On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation

Abstract

Background: In what follows, we consider the Proper Orthogonal Decomposition (POD) technique of model order reduction, for a parameterized quasi-nonlinear parabolic equation. Methods: A POD basis associated with a set of reference values of the characteristic parameters is considered. From this basis, a parametric reduced order model (ROM) projecting the initial equation is constructed. Results: A mathematical a priori estimate of the parametric squared L2-error induced by this projection is developed. This later estimate is based on both, the parametric behavior of the squared L2-ROM-error thanks to the resolution of a Ricatti differential inequality in the parametric ROM-error, and the convergence rate of the parametric ROM to the full problem, via the augmentation of the basis dimension. Indeed, under restrictive conditions on the solutions regularity of such equations, we are able to precise the slope of the logarithm of the squared L2-norm of the ROM error, as a function of the logarithm of the basis modes number. Numerical experiments of our theoretical estimate, are presented for the 2D Navier-Stokes equations in the case of an unsteady and incompressible fluid flow in a channel around a circular cylinder. Conclusion: A mathematical a priori estimate of the parametric squared L2-error induced by the model reduction by POD is developped for a parameterized quasi-nonlinear parabolic equation. This estimate is obtained thanks to the resolution of a Ricatti differential inequality.