Description du projet
Modéliser les systèmes physiques complexes à de multiples échelles
Les microstructures de particules présentant des formes arbitraires peuvent être à l’origine de phénomènes physiques étonnants. Les simulations numériques offrent aux scientifiques la possibilité de mieux comprendre et contrôler ces systèmes physiques. Toutefois, la capacité de calcul requise pour simuler les intrications complexes de leurs effets à de nombreuses échelles non séparables, voire à un continuum d’échelles caractéristiques, dépasse les ressources de calcul actuelles de plusieurs ordres de grandeur. Modéliser des phénomènes physiques à de multiples échelles nécessite une nouvelle génération de méthodes de calcul tenant compte, de manière hiérarchique et adaptative, de leur caractère aléatoire et désordonné. En quête d’un tel paradigme, le projet RandomMultiScales, financé par l’UE, entend concevoir des méthodes d’analyse numérique efficaces pour étudier ces problèmes à plusieurs échelles. Les recherches menées se situent au carrefour entre la quantification de l’incertitude et la physique computationnelle.
Objectif
Geometrically or statistically heterogeneous microstructures and high physical contrast are the key to astonishing physical phenomena such as invisibility cloaking with metamaterials or the localization of quantum waves in disordered media. Due to the complex experimental observation of such processes, numerical simulation has very high potential for their understanding and control. However, the underlying mathematical models of random partial differential equations are characterized by a complex interplay of effects on many non-separable or even a continuum of characteristic scales. The attempt to resolve them in a direct numerical simulation easily exceeds today's computing resources by multiple orders of magnitude. The simulation of physical phenomena from multiscale models, hence, requires a new generation of computational multiscale methods that accounts for randomness and disorder in a hierarchical and adaptive fashion.
This proposal concerns the design and numerical analysis of such methods. The main goals are connected to fundamental mathematical and algorithmic challenges at the intersection of multiscale modeling and simulation, uncertainty quantification and computational physics:
(A) Numerical stochastic homogenization beyond stationarity and ergodicity,
(B) Uncertainty quantification in truly high-dimensional parameter space,
(C) Computational multiscale scattering in random heterogeneous media,
(D) Numerical prediction of Anderson localization and quantum phase transitions.
These objectives base upon recent breakthroughs of deterministic numerical homogenization beyond periodicity and scale separation and its deep links to seemingly unrelated theories ranging all the way from domain decomposition to information games and their Bayesian interpretation. It is this surprising nexus of classical and probabilistic numerics that clears the way to the envisioned new computational paradigm for multiscale problems at randomness and disorder.
Champ scientifique
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencescomputer and information sciencescomputational sciencemultiphysics
- natural sciencesmathematicsapplied mathematicsnumerical analysis
- natural sciencesmathematicsapplied mathematicsmathematical model
Programme(s)
Régime de financement
ERC-COG - Consolidator GrantInstitution d’accueil
86159 Augsburg
Allemagne