Multiscale systems, characterized by interactions between different scales in space and time, offer valuable insights into diverse scientific disciplines, including physics, biology, materials science, and engineering. With advances in hardware technologies enabling simulations of larger and more complex systems, the development of robust and scalable numerical methods becomes crucial to fully unlock the potential of modern High-Performance Computing (HPC) architectures. This minisymposium aims to address key topics in this field, including multilevel solvers for partial differential equations, nonlinear optimization, and multi-fidelity techniques. A fundamental, but usually neglected, part of the solution procedure is the choice of the hardware and the representation of infinite dimensional objects such as irrational numbers. Because of this, we also welcome contributions related to hybrid computing, matrix-free solvers and mixed-precision algorithms.