Bridging Scales in Random Materials

Projektbeschreibung

Zufälligkeit in mehrskaligen partiellen Differentialgleichungen – Mathematik der Zufallsmedien

Bei der mathematischen Beschreibung vieler physikalischer Phänomene entstehen partielle Differentialgleichungen mit mehreren Skalen. Das EU-finanzierte Projekt RandSCALES befasst sich mit der zentralen Rolle der Zufälligkeit in mehrskaligen partiellen Differentialgleichungen. Mathematische Modelle für Materialien mit zufälliger Mikrostruktur führen naturgemäß zu derartigen mehrskaligen partiellen Differentialgleichungen, in denen Zufälligkeit herrscht, jedoch harren wichtige Fragen auf diesem Gebiet noch einer Antwort. Das Projekt wird sich insbesondere auf die quantitative Ableitung effektiver Modelle für nichtlineare Materialien mit zufälliger Mikrostruktur, den Entwurf numerischer Homogenisierungsschemata bei fehlender Skalentrennung und die stabilisierende Wirkung der Zufälligkeit auf evolutionäre partielle Differentialgleichungen konzentrieren.

Ziel

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.