Speaker
Description
Various phenomena simulated using partial differential equations (PDEs) give rise to constrained systems of equations. These include models of optimal control with constraints given by elliptic PDEs, as well as fundamental models of fluid dynamics such as the Stokes equations where the constraints correspond to the incompressibility (divergence-free) condition. If these models also depend on random (parametrized) input data, than it is important to develop reduced-order models (ROMs) to reduce the computational costs associated with multiple solutions of the large-scale algebraic systems that arise from discretization. Several approaches have been developed to construct ROMs for constrained problems. These approaches supplement greedy search strategies [4] with methods that augment the spaces obtained from searching in order to enforce inf-sup stability, which otherwise does not hold in the reduced spaces. In this work, we present two sets of results. The first is a comparison of the effectiveness of two such methods for augmentation, known as aggregation methods [3] and supremizing methods [1, 2]. The second is an introduction of a new approach that avoids the difficulties caused by lack of inf-sup stabiity by forcing the reduced model to have a simpler structure not of saddle-point form.
References
[1] A. Quarteroni and G. Rozza, Numerical solution of parametrized Navier-Stokes equations by reduced basis methods, Numerical Methods for Partial Differential Equations 23(4):923–948, 2007.
[2] G. Rozza and K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains, Computer Methods in Applied Mechanics and Engineering 196(7):1244–1260, 2007.
[3] F. Negri and G. Rozza and A. Manzoni and A. Quarteroni, Reduced basis method for parametrized optimal control problems, SIAM Journal on Scientific Computing 35(5): A2316-A2340, 2013
[4] A. Buffa, Y. Maday, A. T. Patera, C. Prud’homme, and G. Turinici. A priori convergence of the greedy algorithm for the parametrized reduced basis method ESAIM: Mathematical Modelling and Numerical Analysis 46(03):595–603, 2012.
