对洛伦兹系统的可计算性进行量化
Quantifying the computability of the Lorenz system
AI总结：
本文通过后验误差分析技术，量化了洛伦兹系统在浮点精度下的可计算区间长度，探讨了计算稳定性与初始数据、离散化和舍入误差的关系。
摘要：
众所周知，计算洛伦兹系统的准确轨迹是一个困难的问题。计算解对时间步长大小和方法的多项式阶数决定的离散误差以及舍入误差非常敏感。在本文中，我们展示了舍入误差如何限制洛伦兹系统的可计算性，并精确量化了可以计算的区间长度，以浮点精度表示。利用基于伴随的后验误差分析技术，我们分别估计了计算相对于初始数据、离散化和舍入误差的稳定性。分析通过使用非常高的阶数（阶数200）有限元方法和非常高的浮点精度（400位）计算时间区间[0, 1000]上的准确解来验证。
英文摘要：
It is well known that the computation of accurate trajectories of the Lorenz system is a difficult problem. Computed solutions are very sensitive to the discretization error determined by the time step size and polynomial order of the method, as well as round-off errors. In this work, we show how round-off errors limit the computability of the Lorenz system and quantify exactly the length of intervals over which solutions can be computed, expressed in terms of the floating point precision. Using adjoint-based \emph{a posteriori} error analysis techniques, we estimate the stability of computations with respect to initial data, discretization, and round-off errors, respectively. The analysis is verified by computing an accurate solution on the time interval $[0, 1000]$ using a very high order (order 200) finite element method and very high floating point precision 400 digits).
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