates in the affine space and the domain mapping that minimize the residual of the unreduced PDE discretization. This is analogous to standard minimum-residual reduced-order models. Instead of only minimizing the residual over the affine subspace of PDE states, the method enriches the optimization space also to include admissible domain mappings. The nonlinear trial manifold is constructed using the proposed residual minimization formulation to determine domain mappings that cause parameterized features to align in a reference domain for training parameters. Because the feature is stationary in the reference domain, i.e., the convective nature of the solution is removed, the snapshots are effectively compressed to define an affine subspace. The space of domain mappings, originally constructed using high-order finite elements, is also compressed to ensure the boundaries of the original domain are maintained.

The presented method uses the empirical quadrature procedure (EQP) \cite{yano2019discontinuous} to reduce the cost of the ROM-IFT method for convection-dominated problems containing shocks. EQP is used to select an optimal subset of the mesh and the method utilizes either the Galerkin or Petrov-Galerkin projections-based reduced model over the deformed mesh to achieve significant cost reduction.

Moreover, to have an optimal selection of the parameters to make a basis, we conjugate an accelerated greedy search with the hyperreduction method to have a fast computation.

The EQP weight vector is computed over the hyperreduced solution and the deformed mesh, allowing the mesh to be dependent on the parameters and not fixed. The greedy search is also applied to the hyperreduced solutions, further reducing computational costs and speeding up the process.

The minimum residual is applied to a small, optimal subset of mesh elements to align the new configuration and reduce the cost. The method's effectiveness is demonstrated through numerical experiments for various convection-dominated problems, including advection-reaction, inviscid Burgers' equation, and transonic flow over a NACA0012 airfoil. The results show that the method can produce accurate approximations with a small size basis. The cost of ROM-IFT with and without the hyperreduction procedure is compared, revealing significant computational cost savings without sacrificing accuracy.