Periodic Reporting for period 4 - RandomMultiScales (Computational Random Multiscale Problems)
Periodo di rendicontazione: 2025-02-01 al 2025-07-31
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.
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.
From the outset, the project advanced the theory and practice of computer simulations for complex systems with multiple interacting scales and random effects. New mathematical methods were designed that make it possible to capture microscopic structures and their impact on macroscopic behaviour far more efficiently than before. These included new approaches to stochastic homogenization, a breakthrough super-localization technique that increases both accuracy and robustness, and hierarchical algorithms that can represent a continuum of scales. Together, these methods significantly extend the reach of multiscale simulations in random and disordered settings.
The results were applied to several areas of science and engineering. Robust solvers were developed for wave propagation in heterogeneous and high-contrast media, including scattering problems where classical methods break down. In mathematical physics, the project achieved the first quantitative prediction of localization and delocalization in nonlinear Bose–Einstein condensates and culminated in the first three-dimensional multiscale simulation of quantum phase transitions, directly connected to experiments. Complementary work introduced reliable solvers for nonlinear eigenvalue problems, which are relevant not only for quantum physics but also for computational chemistry. Unexpectedly, neural networks were successfully integrated into multiscale discretizations, and first steps were taken in the numerical analysis of quantum algorithms.
The project’s findings were disseminated through numerous publications in leading international journals, invited and plenary conference presentations, and dedicated workshops and schools. Several PhD theses were completed within the project, ensuring transfer of knowledge and training of young researchers. The methods are already being used in collaborations with partners in engineering, demonstrating their exploitation potential well beyond mathematics.
The results were applied to several areas of science and engineering. Robust solvers were developed for wave propagation in heterogeneous and high-contrast media, including scattering problems where classical methods break down. In mathematical physics, the project achieved the first quantitative prediction of localization and delocalization in nonlinear Bose–Einstein condensates and culminated in the first three-dimensional multiscale simulation of quantum phase transitions, directly connected to experiments. Complementary work introduced reliable solvers for nonlinear eigenvalue problems, which are relevant not only for quantum physics but also for computational chemistry. Unexpectedly, neural networks were successfully integrated into multiscale discretizations, and first steps were taken in the numerical analysis of quantum algorithms.
The project’s findings were disseminated through numerous publications in leading international journals, invited and plenary conference presentations, and dedicated workshops and schools. Several PhD theses were completed within the project, ensuring transfer of knowledge and training of young researchers. The methods are already being used in collaborations with partners in engineering, demonstrating their exploitation potential well beyond mathematics.
The project has moved well beyond the existing state of the art in several directions. In multiscale mathematics, it introduced a new super-localization technique that provides far more accurate and efficient simulations than classical approaches, making it possible to model complex random materials in regimes that were previously inaccessible. In stochastic settings, the development of hierarchical multiresolution algorithms opened the door to reliable uncertainty quantification for highly irregular problems, something that standard methods could not achieve. For wave propagation, the project produced the first robust multiscale solvers capable of treating high-frequency scattering in strongly disordered and high-contrast media, a long-standing challenge in computational physics. In quantum applications, the project delivered the first quantitative prediction of localization and delocalization in nonlinear Bose–Einstein condensates and culminated in a full three-dimensional multiscale simulation of quantum phase transitions, directly linked to experiments.
In addition to these planned breakthroughs, the project also generated unexpected advances. Neural networks were integrated into multiscale simulations, providing efficient surrogates that retain accuracy. New optimization methods were developed for nonlinear eigenvalue problems, ensuring reliable solvers for quantum and chemical models. The project also contributed foundational insights into the numerical analysis of quantum algorithms, opening a bridge between numerical mathematics and quantum computing.
By the end of the project, the ambitious objectives set out in the ERC proposal were fully achieved, and in several areas substantially exceeded. The results are already being disseminated through international publications, conferences, and collaborations, and they provide a sustainable basis for future progress on remaining challenges such as rigorous theory for arbitrary random coefficients and the extension of multiscale scattering methods to Maxwell’s equations. These advances establish a foundation for continued impact in mathematics, physics, and engineering.
In addition to these planned breakthroughs, the project also generated unexpected advances. Neural networks were integrated into multiscale simulations, providing efficient surrogates that retain accuracy. New optimization methods were developed for nonlinear eigenvalue problems, ensuring reliable solvers for quantum and chemical models. The project also contributed foundational insights into the numerical analysis of quantum algorithms, opening a bridge between numerical mathematics and quantum computing.
By the end of the project, the ambitious objectives set out in the ERC proposal were fully achieved, and in several areas substantially exceeded. The results are already being disseminated through international publications, conferences, and collaborations, and they provide a sustainable basis for future progress on remaining challenges such as rigorous theory for arbitrary random coefficients and the extension of multiscale scattering methods to Maxwell’s equations. These advances establish a foundation for continued impact in mathematics, physics, and engineering.