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

TL;DR

This survey reviews recent computational methods for solving PDE-constrained inverse problems with state-dependent parameters, focusing on mathematical theory, algorithms, applications, and open challenges in the field.

Contribution

It provides a comprehensive overview of variational, adjoint-based, and regularization techniques for state-dependent inverse problems in PDEs, highlighting recent advances and future research directions.

Findings

Emphasizes the importance of identifiability and ill-posedness in inverse problems.

Discusses computational frameworks and their applications in physics and engineering.

Identifies open challenges such as nonconvexity and data limitations.

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.