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

Federico Pichi, Maria Strazzullo

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.

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

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.
Paper Structure (21 sections, 32 equations, 11 figures, 8 algorithms)

This paper contains 21 sections, 32 equations, 11 figures, 8 algorithms.

Table of Contents

  1. Introduction and motivation
  2. Certified ROMs for nonlinear parametric PDEs
  3. Problem formulation: continuous and discrete settings
  4. The greedy algorithm
  5. A-posteriori error estimator for nonlinear problems
  6. State of the art for bifurcating phenomena
  7. Bifurcation and stability analysis
  8. Deflation method
  9. ROMs for bifurcation systems
  10. The adaptive-greedy algorithm
  11. The deflated-greedy algorithm
  12. Reduced deflation method
  13. Certification of multiple branches
  14. Numerical results
  15. Numerical setting: the Coanda effect
  16. ...and 6 more sections

Figures (11)

  • Figure 1: Schematic representation of the solution ensemble in the case of a pitchfork bifurcation.
  • Figure 2: Comparison of vanilla-greedy methodology and the proposed deflated-greedy approach.
  • Figure 3: Sudden-expansion channel domain $\Omega$.
  • Figure 4: Pitchfork bifurcation for the Coanda effect via point-wise evaluation of the vertical velocity in the channel's centerline, and the corresponding coexisting solutions for the symmetric and asymmetric branches at $Re = 156$, i.e. $\mu = 0.5$.
  • Figure 5: Evolution of the refinement strategy for the parametric space versus the number of basis/iterations.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Remark 1: Continuation methods
  • Remark 2: Error estimator
  • Remark 3: Computational times
  • Remark 4: Bifurcation agnostic