Abstract
In what follows, we consider the Proper Orthogonal Decomposition (POD) technique of model order reduction, for a parameterized quasi-nonlinear parabolic equation. 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. 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. 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.
Highlights
In what follows, we consider the Proper Orthogonal Decomposition (POD) technique of model order reduction, for a parameterized quasi-nonlinear parabolic equation
Two important questions can be asked: What is the parametric confidence region of a given reduced order model? How can we improve the performance of a reduced order model when parameters are variying significantly? Several techniques of model reduction exist to build a good candidate within the parametric ROMs: We consider first the reduced basis (RB) method
They were interested in showing an a posteriori bound of the squared parametric L2-ROM-RB error, of which computation at each parameter value is less expensive than the parametric ROM-error itself as it is usually done in the original Greedy algorithm
Summary
We consider the Proper Orthogonal Decomposition (POD) technique of model order reduction, for a parameterized quasi-nonlinear parabolic equation. A very adaptive technique, for building a parametric ROM is the Proper Generalized Decomposition (PGD) method It is based on building an approximation of the initial PDE as a finite combination of functions of separate variables, including the space and time variables, and all eventual parameters that could be associated with the initial equations. These functions and their coefficients in the later expression are obtained by an iterated algorithm which minimizes the error with respect to the initial problem. It was generalised by Chinesta et al for multidimensional problems [23,24,25,26,27,28]
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