Periodic Reporting for period 3 - RandomMultiScales (Computational Random Multiscale Problems)
Okres sprawozdawczy: 2023-08-01 do 2025-01-31
Computer simulations have long played a significant role in scientific and technological progress. They are not only the basis for the further development of high-performance materials (e.g. fibre composites, which replace conventional materials in large parts of product development), they also allow the exploration of new physical phenomena for high-tech products of the future (e.g. peculiar aggregate states such as Bose-Einstein condensates with applications in technologies such as atomic lasers or quantum computers). Like the physical processes themselves, the mathematical models underlying their simulation are characterized by a complex interplay of effects and mechanisms on a variety of length and time scales. Attempting to represent this inherent multiscale nature in standard computer models pushes even the most advanced supercomputers to their limits. The simulation of such multiscale phenomena therefore requires a new generation of algorithms that use hierarchical and adaptive solution strategies to reduce the complex mathematical models to a computable and economical scale without significantly compromising their expressiveness.
In this project, we aim to construct and analyse computational methods for those multiscale problems under randomness and disorder. For the design and numerical analysis of such methods, we bridge research areas such as multiscale modelling and simulation, uncertainty quantification, and computational physics. Firstly, the essential challenges of random multiscale problems are studied and decoded in prototypical model scenarios involving randomness and disorder. We study a class of methods called numerical homogenization to capture the impact of unresolved microscopic effects of the models effectively and accurately. To ensure that the corresponding developments are of relevance in applications, we generalize the methods to relevant linear and nonlinear problems in the context of wave phenomena in disordered media, aiming to provide reliable computational methods for wave simulation in extreme parameter regimes. This lays the basis for reliable simulation of physical processes for technologies like the ones mentioned above.
In this project, we aim to construct and analyse computational methods for those multiscale problems under randomness and disorder. For the design and numerical analysis of such methods, we bridge research areas such as multiscale modelling and simulation, uncertainty quantification, and computational physics. Firstly, the essential challenges of random multiscale problems are studied and decoded in prototypical model scenarios involving randomness and disorder. We study a class of methods called numerical homogenization to capture the impact of unresolved microscopic effects of the models effectively and accurately. To ensure that the corresponding developments are of relevance in applications, we generalize the methods to relevant linear and nonlinear problems in the context of wave phenomena in disordered media, aiming to provide reliable computational methods for wave simulation in extreme parameter regimes. This lays the basis for reliable simulation of physical processes for technologies like the ones mentioned above.
At this stage of the project, we have obtained numerous results, published in relevant scientific journals. In particular, we want to highlight six achievements, which we believe to have the highest impact on the associated research fields so far:
1. We bridged the two recent theories of modern numerical homogenization and the quantitative stochastic homogenization and derived for the first time rigorous error estimates for a practical algorithm which allow to assess its performance a priori.
2. We have taken a major step toward understanding localized states in the context of random nonlinear Schrödinger equations, giving the first quantitative prediction of localization and delocalization regimes in the nonlinear setting.
3. We employed a problem-adaptive approach to derive algorithms for a class of nonlinear eigenvector problems using their representation as a constrained optimization problem. These methods are competitive for the efficient and reliable simulation of models of quantum physics and computational chemistry.
4. We developed a new super-localization method for numerical homogenization which marks an algorithmic breakthrough as it outperforms the best-known decay rates in practice. This development sheds light on the long-standing question in numerical analysis, whether truly local numerical homogenization without logarithmic computational overhead is possible. In one dimension, true locality was confirmed, whereas in higher dimensions the overhead is hardly observable in practice.
5. We derived a non-trivial multi-resolution extension of an existing multiscale method, known as Localized Orthogonal Decomposition, to the simulation of sound scattering in heterogenous media. It marks the first step in the direction of a homogenization-based multilevel sampling scheme beyond well-behaved linear diffusion problems.
6. We successfully trained deep neural networks to obtain surrogate models for problems involving multiple scales. These models have been used to simulate the effective behaviour of the problems’ solutions on a macroscopic scale of interest.
1. We bridged the two recent theories of modern numerical homogenization and the quantitative stochastic homogenization and derived for the first time rigorous error estimates for a practical algorithm which allow to assess its performance a priori.
2. We have taken a major step toward understanding localized states in the context of random nonlinear Schrödinger equations, giving the first quantitative prediction of localization and delocalization regimes in the nonlinear setting.
3. We employed a problem-adaptive approach to derive algorithms for a class of nonlinear eigenvector problems using their representation as a constrained optimization problem. These methods are competitive for the efficient and reliable simulation of models of quantum physics and computational chemistry.
4. We developed a new super-localization method for numerical homogenization which marks an algorithmic breakthrough as it outperforms the best-known decay rates in practice. This development sheds light on the long-standing question in numerical analysis, whether truly local numerical homogenization without logarithmic computational overhead is possible. In one dimension, true locality was confirmed, whereas in higher dimensions the overhead is hardly observable in practice.
5. We derived a non-trivial multi-resolution extension of an existing multiscale method, known as Localized Orthogonal Decomposition, to the simulation of sound scattering in heterogenous media. It marks the first step in the direction of a homogenization-based multilevel sampling scheme beyond well-behaved linear diffusion problems.
6. We successfully trained deep neural networks to obtain surrogate models for problems involving multiple scales. These models have been used to simulate the effective behaviour of the problems’ solutions on a macroscopic scale of interest.
All the results obtained so far go beyond the state of the art by introducing new methods and their numerical analysis or by identifying new surprising links between existing techniques. In addition to our achievements to date, the new super-localization method for numerical homogenization stands out as a powerful methodology on its own. In several publications, its large scope and high potential have been demonstrated. Moreover, we believe that we can use it for, e.g. multiscale simulation of Bose-Einstein condensates in quantum physics that goes beyond current capabilities and even allows the exploration of new physical phenomena.
From here, we are confident that by the end of the project, we will be able to overcome the current limitations of fast methods for the uncertainty quantification of mathematical models of physical processes involving randomness at short correlation lengths. Moreover, we expect to take a key step toward understanding how to deal with randomness and high contrast in indefinite scattering problems at high frequencies.
From here, we are confident that by the end of the project, we will be able to overcome the current limitations of fast methods for the uncertainty quantification of mathematical models of physical processes involving randomness at short correlation lengths. Moreover, we expect to take a key step toward understanding how to deal with randomness and high contrast in indefinite scattering problems at high frequencies.