確率微分方程式の数値解法の研究

首席调查员在齐藤等人的密切协调下，开展了上述及其他相关课题的数值分析研究工作。主要关注的是通过微分方程组的数学建模以及这种方程的数值解。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组(SDE_S)、时滞微分方程组(DDE_S)等。该项目主要致力于其数值算法的稳定性和分析。(A)对于SDE_S的数值解，我们可以建立行型格式的代数结构，特别是它们的根树分析、它们的计算机实现问题、SDE_S的数值稳定性概念的排列以及我们提出的M-和T-稳定性的判据。(B)我们研究了ODE_S的隐式Runge-Kutta方法的并行实现，特别是我们提出了配置型两步Runge-Kutta方法作为它们的并行实现。(C)研究了线性多步法和龙格-库塔法的数值稳定性和解析稳定性，建立了线性多步法和龙格-库塔法应用于离散微分方程组的绝对稳定性判据。(D)研究了系数矩阵稀疏的线性代数方程组的迭代解，指出了由共轭梯度法衍生的一系列乘积型迭代方法的收敛性质。然而，我们的成就仍然存在一些问题，特别是在执行问题上。从今以后，我们希望一个新的研究项目能继承我们的研究项目，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.