Variational State-Dependent Inverse Problems in PDE-Constrained Optimization: A Survey of Contemporary Computational Methods and Applications

TL;DR

This survey consolidates the theory and computation of state-dependent inverse problems where constitutive laws p(u) depend on the system state in PDEs. It synthesizes foundational variational and adjoint frameworks, emphasizes identifiability constrained to the attained state range, and outlines gradient-based optimization in infinite-dimensional function spaces with Sobolev regularization. Key contributions include adjoint-derived gradient representations that accumulate over state level sets, strategies to enforce physical admissibility, and mechanisms to expand the identifiability region via adaptive experimental design. The work also surveys a broad set of applications across heat transfer, fluid and multiphysics flows, turbulence closures, reduced-order models, and electrochemical batteries, illustrating both methodological maturity and intrinsic limits such as objective-dependent nonexistence of minimizers and model-class dependence. Collectively, the article provides a rigorous reference for PDE-constrained state-dependent inversion, while identifying open challenges in theory, computation, reproducibility, and integration with data-driven approaches for reliable, large-scale predictive modeling.

Abstract

State-dependent parameter identification, where unknown model parameters depend on one or more state variables in partial differential equations (PDEs) or coupled PDE systems, is fundamental to a wide range of problems in physics, engineering, and materials science. This review surveys PDE-constrained optimization approaches for such inverse problems, emphasizing the underlying mathematical theory and key computational advances developed since 2011. We discuss variational formulations, adjoint-based gradient methods, regularization strategies, and modern computational frameworks, and highlight representative applications, with particular emphasis on identifiability, ill-posedness, and structural limits of state-dependent inverse problems. The review concludes with major open challenges and emerging research directions related to nonconvexity, identifiability, regularization, adjoint computation, data limitations, and model-class dependence.

Figures (2)

• Figure 1: Schematic showing (left) the solution $\zeta(x, t_0)$ at some fixed time $t_0$ and (right) the corresponding constitutive relation $p(\zeta)$ defined over their respective domains, i.e., $\Omega = (-1,1)$ and the identifiability region $\mathcal{I}$. The dotted line represents an extension of the constitutive relation $p(\zeta)$ from $\mathcal{I}$ to the (desired) reconstruction interval $\mathcal{D}$. In the Figure on the right, the horizontal axis is to be interpreted as the ordinate. The concept is adopted from Bukshtynov2013.
• Figure 2: Illustration of the hybrid area-contour integration strategy (adopted from Bukshtynov2013). (a) The narrow band $\Omega_{\zeta_0,h_\zeta} = \{{\bf x} \in \Omega : \zeta({\bf x}) \in [\zeta_0 - \tfrac{1}{2} h_\zeta,\, \zeta_0 + \tfrac{1}{2} h_\zeta]\}$ between the neighboring level sets $\Gamma_{\zeta_0-\frac{1}{2} h_\zeta}$ and $\Gamma_{\zeta_0+\frac{1}{2} h_\zeta}$. (b) Finite-element approximation of the narrow band used for area-based quadrature. The computational mesh $\Omega^{\square}$ consists of quadrilateral elements; shaded cells represent elements lying entirely within $\Omega_{\zeta_0,h_\zeta}$, forming $\tilde{\Omega}^{*}_{\zeta_0,h_\zeta}$, while checked cells indicate elements whose representative points lie inside $\Omega_{\zeta_0,h_\zeta}$ and form $\tilde{\Omega}'_{\zeta_0,h_\zeta}$. These subsets are used to construct the hybrid area-contour approximation of integrals over the level set $\Gamma_{\zeta_0}$. See Section \ref{['sec:level_set_int']} and Bukshtynov2013 for details.

Theorems & Definitions (14)

• Definition 1: Identifiability interval
• Definition 2: Reconstruction interval
• Definition 3: Measurement span
• Theorem 1: Well-posedness of the state equation ChaventLemonnier1974Lions1968Lions1969JosephLundgren1972
• Corollary 1: Identifiable support of the constitutive law
• Proposition 1: Support of parameter sensitivity ChaventLemonnier1974
• Theorem 2: Differentiability of the parameter--to--state map ChaventLemonnier1974
• Theorem 3: Adjoint equation and gradient representation ChaventLemonnier1974Lions1968Cea1971
• Corollary 2: State-space structure of the gradient
• Theorem 4: Minimum principle for temperature-dependent conductivity Bukshtynov2011Grisvard1985