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

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.

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