The recent advancements in numerical methods and computational techniques have significantly enhanced the ability to solve complex problems across various domains. A common theme among the developments is the emphasis on high-order accuracy, stability, and adaptability, particularly in the context of nonlinear and non-convex problems. Notable innovations include the extension of traditional methods like the Kaczmarz algorithm to tensor systems, enabling efficient solutions for high-dimensional data problems such as image deblurring. Adaptive and robust identification algorithms are being advanced to handle saturated observations and non-traditional system signals, crucial for applications like judicial sentencing prediction. Optimal control problems constrained by complex kinetic equations are being addressed using novel hypocoercivity frameworks, ensuring stability and robustness. Low-rank matrix factorization and robust PCA have seen improvements with new models integrating adaptive weighted least squares. Neural adaptive spectral methods are emerging as powerful tools for solving optimal control problems efficiently, offering substantial speedups and high generalization capabilities. The application of optimal transport theory to ensemble control problems provides new computational efficiencies, particularly in state tracking scenarios with limited observations. In the realm of numerical methods for stochastic partial differential equations (SPDEs), domain decomposition techniques and boundary-preserving schemes are enhancing computational efficiency and error bounds. Long-term stable algorithms for non-Newtonian Stokes equations with transport noise have opened new avenues for studying noise influence on fluid dynamics. Significant progress has been made in computing rough solutions of the stochastic nonlinear wave equation, with novel time discretization methods achieving robust convergence under relaxed regularity conditions. Advances in numerical methods for PDEs on surfaces include meshfree methods with Radial Basis Function-Finite Difference (RBF-FD) approximations, offering high-order convergence and flexibility. New interior penalty methods for higher-order formulations of surface Stokes problems address challenges associated with surface curvature. Optimization-based strategies for coupling 3D-1D problems have been extended to nonlinear contexts, enhancing geometric complexity management. Predictor-corrector time-stepping methods in parametric finite element approaches for surface diffusion achieve second-order temporal accuracy without mesh regularization. Entropy-conservative and entropy-stable methods are being extended to new domains like cut meshes and magnetohydrodynamic equations, ensuring stability and accuracy. Matrix-free implementations and non-nested multigrid methods offer flexibility and scalability for large-scale problems. High-order finite element methods and discontinuous Galerkin approximations focus on error estimates and conservation properties. Novel preconditioning techniques and hierarchical solvers improve convergence rates in eigenvalue problems. Second-order accurate methods are being introduced to enhance efficiency in micromagnetics simulations, and the use of harmonic functions and discrete harmonics achieves higher convergence rates in fluid dynamics problems. Discontinuous Galerkin isogeometric methods are applied to electronic structure calculations, providing a unified analysis framework for elliptic eigenvalue problems. A priori and a posteriori error estimates for discontinuous Galerkin approximations of time-harmonic Maxwell's equations ensure minimal regularity assumptions and optimal accuracy. These advancements collectively push the boundaries of computational methods, making them more robust and applicable to a wider range of scientific and engineering challenges.