Most theoretical and real world systems can be understood at multiple scales of detail. Consider a system of gas particles. This system could be understood in terms of:
- quantum mechanical wavefunctions for the particles comprising each atom,
- trajectories of individual gas molecules,
- probability density functions of particle positions and velocities,
- hydrodynamic quantities like pressure and bulk velocity.
Hierarchies of description linked by assumptions and simplifications exist in biology, chemistry, engineering, physics, and theoretical mathematics systems. However, there exist systems for which the assumptions necessary to model them at one level are not met, but which are also impractical or impossible to analyze and simulate at lower levels. There are also cases in which the different scales of detail cannot be neatly separated.
In both of these cases, a multiscale approach to the problem is necessary. Multiscale techniques leverage the different information made available by viewing the same problem at multiple scales. My research focuses on multiscale techniques as both a practical simulation tool and as a means of understanding the structure of complex systems with multiple scales.