Efficient computation of bifurcation diagrams via adaptive ROMs
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Various ideas concerning model reduction based on proper orthogonal decomposition are discussed, exploited, and suited to the approximation of complex bifurcations in some dissipative systems. The observation that the most energetic modes involved in these low dimensional descriptions depend only weakly on the actual values of the problem parameters is firstly highlighted and used to develop a simple strategy to capture the transitions occurring over a given bifurcation parameter span. Flexibility of the approach is stressed by means of some numerical experiments. A significant improvement is obtained by introducing a truncation error estimate to detect when the approximation fails. Thus, the considered modes are suitably updated on demand, as the bifurcation parameter is varied, in order to account for possible changes in the phase space of the system that might be missed. A further extension of the method to more complex (quasi-periodic and chaotic) attractors is finally outlined by implementing a control of truncation instabilities, which leads to a general, adaptive reduced order model for the construction of bifurcation diagrams. Illustration of the ideas and methods in the complex Ginzburg-Landau equation (a paradigm of laminar flows on a bounded domain) evidences a fairly good computational efficiency. © 2014 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.
laminar flow; phase space methods; principal component analysis; bifurcation parameter; complex bifurcation; complex ginzburg-landau equation; dissipative systems; efficient computation; numerical experiments; proper orthogonal decompositions; reduced order models; bifurcation (mathematics)