The current developments in the research area are significantly advancing the field through innovative methods and preconditioners for solving large-scale problems. There is a notable trend towards adaptive and parallelizable algorithms, particularly in the context of block coordinate descent methods and preconditioners for optimal control problems. These methods are being enhanced with momentum and column sketching techniques to improve convergence rates and computational efficiency. Additionally, there is a growing interest in extending classical methods like the Nystrom method to nonsymmetric matrices, broadening their applicability. The integration of domain decomposition methods with nonlinear solvers is also gaining traction, offering improved convergence for challenging nonlinear problems. Notably, the coupling of software packages like deal.II and FROSch is making advanced preconditioners more accessible and sustainable for complex PDE systems. Overall, the field is moving towards more adaptive, parallel, and robust solutions, with a focus on scalability and efficiency.

Noteworthy Papers:

• The introduction of an adaptive block coordinate descent method with momentum and count sketch technology significantly enhances the solution of large linear least-squares problems.
• A novel parallel-in-time preconditioner for parabolic optimal control problems demonstrates a relaxed convergence condition compared to existing methods.