Mathematical Sciences: Computation of Stability Information and Global Error Estimation with Applications

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Van Vleck The investigator analyzes the behavior and stability properties of differential equations under discretization. In particular, methods are developed for the efficient computation of stability information and global error estimation for large systems of ordinary differential equations. The research is focused in two areas. The first area is to determine methods for accurately computing Lyapunov exponents and related quantities. This includes analysis of methods and their errors and the development of robust software to determine not only the Lyapunov exponents but also an estimate of the error in their computation. The second area is in the study of backward and shadowing error analysis as a means for characterizing global error when solving nonlinear differential equations numerically. The investigator applies these concepts to discrete in space, continuous in time difderential equations that arise in the biological and physical sciences. The ultimate goal of this project is to analyze, develop and implement algorithms for the long-time, large-scale computation of approximate trajectories for systems of lattice differential equations. While computing trajectories the codes simultaneously obtain global error estimates and finite time Lyapunov exponents or kinematic eigenvalues and the associated growth/decay directions. This research is expected to yield contributions in the areas of dynamical systems, numerical analysis and applied modeling. The goal of this project is to provide improved accuracy in simulations of models corresponding to large scale physical and biological processes. With more accurate simulations these models may be studied in greater detail and this should lead to improvements in the models. Because of the large scale nature of the models the investigator employs distributed and parallel computing environments. The techniques to be used to increase the accuracy of the simulations rely on mathematical techniques in dynamical systems and numerical analysis. In particular, the investigator develops techniques that attempt to isolate the potential for error. He is particularly interested in determining the directions in which approximate solutions, obtained during simulations, expand and contract. Additionally, he wishes to determine the rate at which this contraction and expansion takes place. This allows to determine which directions in the approximate solution are susceptible to error growth (i.e. the directions corresponding to expansion) and which directions are less susceptible to error growth (i.e. the contracting directions). Ultimately, the contraction and expansion information is to be incorporated into the simulation software. This information is used to control the error in the simulation and estimate the final error in the approximate solution. The methods are aimed at showing that near the approximate solution there exists an exact physical solution. This may be an exact solution of a slightly different model or an exact solution for the model being studied but starting from a slightly different sample.