Homogenization and Uncertainty Quantification

Uncertainty quantification is an emerging field that aims to provide precise information about the uncertainty in computed solutions to complex problems, often involving partial differential equations. The uncertainty may originate in lack of information about the environment in which the phenomenon modeled by the partial differential equation takes place, or it may originate from uncertainty in constitutive laws, boundary conditions, etc. Quantifying uncertainty is vastly more demanding than just producing a "solution", both theoretically and computationally. Homogenization theory, which is now more than thirty years old, provides some clues as to how this complexity can be reduced in what at first appear to be very complicated problems, ones that involve randomness on small scales, which are effectively unobservable. I will discuss how homogenization theory can help in reducing some of the complexities in uncertainty quantification.

George C. Papanicolaou, Stanford University

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