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Uncertainty quantification analysis of bifurcations of the Allen--Cahn equation with random coefficients

Christian Kuehn, Chiara Piazzola, Elisabeth Ullmann

TL;DR

This work analyzes bifurcations of the Allen–Cahn equation under random coefficients by modeling the randomness in the linear term as $q(\mathbf{x},\omega)=p+g(\mathbf{x},\omega)$, with $p$ as the mean bifurcation parameter. It establishes that bifurcation points and curves become random quantities and develops two modelling regimes: (i) spatially homogeneous randomness with analytical UQ and shift-relations to a reference branch, and (ii) spatially heterogeneous randomness tackled via a non-intrusive generalized polynomial chaos (gPC) expansion combined with stochastic collocation and numerical continuation. The methodology yields closed-form distributions for homogeneous cases and efficient surrogate models for general cases, demonstrated through 1D numerical experiments using COCO and Sparse Grids Kit. The results integrate dynamical systems and uncertainty quantification to enable probabilistic characterization of bifurcations in random PDEs, with potential applicability to more complex bifurcation scenarios and random-field models.

Abstract

In this work we consider the Allen--Cahn equation, a prototypical model problem in nonlinear dynamics that exhibits bifurcations corresponding to variations of a deterministic bifurcation parameter. Going beyond the state-of-the-art, we introduce a random coefficient in the linear reaction part of the equation, thereby accounting for random, spatially-heterogeneous effects. Importantly, we assume a spatially constant, deterministic mean value of the random coefficient. We show that this mean value is in fact a bifurcation parameter in the Allen--Cahn equation with random coefficients. Moreover, we show that the bifurcation points and bifurcation curves become random objects. We consider two distinct modelling situations: (i) for a spatially homogeneous coefficient we derive analytical expressions for the distribution of the bifurcation points and show that the bifurcation curves are random shifts of a fixed reference curve; (ii) for a spatially heterogeneous coefficient we employ a generalized polynomial chaos expansion to approximate the statistical properties of the random bifurcation points and bifurcation curves. We present numerical examples in 1D physical space, where we combine the popular software package Continuation Core and Toolboxes (CoCo) for numerical continuation and the Sparse Grids Matlab Kit for the polynomial chaos expansion. Our exposition addresses both, dynamical systems and uncertainty quantification, highlighting how analytical and numerical tools from both areas can be combined efficiently for the challenging uncertainty quantification analysis of bifurcations in random differential equations.

Uncertainty quantification analysis of bifurcations of the Allen--Cahn equation with random coefficients

TL;DR

This work analyzes bifurcations of the Allen–Cahn equation under random coefficients by modeling the randomness in the linear term as , with as the mean bifurcation parameter. It establishes that bifurcation points and curves become random quantities and develops two modelling regimes: (i) spatially homogeneous randomness with analytical UQ and shift-relations to a reference branch, and (ii) spatially heterogeneous randomness tackled via a non-intrusive generalized polynomial chaos (gPC) expansion combined with stochastic collocation and numerical continuation. The methodology yields closed-form distributions for homogeneous cases and efficient surrogate models for general cases, demonstrated through 1D numerical experiments using COCO and Sparse Grids Kit. The results integrate dynamical systems and uncertainty quantification to enable probabilistic characterization of bifurcations in random PDEs, with potential applicability to more complex bifurcation scenarios and random-field models.

Abstract

In this work we consider the Allen--Cahn equation, a prototypical model problem in nonlinear dynamics that exhibits bifurcations corresponding to variations of a deterministic bifurcation parameter. Going beyond the state-of-the-art, we introduce a random coefficient in the linear reaction part of the equation, thereby accounting for random, spatially-heterogeneous effects. Importantly, we assume a spatially constant, deterministic mean value of the random coefficient. We show that this mean value is in fact a bifurcation parameter in the Allen--Cahn equation with random coefficients. Moreover, we show that the bifurcation points and bifurcation curves become random objects. We consider two distinct modelling situations: (i) for a spatially homogeneous coefficient we derive analytical expressions for the distribution of the bifurcation points and show that the bifurcation curves are random shifts of a fixed reference curve; (ii) for a spatially heterogeneous coefficient we employ a generalized polynomial chaos expansion to approximate the statistical properties of the random bifurcation points and bifurcation curves. We present numerical examples in 1D physical space, where we combine the popular software package Continuation Core and Toolboxes (CoCo) for numerical continuation and the Sparse Grids Matlab Kit for the polynomial chaos expansion. Our exposition addresses both, dynamical systems and uncertainty quantification, highlighting how analytical and numerical tools from both areas can be combined efficiently for the challenging uncertainty quantification analysis of bifurcations in random differential equations.
Paper Structure (19 sections, 4 theorems, 59 equations, 20 figures)

This paper contains 19 sections, 4 theorems, 59 equations, 20 figures.

Table of Contents

  1. Introduction
  2. The Allen-Cahn equation with deterministic coefficients
  3. Bifurcation analysis
  4. The Allen--Cahn equation with random coefficients
  5. Bifurcation analysis
  6. Spatially-homogeneous random coefficients
  7. Uncertainty quantification analysis of bifurcations
  8. Bifurcation points
  9. Probability of bifurcating
  10. Nontrivial bifurcation curves
  11. Spatially-heterogeneous random coefficients
  12. The generalized Polynomial Chaos expansion
  13. Uncertainty quantification analysis of bifurcations
  14. Bifurcation points
  15. Probability of bifurcating
  16. ...and 4 more sections

Key Result

Proposition 3.1

Under Assumption assump-A, the problem eq:AC_random_general defines $\mathbb{P}$-almost surely pathwise a dynamical system $\{\mathcal{T},\mathcal{X},\varphi^t_{\omega}\}$ with time interval $\mathcal{T}=[0,+\infty)$, state space $\mathcal{X}=L^2(D;\mathbb{R})$ and family of evolution operators

Figures (20)

  • Figure 1: Bifurcation diagram of the Allen--Cahn equation \ref{['eq:AC']} on $D=[0,\pi]$: $(p_i^*,0)$, $i=1,2,3$ are the first three bifurcation points and $\gamma_i$ the corresponding nontrivial branches.
  • Figure 2: $Y \sim \mathcal{U}([-1,1])$.
  • Figure 3: $Y \sim \mathcal{N}(0,1)$ truncated on $[-2,2]$.
  • Figure 5: Samples of $\gamma_1(Y)$ for two observables $\mathcal{U}$ associated to its $u$-component.
  • Figure 6: Pdf along $\gamma_1(Y)$, cf. \ref{['eq:pdf_branch']}.
  • ...and 15 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 3.1: Mean bifurcation curve
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof