研究了大型稀疏线性互补问题的矩阵多分裂同步,
混乱与异步并行算法及其收敛性理论。
对于一般线性代数方程组, 运用 Givens 变换, 构造了一类不完全
正交三角分解预处理子, 并建立了预处理矩阵的存在性, 稳定性
和精确性理论; 给出了分裂迭代算法收敛的充分条件。
对于由二阶自共轭椭圆型偏微分方程的边值问题经有限元离散
所产生的对称正定线性代数方程组, 构造了具并行功能的
代数多层迭代算法, 并建立了其收敛性和计算复杂性理论。
同时, 还研究了求解对称正定线性代数方程组的
非定常多分裂二级迭代算法的收敛性质。
2001年正式发表的论文如下:
- Parallel chaotic multisplitting iterative methods for the
large sparse linear complementarity problem,
Journal of Computational Mathematics,
19:3(2001), 281-292.
- Matrix multisplitting methods with applications to
linear complementarity problems:
Parallel synchronous and chaotic methods,
Calculateurs Paralleles,
13:1(2001), 125-154.
(With D.J. Evans)
- A class of incomplete orthogonal factorization methods.
I: Methods and theories,
BIT,
41:1(2001), 53-70. (With I.S. Duff and A.J. Wathen)
- Modified block SSOR preconditioners for
symmetric positive definite linear systems,
Annals of Operations Research,
103(2001), 263-282.
- Sufficient conditions for the convergent splittings of
non-Hermitian positive definite matrices,
Linear Algebra and its Applications,
330(2001), 215-218. (With C.-L. Wang)
- A class of new parallel hybrid algebraic multilevel iterations,
Journal of Computational Mathematics,
19:6(2001), 651-672.
2001年参加国际国内学术会议, 并做邀请报告如下:
- Nested iteration schemes for nonsymmetric
linear systems with a dominant positive definite symmetric part,
International Workshop on Numerical
Linear Algebra, Numerical Methods for
PDEs and Optimization,
Curitiba, Brazil, August 20--23, 2001.
(20 minutes)
2001年学术讲座如下:
- Nested iteration schemes for nonsymmetric
linear systems with a dominant positive definite symmetric part,
Department of Mathematics,
University of Nijmegen,
Nijmegen, The Netherlands,
October 3, 2001.
(45 minutes)
- Nested iteration schemes for nonsymmetric
linear systems with a dominant positive definite symmetric part,
Department of Applied Mathematics,
Delft University of Technology,
Delft, The Netherlands,
October 4, 2001.
(45 minutes)