Uniqueness and stability of an inverse problem for a phase field model using data from one component

Abstract

h i g h l i g h t s• A Carleman estimate for the parabolic-hyperbolic phase field system is proved. • Lipschitz stability and uniqueness for a coefficient inverse problem for this phase field system using data from one component are established.• Lipschitz stability provides theoretical support for numerical methods. a b s t r a c tWe study an inverse problem of determining a spatial varying coefficient in a parabolichyperbolic phase field model with the following observation data of only one component: the order parameter in a subdomain ω satisfying ∂ω ⊃ ∂Ω along a sufficiently large time interval and at a suitable time over the whole spatial domain. Based on a Carleman estimate for the parabolic-hyperbolic phase field system, we prove the Lipschitz stability and uniqueness for this inverse problem.

An energy-stable convex splitting for the phase-field crystal equation

“…The model that is considered in this work can be improved by increasing the number of critical wavelengths one considers in the free energy functional at the expense of computational cost, as the partial differential equation becomes harder to solve [8,28]. Also, molecular dynamics in a multi-scale setting can be used to estimate some of the parameters going into the phase-field crystal equation [30], and inverse formulations of the problem could be considered to validate the calculations [31]. Hopefully, these multi-scale approaches will allow for more complete studies on polycrystalline growth using the PFC equation, such as the ones presented in [32,33] in the setting of phase-field modeling.…”

Fractional physics-informed neural networks for time-fractional phase field models

“…The core of the inverse problem is the estimation of the unknown parameters in the model according to the known model and some observed data. Wu et al [28] proved the Lipschitz stability and uniqueness of the inverse problem of phase field models with an inertial term. Stability results concerning the inverse problem of determining two time-independent coefficients with the observation data were presented in [29].…”