The field of numerical methods for partial differential equation (PDE) models and inverse problems is rapidly advancing, with a focus on developing efficient and accurate algorithms for solving complex problems. Recent research has emphasized the importance of understanding the behavior of numerical methods for non-local transport-dominated PDE models, which are used to describe various collective migration phenomena in cell biology and ecology. The development of new numerical schemes, such as semi-implicit and variational multiscale methods, has improved the accuracy and stability of simulations. Additionally, researchers have made significant progress in developing goal-oriented error estimation frameworks for finite element analysis of convection-diffusion-reaction equations. The application of these methods to real-world problems, such as beam hardening streaks in tomography and optimal insulation problems on non-smooth domains, has also been explored. Notable papers in this area include the development of a new semi-implicit scheme for stabilizing oscillations in non-local equations and a variational multiscale approach to goal-oriented error estimation. Furthermore, researchers have investigated the numerical reconstruction and analysis of backward semilinear subdiffusion problems and developed a mixed finite element method for a class of fourth-order stochastic evolution equations with multiplicative noise. Overall, the field is moving towards the development of more sophisticated and efficient numerical methods for solving complex PDE models and inverse problems. Noteworthy papers include: the development of a new semi-implicit scheme for non-local equations and a variational multiscale approach to goal-oriented error estimation.