TY - JOUR T1 - A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation JF - SIAM Journal on Scientific Computing Y1 - 2020 A1 - Federico Pichi A1 - Annalisa Quaini A1 - Gianluigi Rozza AB -

We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.

UR - https://arxiv.org/abs/1907.07082 ER -