Modern science and technology rely heavily on computer simulations. They are used to design advanced materials such as fibre composites and to explore new physical phenomena like Bose–Einstein condensates, which may form the basis of future technologies in areas such as atomic lasers or quantum computing. The underlying mathematical models involve processes across many different scales, from the microscopic to the macroscopic. Accurately representing this complexity quickly overwhelms even the most powerful supercomputers. The challenge is therefore to create a new generation of algorithms that can capture the essential features of multiscale systems, even under randomness and disorder, without prohibitive computational cost.
The aim of this ERC project was to develop and analyse such computational methods by combining multiscale modelling, uncertainty quantification, and computational physics. The research pursued four objectives: to advance stochastic homogenization methods for random materials, to construct multilevel Monte Carlo solvers based on homogenization, to apply these techniques to scattering problems in high-contrast and disordered media, and to investigate quantum phenomena such as Anderson localization and quantum phase transitions in Bose–Einstein condensates.
Over the course of the project, substantial progress was achieved on all fronts. New methods were created that provide far more efficient multiscale simulations than previously possible. These include “super-localization” techniques that dramatically improve accuracy and efficiency in random media, hierarchical algorithms that enable reliable uncertainty quantification, and robust solvers for wave propagation in complex materials. In mathematical physics, the project delivered the first quantitative prediction of localization and delocalization regimes in nonlinear Bose–Einstein condensates and culminated in the first three-dimensional multiscale simulation of quantum phase transitions, directly linked to experimental settings. Results were published in leading international journals, disseminated through conferences and workshops, and consolidated in doctoral theses, ensuring both scientific impact and training of the next generation of researchers.
The project went well beyond the state of the art. It established methodological breakthroughs such as super-localized multiscale discretizations, hierarchical multiresolution algorithms, and energy-adapted optimization methods for nonlinear eigenvalue problems. Unexpectedly, it also opened new research directions, including the integration of neural networks into multiscale simulations and first contributions to the numerical analysis of quantum algorithms. Together, these advances addressed and in several respects exceeded the challenges set out in the original proposal, and created new bridges to physics, engineering, computational chemistry, and quantum computing. Augsburg has emerged as an internationally visible centre for numerical analysis and multiscale simulation, with a sustainable foundation for future interdisciplinary research.
