信用风险控制中的大型稀疏方程组高性能算法研究

This project will refine large sparse equations based on practical issues in credit risk control to design high-performance algorithms mixing effectiveness, stability and accuracy, verify their effect by numerical tests, apply them to practical problems derived from credit risk control, and assess credit risk in companies or corporations to intervene in time. Problems will be researched from three aspects as follows: Fristly, in view of different properties of several sparse transition matrices within one year, high-performance algorithms used for computing theirs arbitrary roots will be presented, and effectiveness, stability and accuracy will be discussed to define better transition matrices in shorter period. Secondly, based on uniformization method, scaling and squaring method and matrices decompositions and factorizations method, better algorithms used for solving matrices exponential with large sparse origin matrices will be considered to obtain transition matrices in shorter period. Finally, considering properties of exponential matrices with large sparse origin matrices time the given vectors and combining the standard Krylov subspace method, better variants of Krylov subspace method will be presented to achieve matrices exponential.

本项目拟依托信用风险领域的实际问题，提炼出大型稀疏方程组，构造出兼顾有效性、稳定性和准确性的高性能算法，通过数值试验检验其实际效果，将其应用于信用风险控制方面的实际问题，进而评估公司或企业的信用风险，以便及时干预。拟从以下三个方面进行研究：一、针对几类特殊的稀疏年转移矩阵的不同性态，构造出用于计算其任意次方根的高性能算法，研究其有效性、稳定性和准确性，以便更好地确定更短期的转移矩阵；二、基于均匀化法、缩放比例与开方法和矩阵分解，探讨直接计算原矩阵为大型稀疏矩阵的指数矩阵的高性能算法，得到更短期的转移矩阵；三、通过研究原矩阵为大型对称（非对称）稀疏矩阵的指数矩阵与向量乘积的性态，结合标准的Krylov子空间法，拟提出性能更优的改进Krylov子空间法间接计算指数矩阵。

本项目依托信用风险领域的实际问题，提炼出大型稀疏方程组，构造出兼顾有效性、稳定性和准确性的高性能算法，通过数值试验检验其实际效果，将其应用于信用.风险控制方面的实际问题，进而评估公司或企业的信用风险，以便及时干预。从以下三个方面进行研究：一、针对稀疏年转移矩阵为块三对角矩阵和鞍点矩阵，提出相应算法，研究其有效性、稳定性和准确性；针对以上两类稀疏年转移矩阵的不同性态，构造出用于计算其任意次方根的高性能算法，研究其有效性、稳定性和准确性，以便更好地确定更短期的转移矩阵；二、基于均匀化法、缩放比例与开方法和矩阵分解，探讨直接计算原矩阵为大型稀疏矩阵的指数矩阵的高性能算法，得到更短期的转移矩阵; 对其算法进行优化，考察了加权矩阵为稀疏的不定对角矩阵和广义鞍点矩阵的加权线性最小二乘问题，提出了相应算法，并证明该算法为可证的向后稳定；三、通过研究原矩阵为大型对称稀疏矩阵的指数矩阵与向量乘积的性态，结合标准的Krylov子空间法，提出性能更优的改进Krylov子空间法间接计算指数矩阵。