Time and Reduced Basis Methods
Abstract:
Parametrized parabolic problems often occur in industrial or financial applications, e.g. as pricing of options on the stock market. If we want to calibrate an option pricing model, we need several evaluations for different parameters. Fine discretizations, that are needed for these problems, resolve in large scale problems and thus in long computational times. To reduce the size of those problems, we use the Reduced Basis Method (RBM) [QMN16, HRS16]. The ambition of the RBM is to efficiently reduce discretized parametrized partial differential equations given in a variational form.
In this talk, we consider a comparison between space-time methods and the often used time- stepping scheme for the RBM, [GMU16]. Using space-time formulations, we do not use a time- stepping scheme, but take the time as an additional variable in the variational formulation of the problem. Well-posedness for the space-time variational approach has been shown for a wide range of problems. For the general case of a parabolic variational equation, see [SS09]. Combining the RBM with the space-time formulation, we derive a possibly noncoercive Petrov–Galerkin problem, where improved error estimators for parabolic equations could be achieved [UP14].
We conclude with an overview where the space-time methods has been successfully applied to RBM.
References
[GMU16] Silke Glas, Antonia Mayerhofer, and Karsten Urban. Two ways to treat time in reduced basis methods. Preprint, submitted to Springer MoRePaS 3 special edition, 2016.
[HRS16] Jan S. Hesthaven, Gianluigi Rozza, and Benjamin Stamm. Certified reduced basis methods for parametrized partial differential equations. Springer Briefs in Mathematics. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2016. BCAM SpringerBriefs.
[QMN16] Alfio Quarteroni, Andrea Manzoni, and Federico Negri. Reduced basis methods for partial differential equations, volume 92 of Unitext. Springer, Cham, 2016. An introduction, La Matematica per il 3+2.
[SS09] Christoph Schwab and Rob Stevenson. Space-time adaptive wavelet methods for parabolic evolution problems. Math. Comp., 78(267):1293–1318, 2009.
[UP14] Karsten Urban and Anthony T. Patera. An improved error bound for reduced basis approx- imation of linear parabolic problems. Math. Comp., 83(288):1599–1615, 2014.
