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

研究者分析了微分方程在离散化下的行为和稳定性。研究了大型常微分方程系统稳定性信息的高效计算和全局误差估计方法。这项研究集中在两个方面。第一个领域是确定精确计算李雅普诺夫指数和相关量的方法。这包括对方法及其误差的分析，以及开发强大的软件，不仅可以确定李雅普诺夫指数，还可以估计其计算中的误差。第二个领域是研究逆向和阴影误差分析，作为数值解非线性微分方程时表征全局误差的一种手段。研究者将这些概念应用于生物和物理科学中出现的空间离散、时间连续的微分方程。这个项目的最终目标是分析，开发和实现算法，为晶格微分方程系统的近似轨迹的长时间，大规模的计算。在计算轨迹时，代码同时获得全局误差估计和有限时间李雅普诺夫指数或运动学特征值以及相关的增长/衰减方向。这项研究有望在动力系统、数值分析和应用建模等领域做出贡献。该项目的目标是提高模拟模型的准确性，以适应大规模的物理和生物过程。有了更精确的模拟，这些模型可能会得到更详细的研究，这将导致模型的改进。由于模型的大规模性质，研究者采用分布式和并行计算环境。用于提高模拟精度的技术依赖于动力系统和数值分析中的数学技术。特别是，研究者开发的技术，试图隔离潜在的错误。他对确定模拟过程中得到的近似解的扩展和收缩方向特别感兴趣。此外，他希望确定这种收缩和扩张发生的速率。这允许确定在近似解中哪些方向易受误差增长的影响（即与展开相对应的方向），哪些方向不易受误差增长的影响（即收缩方向）。最终，收缩和膨胀信息将被纳入仿真软件。该信息用于控制仿真中的误差，并在近似解中估计最终误差。这些方法旨在证明在近似解附近存在一个精确的物理解。这可能是一个稍微不同的模型的精确解，或者是正在研究的模型的精确解，但从一个稍微不同的样本开始。

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.