M类非对称矩阵微分Riccati方程及其在随机流体模型中的应用

Stochastic fuid flow model is an important class of stochastic models, it has a variety of applications in storage, communications, insurance and so on. Over the past two decades, people mainly studied its steady state analysis and transient analysis. The distribution of maximum peak and that of minimum trough observed within a busy period, which have been noticed in recent years, indeed permit to estimate the worst case performance bounds, but there is no efficient algorithms to compute them. Based on probabilistic arguments, the distribution of maximum peak and that of minimum trough during a busy period satisfy a class of nonsymmetric matrix differential Riccati equation whose four coefficient matrices constitute the infinitesimal generator of a Markov process. This project intends to study such nonsymmetric matrix differential equation in three aspects:(1)give the probabilistic interpretation on the minimal nonnegative solution to the equation, prove the distribution of maximum peak (minimum trough) during a busy period is the minimal nonnegative solution to its corresponding equation; (2)design some stable algorithms to deliver the minimal nonnegative solution by stochastic coupling and matrix decomposition; (3)consider how to use the sparse structure of the coefficient matrices to design efficient algoritms.

随机流体模型是一类重要的随机模型，在存储，通讯，保险等领域有广泛的应用。在过去的二十年里，主要研究了模型的稳态分析和瞬时分析。模型繁忙期最大峰值的分布和最小低谷的分布通常用来估计实际应用中最坏情况下的性能界限，近年来备受关注，但理论上尚无有效的算法来计算它们。由模型的概率意义，繁忙期最大峰值的分布和最小低谷的分布满足一类非对称矩阵微分Riccati方程，其四个系数矩阵构成Markov过程的无穷小生成元。本项目拟对这类非对称矩阵微分Riccati方程进行三方面的研究：（一）给出方程最小非负解的概率意义，证明模型忙期最大峰值的分布（最小低谷的分布）为其对应方程的最小非负解；（二）利用随机耦合的思想构造收敛到最小非负解的迭代序列，并保证序列本身的计算是稳定的，通过矩阵分解的技巧设计直接求解方程的稳定算法；（三）对系数矩阵具有稀疏结构的方程，利用稀疏结构设计有效的算法。

本项目主要研究了Palindromic 多项式特征值问题的结构向后误差分析和M类非对称矩阵微分Riccati 方程。项目所取得的成果如下：给出了四种Palindromic多项式特征值问题中三特征对的向后误差分析，得到了可计算的上界；同时指出了临界情形下，向后误差界存在的条件，展现了Palindromic 多项式特征值问题有别于一般多项式特征值问题的特点。针对M类非对称矩阵微分Riccati 方程设计了高精度的迭代算法，在整个迭代过程中避免了减法运算，使计算解的每一个分量都有很高的相对精度。