Computational Random Multiscale Problems

Descripción del proyecto

Modelización de sistemas físicos complejos a múltiples escalas

Las microestructuras con partículas de forma arbitraria pueden crear fenómenos físicos sorprendentes. La simulación numérica ofrece a los científicos el potencial para comprender y controlar mejor estos sistemas físicos. Sin embargo, la capacidad computacional para simular su compleja interacción de efectos en muchas escalas no separables o incluso en un continuo de escalas características excede los recursos informáticos actuales en varios órdenes de magnitud. La modelización de fenómenos físicos a múltiples escalas requiere una nueva generación de métodos computacionales que considere la aleatoriedad y el desorden de manera jerárquica y adaptativa. Con ese objetivo, el proyecto RandomMultiScales, financiado con fondos europeos, se propone diseñar métodos de análisis numérico eficientes para estudiar esos problemas multiescala. La investigación realizada se encuentra entre la cuantificación de la incertidumbre y la física computacional.

Objetivo

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.