Research on High Dimension Problems in Numerical Analysis.

我们研究了解决高维问题的快速数值算法以及并行或矢量结构的快速计算机的有效实现。在代数计算方面，我们发现了多项式数值分解的新算法和多项式余数序列算法。研究了多项式的Grobner基和商环基的稳定数值算法。在常微分方程领域，提出了适于并行计算的两步龙格-库塔法新格式，并研究了特征曲线法作为并行ODE求解器求解对流和扩散问题的应用。在随机微分方程上，通过根树分析，得到了row型格式的阶条件。我们还提出了新的数值格式稳定性判据。在时滞微分方程上，研究了线性系统的稳定性，提出了几种确定线性系统稳定性的方法和数值求解方法。在孤子方程上，我们找到了表示二维孤子的KP方程的新算法。在高维积分上，提出了高精度的好点法。我们还开发了一种新的方法——稀疏网格法。在高维线性方程组上，我们提出了一种基于Bi-CG的求解非对称线性方程组的广义积型方法GPBi-CG。我们还提出了适合向量架构的旋转替代LU分解。为了高维数学问题的可视化，我们研究了具有正性、单调性和凸性约束的弹簧曲线的生成。

We investigated fast numerical algorithms for solving high dimensional problems and efficient implementation for fast computers with parallel or vector architecture.On the field of algebraic calculation, we found new algorithms for numerical factorization of polynomials and an algorithm for sequence of polynomial remainders. And we also investigate stable numerical algorithms for Grobner basis and basis of quotient rings of polynomials.On the field of ordinary differential equations, we proposed new scheme of two-step Runge-Kutta methods suitable for parallel computing and investigated characteristic curve methods solving convection and diffusion problems as an application of parallel ODE solvers.On stochastic differential equations, we found the order conditions of ROW-type scheme by rooted tree analysis. We also proposed new stability criteria for numerical scheme.On delay-differential equations, we investigate the stability of linear systems and proposed several methods for determine stability of linear systems and numerical solver.On soliton equations, we found new algorithm for KP equations which represents a two dimensional soliton.On high dimensional integration, we proposed high precision good lattice points methods. We also developed a new method named sparse grid method.On high dimensional linear equations, we proposed GPBi-CG,which is a generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems. We also proposed rotated alternative LU decomposition which is suitable for vector architecture.For visualization of high dimensional mathematical problems, we investigate the generation of sprine curves with several constraints, i.e., positivity, monotonisity and convexity.