Bayesian Parameter Identification in the Landau-de Gennes Theory for Nematic Liquid Crystals

TL;DR

The paper tackles the problem of recovering Landau-de Gennes material parameters $(\alpha,\beta)$ from observed $Q$-tensor fields in a two-dimensional reduced model of nematic liquid crystals. It adopts a Bayesian inverse problem approach with Gaussian or improper priors and uses Metropolis-Hastings MCMC to sample the posterior distribution given simulated $Q$-data derived from the forward PDE. Key contributions include a comprehensive inverse-problem formulation, practical MCMC implementation details, and an assessment of identifiability across diagonal and rotated solution classes, along with explicit non-identifiability scenarios. The results show that $\alpha$ and $\beta$ can be recovered with percent-level accuracy in the tested ranges, yielding credible intervals and convergence diagnostics, while revealing regimes where the model or data prevent unique parameter recovery. This work provides uncertainty-quantified pathways to infer fundamental LC material properties from $Q$-tensor data, offering a principled link between experiments and LdG modelling for planar devices.

Abstract

This manuscript establishes a pathway to reconstruct material parameters from measurements within the Landau-de Gennes model for nematic liquid crystals. We present a Bayesian approach to this inverse problem and analyse its properties using given, simulated data for benchmark problems of a planar bistable nematic device. In particular, we discuss the accuracy of the Markov chain Monte Carlo approximations, confidence intervals and the limits of identifiability.

Figures (14)

• Figure 1: Experimental setup used at Hewlett Packard Laboratories, Bristol, as reported in ln07.
• Figure 2: Chain for $\alpha_\ast = 0.0008$ in case of the diagonal solution with uniform prior: first 1000 entries (left), burn-in phase (right).
• Figure 3: Mesh with $h=0.0442$; the diagonal and rotated solutions computed on this mesh.
• Figure 4: Diagonal solution, reference value $\alpha_{\ast}=0.004$: (a) plot of reference solution; (b) Markov chain and (c) histogram of the posterior distribution of parameter $\alpha$ for uniform prior (UP) distribution; (d) Markov chain and (e) histogram of the posterior distribution of parameter $\alpha$ for Gaussian prior (GP) distribution.
• Figure 5: Chain for $\alpha_{\ast} = 0.004$, diagonal, uniform prior: mean, median and 95% confidence interval for mean depending on length of chain segment.