Using Gappy-POD to derive a reduced quadrature rule

Among various reduced-order modeling techniques, the combination of low-dimensional approximation using proper orthogonal decomposition (POD) and the Galerkin method is a promising approach. However, the POD–Galerkin method has a well-known drawback that the computation of the Galerkin projection is heavy, which overshadows the reduction of computational cost for solving simultaneous equations. To speed up the reduced-order model analysis, a hyper-reduction method, which approximately calculates the Galerkin projection, has been introduced. Although several hyper-reduction methods have been proposed up to date, currently, a reduced quadrature (RQ) method is widely used because of its stability. In the conventional RQ method, a sparse representation problem with ${\ell }_0$ pseudo-norm minimization under the non-negativity constraint is solved to derive an RQ rule. However, it is difficult to control the number of non-zero entries in the weight vector of RQ and the error of least-squares fitting. The purpose of the present study was to develop a new RQ derivation method to overcome this difficulty. The formulation of the new method is not based on the sparse representation but on Gappy-POD, which is a sparse sampling technique and was originally proposed for image reconstruction. To demonstrate the new method, we applied it to nonlinear dynamic structural analysis with geometrical nonlinearity and to incompressible viscous flow analysis. The results confirmed that the new method can provide a more accurate RQ rule than can the conventional method.