Research on unifying techniques for sensitivity analysis of large-scale systems

在工程、科学、社会学等许多领域，敏感性分析是系统设计/运行和验证解释自然或社会现象的数学模型适用性的关键技术。然而，许多研究人员在不同的国家进行了独立的研究，他们的主题广泛传播，术语差异很大（如共轭系统，伴随系统，摄动计算）。结果，他们每个人都在不同的语境中分别“发现”了几乎相同的想法。我们的研究目标是：(1)统一和系统化提出的思想；(2)构建具有快速自动微分的“敏感性分析综合理论”；(3)建立通用算法；(4)开发算法的软件系统；(5)通过计算实验进行评价。我们获得了以下预期结果。首先，提出了一种新的非线性系统灵敏度分析理论，并建立了一种普遍适用于任何模型的灵敏度分析算法。它既可以进行灵敏度分析，也可以进行数学模型分析。特别是，其复杂性与单独分析模型的复杂性是相同的。其次，用数值计算方法统一了伴随系统、共轭系统等概念。我们正在开发一个可以处理区间数、指数变长浮点数、自动微分数等的实验系统。最后，我们认识到表征系统结构的一个问题，其灵敏度易于分析，即，通过它可以很容易地预测元素的摄动对整个系统的影响，并以串并联系统的结构为例。这种认识对于通过系统分析来设计系统的“结构”具有重要的工程意义。

In many fields of engineering, science, sociology, etc., the sensitivity analysis is a key technique for system design/operation and verification of suitability of mathematical models explaining natural or sociological phenomena. Many researchers, however, had studied on it independently, who lived in different countries, whose themes were widely spread, whose terminologies were quite different (e.g., conjugate system, adjoint system, perturbation computation). As a result, each of them had separately "found" almost same idea in the different contexts. Our research aims were (1) unification and systematization of the proposed ideas, (2) construction of "synthetic theory on sensitivity analysis" with fast automatic differentiation, (3) establishment of a general algorithm, (4) development of software system for the algorithm, and (5) evaluation through computational experiments.We have obtained the following expected results. Firstly, we proposed a new theory for the sensitivity analysis on nonlinear systems, and established an algorithm which could be generally applied to any model. It can perform the sensitivity analysis as well as analyzing the mathematical model. In particular its complexity was the same order as that for analyzing the model alone. Secondly, we unified the concepts referred as adjoint system, conjugate system, etc. with a numerical computational approach. We are developing an experimental system by which we can treat interval numbers, floating-point numbers with variable length of exponent, automatic differentiation numbers, etc. Finally, we have recognized an issue for characterizing system structures whose sensitivity can be easily analyzed, i.e., by which influences of perturbation of elements to the whole system can be easily anticipated, giving the structure of the serial-parallel system as an example. This recognition is significance in engineering in order to design "structure" of system through system analysis.