確率微分方程式の数値解法の研究

首席研究员与SAITO等人密切合作，开展了上述和其他数值分析相关课题的研究工作。主要关注的是通过微分方程的数学建模以及对这些方程的数值解。它们包括大型常微分方程组（ODE_S）、作为ODE_S的随机推广的随机微分方程（ODE_S）、时滞微分方程（DDE_S）等。该项目主要致力于其数值算法的稳定性和分析。成就随之而来。(a)在数值解方面，我们建立了ROW型格式的代数结构，特别是根树分析，计算机实现问题，对数值稳定性概念的整理，以及我们提出的M-稳定性和T-稳定性判据。(b)研究了常微分方程隐式Runge-Kutta方法的并行实现。特别地，我们提出了配置型两步龙格库塔方法作为它们的并行化。并对其进行了初步的性能测试。(c)研究了DDE_S的数值稳定性和解析稳定性，建立了线性多步法和Runge-Kutta法应用于DDE_S时的绝对稳定性判据。(d)研究了具有稀疏系数矩阵的大型线性代数方程组的迭代解法，指出了由共轭梯度法导出的一系列乘积型迭代法的收敛性特征，这些问题在理论和实践上都将引起人们的极大兴趣.然而，我们的成就仍然存在一些问题，特别是在执行问题上。尽管如此，我们希望有一个新的研究项目接替我们的项目，以取得更丰硕的成果。

Head investigator, in the close coordination with SAITO and other people, has carried out the research work of the above and other relevant topics of numerical analysis. The main concern is on the mathematical modelling through differential equations as well as on the numerical solution of such equations. They include large system of ordinary differential equations (ODE_S), stochastic differential equations (SDE_S) as a stochastic extension of ODE_S, delay-differential equations (DDE_S) and others. The project has been mainly devoted in the stability and analysis of their numerical algorithms. The achievement follows.(a) As for the numerical solution of SDE_S, we can establish the algebraic structure of ROW-type schemes, especially their rooted tree analysis, the issues of their implementation on computer, arrangement of the concepts of numerical stability for SDE_S, and criteria of M- and T-stabilities which were proposed by ourselves.(b) We studied the parallel implementation of the implicit Runge-Kutta methods for ODE_S. In particular, we proposed the collocation-type two-step Runge-Kutta methods as a parallelization of them. Some initial performance tests were also carried out for them.(c) We studied the numerical as well as analytical stability of DDE_S, and established criteria of absolute stability for linear multistep and Runge-Kutta methods when applied to DDE_S.(d) Studying the iterative solution of large system of linear algebraic equations with sparse coefficient matrices, we pointed out features of the convergence behavior of a series of the product-type iterative methods derived from the conjugate gradient method.The topics described above are strongly expected to attract further interests in both theoretical and practical point of view. However, our achievement remains some questions, in particular in their implementation issues. Henceforth we hope a new research project succeeds ours to obtain more fruitful achievement.