Bridging Scales in Random Materials

Descripción del proyecto

Aleatoreidad en ecuaciones diferenciales parciales multiescala: las matemáticas de medios aleatorios

Las ecuaciones diferenciales parciales (EDP) con múltiples escalas se plantean en la descripción matemática de una gran variedad de fenómenos físicos. El objetivo del proyecto financiado con fondos europeos RandSCALES es estudiar el papel esencial de la aleatoriedad en las EDP multiescala. Los modelos matemáticos de materiales con microestructura aleatoria conducen naturalmente a tales EDP multiescala con aleatoriedad. Aun así, existen muchas preguntas sin respuesta en este ámbito. En concreto, el proyecto se centrará en la derivación cuantitativa de modelos eficaces para materiales no lineales con microestructura aleatoria, el diseño de esquemas de homogeneización numérica en ausencia de separación de escalas y en el efecto estabilizador de la aleatoreidad en las EDP evolutivas.

Objetivo

The identification and justification of scaling limits is a central theme in modern PDE theory, reflected for instance in the theories of homogenization and singular limits. In many multiscale PDE models, randomness plays a crucial role: In random media, the quantitative homogenization process is driven by decorrelation and concentration of measure; for ill-posed evolution problems like many interface evolution equations, random noise may provide a regularization, potentially restoring well-posedness and hence approximability by numerical schemes. In the present project, we pursue a program to achieve a deeper understanding of the role of randomness in multiscale PDEs. We focus on three important, yet largely unexplored, aspects:

A) We develop a quantitative stochastic homogenization theory for nonlinear material models, ranging from variational models for brittle fracture over models from statistical mechanics to the motion of interfaces in random media. A key challenge is posed by the non-convex structure of the models, giving rise to rough energy landscapes and the emergence of complex physical phenomena.

B) We establish generalizations of homogenization in the absence of scale separation, a problem naturally posed in the framework of random media. By developing new high-dimensional approximability results, we will contribute to uncertainty quantification and the design of numerical homogenization schemes with lower computational complexity.

C) We develop a theory of stability and approximability of interface evolution problems past topology changes, a setting in which randomness may lead to the regularization of ill-posed evolutions and thereby allow for the derivation of error estimates for numerical approximation schemes. By relying on energy methods, we avoid the use of comparison principles, greatly enhancing the scope of applicability of our theory.