# Multiscale Mathematics

Building on our established strengths in multiscale science, PNNL is part of a coordinated effort in Multiscale Mathematics to develop the mathematical and computational theory and algorithms that will enable true multiscale simulation and analysis.

Specific multiscale math capabilities being developed include

- Adaptive methods and time-step acceleration
- Hybrid ab-initio and QC methods
- new adaptive evolution algorithms
- wavelets
- multivariational analysis
- adaptive multiscale particle methods
- accelerated stochastic kinetic simulation with adaptive spatial meshing

- Optimization, inverse modeling, and data integration
- Large-scale parameter estimation
- clustering and probabilistic model selection
- constrained stochastic optimization
- multiscale likelihood calculations.

Through its partnerships with national laboratories and universities in the US and abroad, PNNL is positioned to develop and apply multiscale mathematics to solve multidisciplinary scientific problems:

**Climate: Modeling Framework (MMF) for Climate Research** Climate modelers are using grid resolution developments in multiscale mathematics to connect the physics and chemistry of atmospheric processes across naturally occurring scales, from hundreds of nanometers to thousands of kilometers.

**Uncertainty Quantification: Design and Analysis of Computer Experiments **This adaptive methodology will enable efficient, global analysis of uncertainty and sensitivity of predictions from large-scale scientific simulations.

**Crosscutting: Hierarchy Theory** This work concerns finding the mathematical conditions under which model reduction can be achieved by collapsing discrete collections of objects into continuous representation, while retaining the accuracy of information flow across levels of scale and organization.

**Bioremediation: Multiscale Modeling and Uncertainty Quantification for Bioremediation** Advances in these mathematical methods will result in more accurate modeling of the wettability properties of various fluids, thereby facilitating realistic subsurface flow predictions important for groundwater remediation and oil extraction efforts.

**Biology: Multicell Models of Cell-to-Cell Communication** This software will facilitate linkage of stochastic receptor binding events with intercellular processes of protein-to-protein interactions, leading to the gene expression regulation.

Contact: Alexandre Tartakovsky