An investigation of the approximate optimal projection reduced-order models for elliptic partial differential equations

Reduced-order modeling is often utilized in obtaining an approximate solution to a complex high-fidelity system. An essential step in constructing a reduced-order system is to project the original high-fidelity system onto the reduced-order bases. Recently, a projection that is optimal in an operator-independent norm has been developed. This projection is composed of the standard Galerkin projection and other components that represent the effect of the scales that are not represented by the reduced-order bases. Unfortunately, the use of the exact formulation of this projection is not practical because it involves computing the inverse of the original high-fidelity operator. Therefore, this projection is approximated by using inexpensive preconditioners to approximate the inverse. This approximate optimal projection (AOP), with the inverse approximated by the incomplete LU (ILU) preconditioner, is tested on three elliptic partial differential equations (PDEs): modified Helmholtz equation, dominant advection-reaction equation with weak diffusion, and Helmholtz equation. The performance of the AOP in the three equations is quantified and compared to that of the Galerkin projection. For all but the Helmholtz equation, it is observed that the AOP generates solutions closer to the optimally projected exact solution than the Galerkin projection. It is argued that for the Helmholtz equation the ILU preconditioner does not provide accurate approximation to the inverse operator, and this leads to the relatively poor performance of the AOP. Hence, the symmetric successive over-relaxation (SSOR) preconditioner is used to approximate the inverse in AOP for the Helmholtz equation. It is observed that the AOP using the SSOR preconditioner yields more accurate and more stable solutions than the Galerkin projection.