Recent Developments in Regularization and Inverse Problems

This volume contains the proceedings of the special session on ""Recent Developments in Regularization Methods for Nonlinear Inverse Problems"" at the Joint Mathematics Meetings held in Seattle, Washington, on January 8-11, 2025. Inverse problems constitute a vibrant and rapidly expanding branch of applied mathematics with wide-ranging applications in science, engineering, medicine, and data science. Because such problems are typically ill-posed, their analysis and resolution require sophisticated mathematical tools that bridge theory, computation, and practice. This edited volume brings together a diverse collection of contributions highlighting recent advances in the theory, numerical analysis, and application of inverse problems. The chapters cover a broad spectrum of contemporary topics, including modern developments in regularization techniques, $\mathbf{L}^1$ data fitting with discretized white Gaussian noise using stochastic optimization methods, and the analysis of convergence rates. The volume also explores inverse tensor variational inequalities, iterative regularization methods for ill-posed problems with low-regularity solutions, and the classification of ill-posed problems. In addition to theoretical developments, the book presents innovative applications and computational approaches. These include model calibration and forecasting through cholera network models, domain decomposition techniques for certain online partial differential equations, and uncertainty quantification for stochastic partial differential equations using first-order adjoint methods.