Stephen Bond

Ph.D. University of Kansas, 2000

Research Statement

My primary research interest is in the study of numerical algorithms for the simulation of systems arising in Biochemistry and Statistical Mechanics. One of the major challenges in this field is in the design of efficient scalable algorithms which capture the multiscale behavior correctly, without introducing nonphysical numerical artifacts. This is a difficult task, due to the wide range of time and length scales, and the nonlinear coupling between the motion on each scale.

Biomolecular simulation serves as an interesting paradigm for the difficulties in effectively modeling multiscale behavior. On the shortest classical scale, the motion is governed by the vibrations of bonded atoms which occur on a time scale of 10-14 seconds, and a length scale of 10-10 meters. In contrast, larger scale conformational (e.g. allosteric and denaturing) changes occur on a time scale of 10-5 to 101 seconds, and often involve the collective motion of hundreds of thousands (to millions) of atoms. It is for this reason that a typical molecular dynamics simulation can take months of supercomputer time, only to capture a very small segment (in space and time) of a larger biochemical process.

Unfortunately, classical "forward" error analysis is not useful in the context of molecular dynamics, due to the extremely long simulation time, large time step size, and quasiergodic nature of the underlying system. Efficiency of a numerical method must be measured with respect to averages (from Statistical Mechanics), since there is no meaningful concept of an exact trajectory in these systems. My research in this field has been centered around "geometric integrators" and "backward" error analysis, which has proved useful in designing numerical methods which preserve important statistical properties over long time intervals. In addition to classical atomistic methods (ODEs), I am interested in coarse-grain continuum models (PDEs) and hybrid methods which interface continuum and atomistic/quantum (ODEs/PDEs) models.