, Los Alamos National Laboratory

Pages: pp. 14-15

Fueled by breakthroughs in both hardware and algorithm development, the past few decades have witnessed an explosive growth in computational power, which has led to remarkable advances in various fields of science and technology such as the mapping of the human genome. It also set the stage for addressing even more ambitious goals, many of which can't be achieved through hardware developments alone. Because conventional modeling strategies are typically based on single physical descriptions, they're simply unable to capture the relevant phenomena occurring over many space and time scales. Problems of this type include modeling space weather (where plasma kinetics meets magneto-hydrodynamics), materials fracture (where *ab initio* molecular dynamics meets solid mechanics), distributed network performance (where discrete event dynamics meets stochastic fluid models), and nanodevice electronics (where quantum kinetic theory meets quantum hydrodynamics).

The importance of these problems and the need for major breakthroughs in multiscale, multiphysics modeling tools were highlighted in a series of recent workshops sponsored by the US Department of Energy (DOE). These workshops resulted in a report called "Multiscale Mathematics Initiative: A Roadmap" ( www.sc.doe.gov/ascr/mics/ams/index.html), which identifies the need for multiscale mathematics—from Earth and environmental sciences, to materials sciences, to biosciences and chemistry, to power grid and information networks.

This report comes on the heels of another high-profile program, "Interagency Opportunities in Multiscale Modeling in Biomedical, Biological, and Behavioral Systems," funding for which is provided jointly by the US National Science Foundation, the US National Institutes of Health, NASA, and the DOE. It should therefore come as no surprise that interest in multiscale, multiphysics modeling is at a peak, and the number of research papers and scientific conferences devoted to the subject is constantly growing.

This special issue of *Computing in Science & Engineering* contains articles that highlight various aspects and outstanding problems in multiscale mathematics. The techniques covered involve solving the same equations at widely different scales, solving different equations at different scales, and solving equations at a coarse scale (which are available only through the data provided at the fine scale).

"An Equation-Free, Multiscale Computational Approach to Uncertainty Quantification" describes a computational approach for uncertainty quantification in the behavior of nonlinear dynamical systems, whose parameters—and initial conditions—are uncertain (random).

"Adaptive Mesh Refinement for Multiscale Nonequilibrium Physics" introduces a computational strategy for modeling physical phenomena in which proper space–time resolutions are critical, yet standard fixed-grid integration approaches are computationally prohibitively expensive.

"Noise in Algorithm Refinement Methods" highlights the importance of capturing small-scale noise in hybrid models, which combine a physical phenomenon's fine- and coarse-scale descriptions.

"A Seamless Approach to Multiscale Complex Fluid Simulation" provides an example of hybrid algorithms for modeling complex fluids and dense suspensions, whose accurate characterization requires a multiplicity of time and length scales ranging from the atomic level to millimeters and beyond.

Finally, "Patch Dynamics for Multiscale Problems" reviews recent advances in patch dynamics. This computational approach uses the microscopic models at the patches as well as the interpolated boundary conditions for the patches in the same way that finite difference methods use partial differential equations to define the time derivatives of the macroscopic variables at the grid points.

These few articles only begin to sample some of the methods being developed and used today. Our list of important contributions to both theoretical and algorithmic developments in the field of multiscale multiphysics mathematics is by no means complete. The field of multiscale mathematics is still rather new, and in the near future, we expect to see many other novel approaches to solving problems in this exciting and important area.

Daniel M. Tartakovsky is a team leader at Los Alamos National Laboratory. His technical interests include stochastic partial differential equations, uncertainty quantification, inverse modeling, and hybrid methods. Tartakovsky has a PhD in hydrology from the University of Arizona. He is a member of SIAM, the American Geophysical Union, and the International Association of Hydrological Sciences. Contact him at dmt@lanl.gov.

Francis J. Alexander is a deputy group leader at Los Alamos National Laboratory. His technical interests include statistical mechanics, computational physics, and estimation theory. Alexander has a PhD in physics from Rutgers University. He is a member of the American Physical Society and SIAM. Contact him at fja@lanl.gov.

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