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Bridging Scales in Random Materials

Project description

Randomness in multiscale partial differential equations - mathematics of random media

TPartial differential equations (PDEs) with multiple scales arise in the mathematical description of many physical phenomena. The EU-funded RandSCALES project is concerned with the pivotal role of randomness in multiscale PDEs. Mathematical models for materials with random microstructure naturally lead to such multiscale PDEs with randomness, yet central questions in the field are still unresolved. The project will focus in particular on the quantitative derivation of effective models for nonlinear materials with random microstructure, the design of numerical homogenisation schemes in the absence of scale separation, and the stabilising effect of randomness on evolutionary PDEs.

Objective

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.

Host institution

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA
Net EU contribution
€ 1 142 899,00
Address
Am Campus 1
3400 Klosterneuburg
Austria

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Region
Ostösterreich Niederösterreich Wiener Umland/Nordteil
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 1 142 899,00

Beneficiaries (1)