Approximate optimal projection for reduced-order models

A critical component of a reduced-order model is the projection that maps the original high-fidelity system on to the reduced-order basis. In this manuscript, we develop a projection for linear systems that is optimal in an operator-independent norm. We derive an expression for this projection as a Galerkin projection plus another component that is interpreted as the effect of the scales that live outside the reduced-order basis. We note that the exact form of this projection does not lead to a viable computational method because it involves the inverse of the original high-fidelity operator. We approximate this inverse by an inexpensive preconditioner and create a practical method whose costs are of the same order as the Galerkin method. We test the performance of this method on heat conduction and advection-diffusion problems while using the incomplete LU preconditioner as an approximate to the inverse of the original operator, and conclude that it provides more accurate results than the Galerkin projection. Copyright (C) 2015 John Wiley & Sons, Ltd.