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

Van Vleck等人分析了离散化条件下微分方程组的性态和稳定性。特别地，发展了大型常微分方程组的稳定性信息的有效计算和全局误差估计的方法。研究主要集中在两个方面。第一个领域是确定准确计算李亚普诺夫指数和相关量的方法。这包括分析方法及其误差，以及开发稳健的软件，不仅确定李亚普诺夫指数，而且估计其计算中的误差。第二个领域是研究向后和跟踪误差分析，作为一种在数值求解非线性微分方程组时表征全局误差的手段。研究人员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式。这个项目的最终目标是分析、开发和实现算法，用于长时间、大规模地计算格点微分方程组的近似轨迹。在计算轨迹时，代码同时获得全局误差估计和有限时间Lyapunov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领域做出贡献。该项目的目标是提高与大规模物理和生物过程相对应的模型的模拟精度。有了更准确的模拟，这些模型可能会被更详细地研究，这应该会导致模型的改进。由于模型的大规模性质，研究人员采用了分布式和并行计算环境。用来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，研究人员开发了一些技术，试图隔离出错的可能性。他对确定在模拟过程中获得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的速度。这允许确定近似解中的哪些方向容易受到误差增长的影响(即，对应于扩展的方向)以及哪些方向较不容易受到误差增长的影响(即，收缩方向)。最终，收缩和膨胀信息将被合并到模拟软件中。该信息用于控制模拟中的误差，并估计近似解中的最终误差。这些方法的目的是证明在近似解附近存在精确的物理解。这可能是略有不同的模型的精确解，也可能是正在研究的模型的精确解，但从略有不同的样本开始。

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.