生命科学中两类反问题的计算方法及应用

Inverse problem is the most rapidly developing and growing fields of applied mathematics in recent years, which holds a broad application background. Moreover, the popularity of big data sets an even higher standard for the designing of algorithms to tackle inverse problems. In this project, we will mainly concentrate on solving two special types of differential equation inverse problems, i.e. parametric ordinary differential equation inverse problem and underdetermined sparse ordinary differential equation inverse problem. These two problems originate from the field of life sciences, with very strong application background. We would analyze deeply on the theoretical properties of the problems, e.g. the separability, to figure out the proper reformulation of the model, and materialize dimension reduction on the problem, to finally design efficient computational algorithms for solving extremely large-scale differential equation inverse problems. In the meantime, in this project, we would research on the theoretical properties of the algorithms designed, analyze their complexity, and propose the proof of global convergence and the conditions of converging to the global optimizer. Finally, inasmuch as the problems we analyze in this project originate from applications, the algorithms above should return to the applications to be tested for performances. We would apply the newly designed algorithms to problems in fields like life sciences, which are hardly conquered by traditional methods, and try to answer several core questions highly concerning researchers, to stimulate the development of interdisciplinary research. We look forward to establishment of the project, to carry more favorable theoretic research and practical applications.

反问题是近年来应用数学领域中发展和成长最快的领域之一，具有广泛的应用背景。而近年来大数据的流行，更是为反问题的算法设计提出了更高的要求。本项目将主要围绕两类特殊的微分方程反问题展开研究，即参数型常微分方程反问题与欠定稀疏常微分方程反问题的求解。这两类问题均来源于生命科学领域，有着很强的实际应用背景。我们将深入分析问题的可分性等理论性质，寻找对模型的合理变换，实现对问题的降维，从而设计出可以求解超大规模微分方程反问题的高效数值算法。同时，本项目还将对所设计算法的理论性质展开研究，分析算法的复杂度，并证明算法的全局收敛性及给出收敛到全局解的条件。最后，本项目的问题来源于实际，最终也需要回到实际中去检验算法的实际表现。我们将把这些新算法应用在生命科学等领域中的一些传统算法难以解决的问题上，回答这些领域中所关心的几个核心问题，促进交叉学科的发展。我们期待该课题的立项，以便开展理论研究和实际应用。