Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method

Abstract

The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.

Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations

“…In this investigation, we do not deal with recovering all bifurcation branches, but limit our attention to the stable branch of solutions with jets hugging the bottom wall. However, we remark that recovering all bifurcating solutions with model reduction methods is also possible, see, e.g., ( [10]) and ( [19]).…”

A comparison of reduced-order modeling approaches using artificial neural networks for PDEs with bifurcating solutions

“…Finally, we would like to mention that machine learning techniques based on sparse optimization have been applied to detect bifurcating branches of solutions for a two-dimensional laterally heated cavity and Ginzburg-Landau model in [2,12], respectively. Finding all branches after a bifurcation occurs can be done with deflation methods (see [21]), which require introducing a pole at each known solution. In principle, the ROM approaches under investigation here can be combined with deflation techniques.…”

Model order reduction for bifurcating phenomena in Fluid-Structure Interaction problems

“…In the following, we will be using the simplest version of this methodology which consists in choosing directly (X h (µ j ), µ j + ∆µ) as initial guess. However, there are more complex strategies which can be exploited to automatically discover new branches [19,42]. We remark that a lot of attention must be paid to the choice of the discretization step; if it is too large, we risk not to capture the bifurcating behaviour, while a too-small step causes a possible waste of computational resources, especially in regions of the parameter space far from the bifurcation points.…”