On uniqueness and stable estimation of multiple parameters in the Cahn-Hilliard equation

TL;DR

This paper investigates the identifiability and stable numerical estimation of multiple parameters in the Cahn-Hilliard equation, providing theoretical guarantees and numerical validation for parameter recovery from phase fraction measurements.

Contribution

It establishes conditions for parameter identifiability in the Cahn-Hilliard model and introduces a regularized method with proven convergence for stable parameter estimation.

Findings

Parameters are identifiable up to scaling invariances.

The regularized approach converges under data noise.

Numerical tests confirm theoretical results.

Abstract

We consider the identifiability and stable numerical estimation of multiple parameters in a Cahn-Hilliard model for phase separation. Spatially resolved measurements of the phase fraction are assumed to be accessible, with which the identifiability of single and multiple parameters up to certain scaling invariances is established. A regularized equation error approach is proposed for the stable numerical solution of the parameter identification problems, and convergence of the regularized approximations is proven under reasonable assumptions on the data noise. The viability of the theoretical results and the proposed methods is demonstrated in numerical tests.