Deflation-based certified greedy algorithm and adaptivity for bifurcating nonlinear PDEs

TL;DR

This paper introduces two novel algorithms for reduced order modeling of bifurcating nonlinear PDEs, enabling detection of bifurcation points and certification of multiple solution branches without prior knowledge.

Contribution

The work presents the adaptive-greedy and deflated-greedy algorithms that improve bifurcation detection and solution certification in reduced models of nonlinear PDEs.

Findings

Successfully detected bifurcation points in Navier-Stokes simulations.

Certified multiple coexisting solution branches with improved accuracy.

Outperformed traditional POD and greedy methods in test cases.

Abstract

This work deals with tailored reduced order models for bifurcating nonlinear parametric partial differential equations, where multiple coexisting solutions arise for a given parametric instance. Approaches based on proper orthogonal decomposition have been widely investigated in the literature, but they usually rely on some \emph{a-priori} knowledge about the bifurcating model and lack any error estimation. On the other hand, standard certified reduced basis techniques fail to represent correctly the branching behavior, since the error estimator is no longer reliable. The main goal of the contribution is to overcome these limitations by introducing two novel algorithms: (i) the adaptive-greedy, detecting the bifurcation point starting from scarce information over the parametric space, and (ii) the deflated-greedy, certifying multiple coexisting branches simultaneously. The former approach takes advantage of the features of the reduced manifold to detect the bifurcation, while the latter exploits the deflation and continuation methods to discover the bifurcating solutions and enrich the reduced space. We test the two strategies for the Coanda effect held by the Navier-Stokes equations in a sudden-expansion channel. The accuracy of the approach and the error certification are compared with vanilla-greedy and proper orthogonal decomposition.